Finite Element Methods for the Simulation of Incompressible Flows

Weierstrass Institute for Applied Analysis and Stochastics

Volker John

Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 Outline of the Lectures

1 The Navier–Stokes Equations as Model for Incompressible Flows

2 Function Spaces For Linear Saddle Point Problems

3 The Stokes Equations

4 The Oseen Equations

5 The Stationary Navier–Stokes Equations

6 The Time-Dependent Navier–Stokes Equations – Laminar Flows

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 2 (151) 1 A Model for Incompressible Flows

• conservation laws ◦ conservation of linear momentum ◦ conservation of mass • ﬂow variables ◦ ρ(t,x) : density [kg/m3] ◦ v(t,x) : velocity [m/s] ◦ P(t,x) : pressure [N/m2] assumed to be sufﬁciently smooth in • Ω ⊂ R3 • [0,T]

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 3 (151) 1 Conservation of Mass

• change of ﬂuid in arbitrary volume V

∂ Z Z Z − ρ dx = ρv · n ds = ∇ · (ρv) dx ∂t V ∂V V | {z } | {z } mass transport through bdry

• V arbitrary =⇒ continuity equation

ρt + ∇ · (ρv) = 0

• incompressibility( ρ = const)

∇ · v = 0

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 4 (151) • acceleration: using ﬁrst order Taylor series expansion in time (board) dv (t,x) = ∂ v(t,x) + (v(t,x) · ∇)v(t,x) dt t

movement of a particle

1 Newton’s Second Law of Motion

• Newton’s second law of motion

net force = mass × acceleration

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 5 (151) 1 Newton’s Second Law of Motion

• Newton’s second law of motion

net force = mass × acceleration

• acceleration: using ﬁrst order Taylor series expansion in time (board) dv (t,x) = ∂ v(t,x) + (v(t,x) · ∇)v(t,x) dt t

movement of a particle

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 5 (151) • principle of Cauchy: internal contact forces depend (geometrically) only on the orientation of the surface t = t(n) n – unit normal vector of the surface pointing outwards of V

1 Newton’s Second Law of Motion

• acting forces on an arbitrary volume V : sum of external (body) forces ◦ gravity and internal (molecular) forces ◦ pressure ◦ viscous drag that a ’ﬂuid element’ exerts on the ’adjacent element’ ◦ contact forces: act only on surface of ’ﬂuid element’

Z Z F(t,x) dx + t(t,s) ds V ∂V t [N/m2] – Cauchy stress vector

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 6 (151) 1 Newton’s Second Law of Motion

Z Z F(t,x) dx + t(t,s) ds V ∂V t [N/m2] – Cauchy stress vector • principle of Cauchy: internal contact forces depend (geometrically) only on the orientation of the surface t = t(n) n – unit normal vector of the surface pointing outwards of V

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 6 (151) 1 Newton’s Second Law of Motion

• it can be shown: conservation of linear momentum results in linear dependency on n t = Sn S(t,x)[N/m2] – stress tensor, dimension 3 × 3 • divergence theorem Z Z t(t,s) ds = ∇ · S(t,x) dx ∂V V • momentum equation

ρ (vt + (v · ∇)v) = ∇ · S + F ∀ t ∈ (0,T], x ∈ Ω

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 7 (151) 1 Newton’s Second Law of Motion

• model for the stress tensor ◦ torque Z Z M0 = r × F dx + r × (Sn) ds [N m] V ∂V T at equilibrium is zero =⇒ symmetry S = S ◦ decomposition S = V + PI V [N/m2] – viscous stress tensor ◦ pressure P acts only normal to the surface, directed into V Z Z Z − Pn ds = − ∇P dx = − ∇ · (PI) dx ∂V V V

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 8 (151) 1 Newton’s Second Law of Motion

• model for the stress tensor (cont.) ◦ viscous stress tensor − friction between ﬂuid particles can only occur if the particles move with different velocities − viscous stress tensor depends on gradient of velocity − because of symmetry: on symmetric part of the gradient: velocity deformation tensor ∇v + (∇v)T (v) = D 2 − velocity not too large: dependency is linear (Newtonian ﬂuids) 2µ = 2µ (v) + ζ − (∇ · v) V D 3 I µ [kg/(m s)] – dynamic or shear viscosity ζ [kg/(m s)] – second order viscosity

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 9 (151) • incompressible ﬂows: incompressible Navier–Stokes equations P F ∂t v − 2ν∇ · D(v) + (v · ∇)v + ∇ = in (0,T] × Ω, ρ0 ρ0 ∇ · v = 0 in (0,T] × Ω

1 Navier–Stokes Equations

• general Navier–Stokes equations

ρ (∂t v + (v · ∇)v) 2µ −2∇ · (µD(v)) − ∇ · ζ − 3 ∇ · vI + ∇P = F in (0,T] × Ω, ρt + ∇ · (ρv) = 0 in (0,T] × Ω

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 10 (151) 1 Navier–Stokes Equations

• general Navier–Stokes equations

ρ (∂t v + (v · ∇)v) 2µ −2∇ · (µD(v)) − ∇ · ζ − 3 ∇ · vI + ∇P = F in (0,T] × Ω, ρt + ∇ · (ρv) = 0 in (0,T] × Ω

• incompressible ﬂows: incompressible Navier–Stokes equations P F ∂t v − 2ν∇ · D(v) + (v · ∇)v + ∇ = in (0,T] × Ω, ρ0 ρ0 ∇ · v = 0 in (0,T] × Ω

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 10 (151) 1 Navier–Stokes Equations

• Claude Louis Marie Henri Navier (1785 – 1836) George Gabriel Stokes (1819 – 1903)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 11 (151) 1 Dimensionless Incompressible Navier–Stokes Equations

• dimensionless equations needed for (numerical) analysis and numerical simulations • reference quantities of ﬂow problem ◦ L [m] – a characteristic length scale ◦ U [m/s] – a characteristic velocity scale ◦ T ∗ [s] – a characteristic time scale • transform of variables x0 v t0 x = , u = , t = L U T ∗ • rescaling

L 2ν P L ∗ ∂t u − ∇ · D(u) + (u · ∇)u + ∇ 2 = 2 F in (0,T] × Ω, UT UL ρ0U ρ0U ∇ · u = 0 in (0,T] × Ω,

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 12 (151) 1 Dimensionless Incompressible Navier–Stokes Equations

• deﬁning P UL L L p = 2 , Re = , St = ∗ , f = 2 F ρ0U ν UT ρ0U p – new pressure Re – Reynolds number St – Strouhal number f – new right hand side • result 2 St∂ u − ∇ · (u) + (u · ∇)u + ∇p = f in (0,T] × Ω, t Re D ∇ · u = 0 in (0,T] × Ω

• generally T ∗ = L/U =⇒ St = 1

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 13 (151) 1 Dimensionless Incompressible Navier–Stokes Equations

• dimensionless Navier–Stokes equations ◦ conservation of linear momentum ◦ conservation of mass

−1 T ut − 2Re ∇ · D(u) + ∇ · (uu ) + ∇p = f in (0,T] × Ω ∇ · u = 0 in [0,T] × Ω u(0,x) = u0 in Ω + boundary conditions • given: • to compute: d ◦ Ω ⊂ R ,d ∈ {2,3}: domain ◦ velocity u, with ◦ T : ﬁnal time ∇u + ∇uT ◦ u : initial velocity (u) = , 0 D 2 ◦ boundary conditions velocity deformation tensor ◦ pressure p • parameter: Reynolds number Re

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 14 (151) 1 The Reynolds Number

• Reynolds number LU Re = ν convective forces = viscous forces

Osborne Reynolds (1842 – 1912) • rough classification of flows: ◦ Re small: steady-state flow field (if data do not depend on time) ◦ Re larger: laminar time-dependent flow field ◦ Re very large: turbulent flows

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 15 (151) • alternative expression of viscous term (due to ∇ · u = 0)

2∇ · D(u) = ∆u

• alternative expression of convective term (due to ∇ · u = 0)

(u · ∇)u = ∇ · (uuT )

1 Dimensionless Incompressible Navier–Stokes Equations

• simpliﬁed form (for mathematics)

∂t u − 2ν∇ · D(u) + (u · ∇)u + ∇p = f in (0,T] × Ω, ∇ · u = 0 in (0,T] × Ω

ν = Re−1 – dimensionless viscosity

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 16 (151) 1 Dimensionless Incompressible Navier–Stokes Equations

• simpliﬁed form (for mathematics)

∂t u − 2ν∇ · D(u) + (u · ∇)u + ∇p = f in (0,T] × Ω, ∇ · u = 0 in (0,T] × Ω

ν = Re−1 – dimensionless viscosity • alternative expression of viscous term (due to ∇ · u = 0)

2∇ · D(u) = ∆u

• alternative expression of convective term (due to ∇ · u = 0)

(u · ∇)u = ∇ · (uuT )

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 16 (151) ◦ Oseen equations: convection ﬁeld known (only for analysis)

−ν∆u + (u0 · ∇)u + ∇p + cu = f in Ω ∇ · u = 0 in Ω

◦ Stokes equations: no convection

−∆u + ∇p = f in Ω ∇ · u = 0 in Ω

1 Incompressible Navier–Stokes Equations

• special cases ◦ steady-state Navier–Stokes equations: stationary ﬂow ﬁelds

−ν∆u + (u · ∇)u + ∇p = f in Ω ∇ · u = 0 in Ω

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 17 (151) ◦ Stokes equations: no convection

−∆u + ∇p = f in Ω ∇ · u = 0 in Ω

1 Incompressible Navier–Stokes Equations

• special cases ◦ steady-state Navier–Stokes equations: stationary ﬂow ﬁelds

−ν∆u + (u · ∇)u + ∇p = f in Ω ∇ · u = 0 in Ω

◦ Oseen equations: convection ﬁeld known (only for analysis)

−ν∆u + (u0 · ∇)u + ∇p + cu = f in Ω ∇ · u = 0 in Ω

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 17 (151) 1 Incompressible Navier–Stokes Equations

• special cases ◦ steady-state Navier–Stokes equations: stationary ﬂow ﬁelds

−ν∆u + (u · ∇)u + ∇p = f in Ω ∇ · u = 0 in Ω

◦ Oseen equations: convection ﬁeld known (only for analysis)

−ν∆u + (u0 · ∇)u + ∇p + cu = f in Ω ∇ · u = 0 in Ω

◦ Stokes equations: no convection

−∆u + ∇p = f in Ω ∇ · u = 0 in Ω

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 17 (151) ◦ free slip boundary condition (e.g. symmetry planes)

u · n = g in (0,T] × Γslip ⊂ Γ, T n Stk = 0 in (0,T] × Γslip, 1 ≤ k ≤ d − 1

1 Incompressible Navier–Stokes Equations

• boundary conditions ◦ Dirichlet boundary conditions (inﬂows)

u(t,x) = g(t,x) in (0,T] × Γdiri ⊂ Γ

g(t,x) = 0 – no slip boundary condition (walls)

u(t,x) = 0 ⇐⇒ u(t,x) · n = 0, u(t,x) · t1 = 0, u(t,x) · t2 = 0

no penetration, no slip

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 18 (151) 1 Incompressible Navier–Stokes Equations

• boundary conditions ◦ Dirichlet boundary conditions (inﬂows)

u(t,x) = g(t,x) in (0,T] × Γdiri ⊂ Γ

g(t,x) = 0 – no slip boundary condition (walls)

u(t,x) = 0 ⇐⇒ u(t,x) · n = 0, u(t,x) · t1 = 0, u(t,x) · t2 = 0

no penetration, no slip ◦ free slip boundary condition (e.g. symmetry planes)

u · n = g in (0,T] × Γslip ⊂ Γ, T n Stk = 0 in (0,T] × Γslip, 1 ≤ k ≤ d − 1

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 18 (151) ◦ periodic boundary conditions (only for analysis, Ω = (0,l)d)

u(t,x + lei) = u(t,x) ∀ (t,x) ∈ (0,T] × Γ

1 Incompressible Navier–Stokes Equations

• boundary conditions (cont.) ◦ do-nothing boundary conditions (outﬂow)

Sn = 0 in (0,T] × Γoutf ⊂ Γ

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 19 (151) 1 Incompressible Navier–Stokes Equations

• boundary conditions (cont.) ◦ do-nothing boundary conditions (outﬂow)

Sn = 0 in (0,T] × Γoutf ⊂ Γ

◦ periodic boundary conditions (only for analysis, Ω = (0,l)d)

u(t,x + lei) = u(t,x) ∀ (t,x) ∈ (0,T] × Γ

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 19 (151) 1 Incompressible Navier–Stokes Equations

• difﬁculties for mathematical analysis and numerical simulations ◦ coupling of velocity and pressure ◦ nonlinearity of the convective term ◦ the convective term dominates the viscous term, i.e. ν is small

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 20 (151) 2 Linear Saddle Point Problems

• motivation ◦ iterative solution of Navier–Stokes equations leads to linear system of equations ◦ linear system have special form: saddle point problem ◦ sufﬁcient and necessary condition on unique solvability needed ◦ can be derived in abstract form, see [1]

[1] Girault, Raviart: Finite Element Methods for Navier-Stokes Equations 1986

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 21 (151) 2 Linear Saddle Point Problems

• spaces: V,Q – real Hilbert spaces • bilinear forms:

a(·,·) : V ×V → R, b(·,·) : V × Q → R

• linear problem: Find (u, p) ∈ V × Q such that for given ( f ,r) ∈ V 0 × Q0

a(u,v) + b(v, p) = h f ,viV 0,V ∀ v ∈ V, b(u,q) = hr,qiQ0,Q ∀ q ∈ Q

• conditions on the spaces and bilinear forms necessary

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 22 (151) 2 Linear Saddle Point Problems

• associated linear operators

0 A ∈ L V,V deﬁned by hAu,viV 0,V = a(u,v) ∀ u,v ∈ V 0 B ∈ L V,Q deﬁned by hBu,qiQ0,Q = b(u,q) ∀ u ∈ V, ∀ q ∈ Q

• dual operator: B0 ∈ L (Q,V 0) deﬁned by

0 B q,v V 0,V = hBv,qiQ0,Q = b(v,q) ∀ v ∈ V, ∀ q ∈ Q

• linear problem in operator form: Find (u, p) ∈ V × Q such that

Au +B0 p = f in V 0 Bu = r in Q0

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 23 (151) 2 The Inf-Sup Condition – Bilinear Form b(·,·)

• spaces ⊥ ◦ V0 := V(0) = ker(B), V = V0 ⊕V0 ˜ 0 0 0 ◦ V = {φ ∈ V : hφ,viV 0,V = 0 ∀ v ∈ V0} ⊂ V • inf-sup condition: The three following properties are equivalent:

i) There exists a constant βis > 0 such that b(v,q) inf sup ≥ βis. q∈Q v∈V kvkV kqkQ

ii) The operator B0 is an isomorphism from Q onto V˜ 0 and

0 B q V 0 ≥ βis kqkQ ∀ q ∈ Q.

⊥ 0 iii) The operator B is an isomorphism from V0 onto Q and

⊥ kBvkQ0 ≥ βis kvkV ∀ v ∈ V0 .

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 24 (151) 2 The Inf-Sup Condition – Bilinear Form b(·,·)

• independently derived in [1,2]: Babuška–Brezzi condition • sometimes: Ladyzhenskaya–Babuška–Brezzi condition, LBB condition • it follows: V(r) = {v ∈ V : Bv = r}

is not empty for all r ∈ Q0

[1] Babuška: Numer. Math. 20, 179–192, 1973 [2] Brezzi: Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8, 129–151, 1974

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 25 (151) 2 Unique Solution of Linear Saddle Point Problem

• sufﬁcient and necessary conditions for unique solution of saddle point problem can be formulated with projection operator, see literature • sufﬁcient conditions

◦ a(·,·) is V0-elliptic, i.e., there is a constant α > 0 such that

2 a(v,v) ≥ α kvkV ∀ v ∈ V0

◦ b(·,·) satisﬁes inf-sup condition

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 26 (151) • pressure space Z 2 2 Q = L0 (Ω) = q : q ∈ L (Ω) with q(x) dx = 0 Ω

with Z (q,r) = (qr)(x) dx, kqkQ = kqkL2(Ω) Ω • dual space: Q0 = Q

2 Continuous Incompressible Flow Problems

• for simplicity: Dirichlet boundary conditions on whole boundary • velocity space

1 1 V = H0 (Ω) = v : v ∈ H (Ω) with v = 0 on ∂Ω

with Z (v,w) = (∇v · ∇w)(x) dx, kvkV := k∇vkL2(Ω) Ω dual space: V 0 = H−1(Ω)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 27 (151) 2 Continuous Incompressible Flow Problems

• for simplicity: Dirichlet boundary conditions on whole boundary • velocity space

1 1 V = H0 (Ω) = v : v ∈ H (Ω) with v = 0 on ∂Ω

with Z (v,w) = (∇v · ∇w)(x) dx, kvkV := k∇vkL2(Ω) Ω dual space: V 0 = H−1(Ω) • pressure space Z 2 2 Q = L0 (Ω) = q : q ∈ L (Ω) with q(x) dx = 0 Ω

with Z (q,r) = (qr)(x) dx, kqkQ = kqkL2(Ω) Ω • dual space: Q0 = Q

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 27 (151) • divergence operator

div : V → range(div), v 7→ ∇ · v

• it can be shown: range(div) = Q0 • associated linear operator: negative divergence operator

B ∈ L (V,Q0), B = −div

2 Continuous Incompressible Flow Problems

• bilinear form for coupling velocity and pressure Z b(v,q) = − q∇ · v dx = −(∇ · v,q) v ∈ V, q ∈ Q Ω

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 28 (151) 2 Continuous Incompressible Flow Problems

• bilinear form for coupling velocity and pressure Z b(v,q) = − q∇ · v dx = −(∇ · v,q) v ∈ V, q ∈ Q Ω • divergence operator

div : V → range(div), v 7→ ∇ · v

• it can be shown: range(div) = Q0 • associated linear operator: negative divergence operator

B ∈ L (V,Q0), B = −div

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 28 (151) • kernel of B: space of weakly divergence-free functions

V0 = Vdiv = {v ∈ V : (∇ · v,q) = 0 ∀ q ∈ Q}

2 Continuous Incompressible Flow Problems

• dual operator: gradient operator

grad : Q → range(grad), q 7→ ∇q

with B0 ∈ L (Q,V 0), B0 = grad

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 29 (151) 2 Continuous Incompressible Flow Problems

• dual operator: gradient operator

grad : Q → range(grad), q 7→ ∇q

with B0 ∈ L (Q,V 0), B0 = grad • kernel of B: space of weakly divergence-free functions

V0 = Vdiv = {v ∈ V : (∇ · v,q) = 0 ∀ q ∈ Q}

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 29 (151) • boundedness and continuity of b(·,·) √ |b(v,q)| ≤ d kvkV kqkQ

◦ proof: board

2 Continuous Incompressible Flow Problems

• estimating divergence by gradient √ 1 k∇ · vkL2(Ω) ≤ d k∇vkL2(Ω) ∀ v ∈ H (Ω)

◦ proof: board ◦ estimate is sharp

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 30 (151) 2 Continuous Incompressible Flow Problems

• estimating divergence by gradient √ 1 k∇ · vkL2(Ω) ≤ d k∇vkL2(Ω) ∀ v ∈ H (Ω)

◦ proof: board ◦ estimate is sharp • boundedness and continuity of b(·,·) √ |b(v,q)| ≤ d kvkV kqkQ

◦ proof: board

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 30 (151) • V and Q fulﬁll the inf-sup condition, i.e. there is a βis > 0 such that (∇ · v,q) inf sup ≥ βis q∈Q v∈V kvkV

◦ proof: board

2 Continuous Incompressible Flow Problems

⊥ • one can show: div is an isomorphism from Vdiv onto Q • corollary: each pressure is the divergence of a velocity ﬁeld: ⊥ for each q ∈ Q there is a unique v ∈ Vdiv ⊂ V such that √ ∇ · v = q and kqkQ ≤ d kvkV , kvkV ≤ C kqkQ

with C independent of v and q ◦ proof: board

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 31 (151) 2 Continuous Incompressible Flow Problems

with C independent of v and q ◦ proof: board

• V and Q fulﬁll the inf-sup condition, i.e. there is a βis > 0 such that (∇ · v,q) inf sup ≥ βis q∈Q v∈V kvkV

◦ proof: board

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 31 (151) h h h • bilinear form b : V × Q → R bh vh,qh := − ∇ · vh,qh ∑ K K∈T h

◦ T h – triangulation of Ω ◦ K ∈ T h – mesh cells ◦ norm in V h vh = ∇vh,∇vh V h ∑ K K∈T h

2 Finite Element Spaces

• finite element spaces ◦ V h – finite element velocity space ◦ Qh – finite element pressure space ◦ V h/Qh – pair • conforming finite element spaces: V h ⊂ V and Qh ⊂ Q

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 32 (151) 2 Finite Element Spaces

• finite element spaces ◦ V h – finite element velocity space ◦ Qh – finite element pressure space ◦ V h/Qh – pair • conforming finite element spaces: V h ⊂ V and Qh ⊂ Q h h h • bilinear form b : V × Q → R bh vh,qh := − ∇ · vh,qh ∑ K K∈T h

◦ T h – triangulation of Ω ◦ K ∈ T h – mesh cells ◦ norm in V h vh = ∇vh,∇vh V h ∑ K K∈T h

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 32 (151) • discrete inf-sup condition

bh vh,qh inf sup ≥ β h > 0 h h h h is q ∈Q vh∈V h kv kV h kq kL2(Ω)

◦ not inherited from inf-sup condition fulﬁlled by V and Q ◦ discussion: board

2 Finite Element Spaces

• space of discretely divergence-free functions

h n h h h h h h ho Vdiv = v ∈ V : b v ,q = 0 ∀ q ∈ Q

h • generally Vdiv 6⊂ Vdiv ◦ ﬁnite element velocities not weakly or pointwise divergence-free ◦ conservation of mass violated

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 33 (151) 2 Finite Element Spaces

• space of discretely divergence-free functions

h n h h h h h h ho Vdiv = v ∈ V : b v ,q = 0 ∀ q ∈ Q

h • generally Vdiv 6⊂ Vdiv ◦ ﬁnite element velocities not weakly or pointwise divergence-free ◦ conservation of mass violated • discrete inf-sup condition

bh vh,qh inf sup ≥ β h > 0 h h h h is q ∈Q vh∈V h kv kV h kq kL2(Ω)

◦ not inherited from inf-sup condition fulﬁlled by V and Q ◦ discussion: board

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 33 (151) 2 Finite Element Spaces

h • Interpolation estimate for Vdiv. Let v ∈ Vdiv and let the discrete inf-sup condition hold. Then √ ! d h h inf ∇ v − v ≤ 1 + h inf ∇ v − w h h L2(Ω) h∈V h L2(Ω) v ∈Vdiv βis w

◦ proof: board

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 34 (151) 2 Finite Elements

• piecewise constant ﬁnite elements P0,(Q0)

one degree of freedom (d.o.f.) per mesh cell • continuous piecewise linear ﬁnite elements P1

d d.o.f. per mesh cell

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 35 (151) 2 Finite Elements

• continuous piecewise quadratic ﬁnite elements P2

(d + 1)(d + 2)/2 d.o.f. per mesh cell • continuous piecewise bilinear ﬁnite elements Q1

2d d.o.f. per mesh cell • and so on for continuous ﬁnite elements of higher order

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 36 (151) 2 Finite Elements

nc • nonconforming linear ﬁnite elements P1 , Crouzeix, Raviart (1973)

continuous only in barycenters of faces

d + 1 d.o.f. per mesh cell

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 37 (151) 2 Finite Elements

rot • rotated bilinear ﬁnite element Q1 , Rannacher, Turek (1992)

◦ continuous only in barycenters of faces ◦ 2d d.o.f. per mesh cell disc • discontinuous linear ﬁnite element P1 ◦ deﬁned by integral nodal functionals h disc h e.g. ϕ ∈ P1 if ϕ is linear on a mesh cell K (2d) and Z Z Z ϕh(x) dx = 0, xϕh(x) dx = 1, yϕh(x) dx = 0 K K K ◦ d + 1 d.o.f. per mesh cell

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 38 (151) • P1/P1 pair of ﬁnite element spaces violates discrete inf-sup condition ◦ counter example: checkerboard instability, board

2 Finite Element Spaces

• criterion for violation of discrete inf-sup condition: there is non-trivial qh ∈ Qh such that bh vh,qh = 0 ∀ vh ∈ V h =⇒ bh vh,qh sup h = 0 vh∈V h kv kV h

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 39 (151) 2 Finite Element Spaces

• criterion for violation of discrete inf-sup condition: there is non-trivial qh ∈ Qh such that bh vh,qh = 0 ∀ vh ∈ V h =⇒ bh vh,qh sup h = 0 vh∈V h kv kV h • P1/P1 pair of ﬁnite element spaces violates discrete inf-sup condition ◦ counter example: checkerboard instability, board

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 39 (151) 2 Finite Element Spaces

• other pairs which violated discrete inf-sup condition

◦ P1/P0 ◦ Q1/Q0 ◦ Pk/Pk, k ≥ 1 ◦ Qk/Qk, k ≥ 1 disc ◦ Pk/Pk−1, k ≥ 2, on a special macro cell • summary: ◦ many easy to implement pairs violate discrete inf-sup condition ◦ different ﬁnite element spaces for velocity and pressure necessary

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 40 (151) 2 Finite Element Spaces

• pairs which fulﬁll discrete inf-sup condition

◦ Pk/Pk−1, Qk/Qk−1: Taylor–Hood ﬁnite elements [1] − proofs: 2D, k = 2 [2] disc ◦ Qk/Qk−1 disc ◦ Pk/Pk−1, k ≥ d, on very special meshes (Scott–Vogelius element) bubble ◦ P1 /P1, mini element bubble disc ◦ Pk /Pk−1 [3] nc ◦ P1 /P0, Crouzeix–Raviart element [4] rot ◦ Q1 /Q0, Rannacher–Turek element [5] . ◦ .

[1] Taylor, Hood: Comput. Fluids 1, 73-100, 1973 [2] Verfürth: RAIRO Anal. Numér. 18, 175–182, 1984 [3] Bernardi, Raugel: Math. Comp. 44, 71–79, 1985 [4] Crouzeix, Raviart: RAIRO. Anal. Numér. 7, 33–76, 1973 [5] Rannacher, Turek: Numer. Meth. Part. Diff. Equ. 8, 97–111, 1992

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 41 (151) 2 Finite Element Spaces

• techniques for proving the discrete inf-sup condition ◦ construction of Fortin operator [1] ◦ using projection to piecewise constant pressure [2] ◦ macroelement technique [3] ◦ survey in [4]

[1] Fortin: RAIRO Anal. Numér. 11, 341–354, 1977 [2] Brezzi, Bathe: Comput. Methods Appl. Mech. Engrg. 82, 27–57, 1990 [3] Stenberg: Math. Comput. 32, 9–23, 1984 [4] Bofﬁ, Brezzi, Fortin: Lecture Notes in Mathematics 1939, Springer, 45–100, 2008

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 42 (151) 3 The Stokes Equations

• continuous equation

−∆u + ∇p = f in Ω, (1) ∇ · u = 0 in Ω

for simplicity: homogeneous Dirichlet boundary conditions • difﬁculty: coupling of velocity and pressure • properties ◦ linear ◦ form −ν∆u + ∇p = f in Ω, ∇ · u = 0 in Ω becomes (1) by rescaling with new pressure, right hand side

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 43 (151) • equivalent formulation: Find (u, p) ∈ V × Q such that

a(u,v) + b(v, p) − b(u,q) = hf,viV 0,V ∀ (v,q) ∈ V × Q

3 The Stokes Equations

1 2 • weak form: Find (u, p) ∈ H0 (Ω) × L0(Ω) such that 1 (∇u,∇v) − (∇ · v, p) = hf,vi −1 1 ∀ v ∈ H (Ω), H (Ω),H0 (Ω) 0 2 −(∇ · u,q) = 0 ∀ q ∈ L0(Ω) • casting into abstract framework ◦ spaces

1 2 V = H0 (Ω), k·kV = |·|H1(Ω) , Q = L0(Ω), k·kQ = k·kL2(Ω)

◦ bilinear forms

a(u,v) = (∇u,∇v), b(v,q) = −(∇ · v,q)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 44 (151) 3 The Stokes Equations

1 2 V = H0 (Ω), k·kV = |·|H1(Ω) , Q = L0(Ω), k·kQ = k·kL2(Ω)

◦ bilinear forms

a(u,v) = (∇u,∇v), b(v,q) = −(∇ · v,q)

• equivalent formulation: Find (u, p) ∈ V × Q such that

a(u,v) + b(v, p) − b(u,q) = hf,viV 0,V ∀ (v,q) ∈ V × Q

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 44 (151) • existence and uniqueness of solution

◦ a(·,·) is Vdiv-elliptic

2 a(v,v) = |v|H1(Ω) ∀ v ∈ V ⊃ Vdiv

◦ b(·,·) satisﬁes inf-sup condition • stability of solution 2 k∇ukL2(Ω) ≤ kfkH−1(Ω) , kpkL2(Ω) ≤ kfkH−1(Ω) βis

◦ proof and discussion: board

3 The Stokes Equations

• Vdiv – space of weakly divergence-free functions • associated problem: Find u ∈ Vdiv such that

(∇u,∇v) = hf,viV 0,V ∀ v ∈ Vdiv

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 45 (151) • stability of solution 2 k∇ukL2(Ω) ≤ kfkH−1(Ω) , kpkL2(Ω) ≤ kfkH−1(Ω) βis

◦ proof and discussion: board

3 The Stokes Equations

• Vdiv – space of weakly divergence-free functions • associated problem: Find u ∈ Vdiv such that

(∇u,∇v) = hf,viV 0,V ∀ v ∈ Vdiv

• existence and uniqueness of solution

◦ a(·,·) is Vdiv-elliptic

2 a(v,v) = |v|H1(Ω) ∀ v ∈ V ⊃ Vdiv

◦ b(·,·) satisﬁes inf-sup condition

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 45 (151) 3 The Stokes Equations

• Vdiv – space of weakly divergence-free functions • associated problem: Find u ∈ Vdiv such that

(∇u,∇v) = hf,viV 0,V ∀ v ∈ Vdiv

• existence and uniqueness of solution

◦ a(·,·) is Vdiv-elliptic

2 a(v,v) = |v|H1(Ω) ∀ v ∈ V ⊃ Vdiv

◦ b(·,·) satisﬁes inf-sup condition • stability of solution 2 k∇ukL2(Ω) ≤ kfkH−1(Ω) , kpkL2(Ω) ≤ kfkH−1(Ω) βis

◦ proof and discussion: board

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 45 (151) • only conforming inf-sup stable ﬁnite element spaces ◦ V h ⊂ V and Qh ⊂ Q ◦ bh vh,qh inf sup ≥ β h > 0 h h h h is q ∈Q vh∈V h kv kV h kq kL2(Ω)

3 Finite Element Methods for the Stokes Equations

• ﬁnite element problem: Find (uh, ph) ∈ V h × Qh such that

ah uh,vh + bh vh, ph = f,vh ∀ vh ∈ V h, bh uh,qh = 0 ∀ qh ∈ Qh

with ah vh,wh = ∇vh,∇wh , bh vh,qh = − ∇ · vh,qh ∑ K ∑ K K∈T h K∈T h

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 46 (151) 3 Finite Element Methods for the Stokes Equations

• ﬁnite element problem: Find (uh, ph) ∈ V h × Qh such that

ah uh,vh + bh vh, ph = f,vh ∀ vh ∈ V h, bh uh,qh = 0 ∀ qh ∈ Qh

with ah vh,wh = ∇vh,∇wh , bh vh,qh = − ∇ · vh,qh ∑ K ∑ K K∈T h K∈T h

• only conforming inf-sup stable ﬁnite element spaces ◦ V h ⊂ V and Qh ⊂ Q ◦ bh vh,qh inf sup ≥ β h > 0 h h h h is q ∈Q vh∈V h kv kV h kq kL2(Ω)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 46 (151) • stability 2 h h ∇u ≤ kfkH−1(Ω) , p ≤ kfkH−1(Ω) L2(Ω) L2(Ω) h βis ◦ same proof as for continuous problem • goal of ﬁnite element error analysis: estimate error by interpolation errors ◦ interpolation errors depend only on ﬁnite element spaces, not on problem ◦ estimates for interpolation error are known • reduction to a problem on the space of discretely divergence-free functions h h h h h h h a u ,v = ∇u ,∇v = f,v ∀ v ∈ Vdiv

3 Finite Element Methods for the Stokes Equations

• existence and uniqueness of a solution ◦ same proof as for continuous problem

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 47 (151) • goal of ﬁnite element error analysis: estimate error by interpolation errors ◦ interpolation errors depend only on ﬁnite element spaces, not on problem ◦ estimates for interpolation error are known • reduction to a problem on the space of discretely divergence-free functions h h h h h h h a u ,v = ∇u ,∇v = f,v ∀ v ∈ Vdiv

3 Finite Element Methods for the Stokes Equations

• existence and uniqueness of a solution ◦ same proof as for continuous problem • stability 2 h h ∇u ≤ kfkH−1(Ω) , p ≤ kfkH−1(Ω) L2(Ω) L2(Ω) h βis ◦ same proof as for continuous problem

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 47 (151) • reduction to a problem on the space of discretely divergence-free functions h h h h h h h a u ,v = ∇u ,∇v = f,v ∀ v ∈ Vdiv

3 Finite Element Methods for the Stokes Equations

• existence and uniqueness of a solution ◦ same proof as for continuous problem • stability 2 h h ∇u ≤ kfkH−1(Ω) , p ≤ kfkH−1(Ω) L2(Ω) L2(Ω) h βis ◦ same proof as for continuous problem • goal of ﬁnite element error analysis: estimate error by interpolation errors ◦ interpolation errors depend only on ﬁnite element spaces, not on problem ◦ estimates for interpolation error are known

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 47 (151) 3 Finite Element Methods for the Stokes Equations

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 47 (151) • polyhedral domain in three dimensions which is not Lipschitz-continuous

3 Finite Element Methods for the Stokes Equations

• ﬁnite element error estimate for the L2(Ω) norm of the gradient of the velocity d ◦ Ω ⊂ R , bounded, polyhedral, Lipschitz-continuous boundary

√ ! d ∇(u − uh) ≤ 2 1 + inf ∇(u − vh) L2(Ω) h h h L2(Ω) βis v ∈V √ + d inf p − qh qh∈Qh L2(Ω) ◦ proof: board

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 48 (151) 3 Finite Element Methods for the Stokes Equations

• ﬁnite element error estimate for the L2(Ω) norm of the gradient of the velocity d ◦ Ω ⊂ R , bounded, polyhedral, Lipschitz-continuous boundary

√ ! d ∇(u − uh) ≤ 2 1 + inf ∇(u − vh) L2(Ω) h h h L2(Ω) βis v ∈V √ + d inf p − qh qh∈Qh L2(Ω) ◦ proof: board • polyhedral domain in three dimensions which is not Lipschitz-continuous

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 48 (151) 3 Finite Element Methods for the Stokes Equations

• ﬁnite element error estimate for the L2(Ω) norm of the pressure ◦ same assumptions as for previous estimate

√ ! 2 d p − ph ≤ 1 + inf ∇(u − vh) L2(Ω) h h h h L2(Ω) βis βis v ∈V √ ! 2 d + 1 + inf p − qh h h h L2(Ω) βis q ∈Q

◦ proof: board

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 49 (151) ˆ 2 • regular dual Stokes problem: For given f ∈ L (Ω), ﬁnd (φ fˆ,ξfˆ) ∈ V × Q such that ˆ −∆φ fˆ + ∇ξfˆ = f in Ω, ∇ · φ fˆ = 0 in Ω ◦ regular if mapping φ fˆ,ξfˆ 7→ −∆φ fˆ + ∇ξfˆ is an isomorphism from H2(Ω) ∩V × H1(Ω) ∩ Q onto L2(Ω) ◦ Γ of class C2 ◦ bounded, convex polygons in two dimensions

3 Finite Element Methods for the Stokes Equations

• error of the velocity in the L2(Ω) norm ◦ by Poincaré inequality not optimal

u − uh ≤ C ∇(u − uh) L2(Ω) L2(Ω)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 50 (151) 3 Finite Element Methods for the Stokes Equations

• error of the velocity in the L2(Ω) norm ◦ by Poincaré inequality not optimal

u − uh ≤ C ∇(u − uh) L2(Ω) L2(Ω)

ˆ 2 • regular dual Stokes problem: For given f ∈ L (Ω), ﬁnd (φ fˆ,ξfˆ) ∈ V × Q such that ˆ −∆φ fˆ + ∇ξfˆ = f in Ω, ∇ · φ fˆ = 0 in Ω ◦ regular if mapping φ fˆ,ξfˆ 7→ −∆φ fˆ + ∇ξfˆ is an isomorphism from H2(Ω) ∩V × H1(Ω) ∩ Q onto L2(Ω) ◦ Γ of class C2 ◦ bounded, convex polygons in two dimensions

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 50 (151) 3 Finite Element Methods for the Stokes Equations

• ﬁnite element error estimate for the L2(Ω) norm of the velocity ◦ same assumptions as for previous estimates

◦ dual Stokes problem regular with solution (φ fˆ,ξfˆ)

u − uh L2(Ω) √ ≤ d ∇ u − uh + inf p − qh L2(Ω) qh∈Qh L2(Ω) " √ ! 1 d h × sup 1 + inf ∇ φ fˆ − φ ˆ h h h L2(Ω) fˆ∈L2(Ω) f L2(Ω) βis φ ∈V h + inf ξfˆ − r rh∈Qh L2(Ω)

◦ proof: board (if time admits)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 51 (151) 3 Finite Element Methods for the Stokes Equations

• finite element error estimates for conforming pairs of finite element spaces ◦ same assumptions on domain as for previous estimates ◦ solution sufficiently regular ◦ h – mesh width of triangulation ◦ spaces bubble − Pk /Pk, k = 1 (mini element), − Pk/Pk−1, Qk/Qk−1, k ≥ 2 (Taylor–Hood element), bubble disc disc − Pk /Pk−1, Qk/Pk−1,k ≥ 2

h k ∇(u − u ) ≤ Ch kukHk+1(Ω) + kpkHk(Ω) L2(Ω) h k p − p ≤ Ch kukHk+1(Ω) + kpkHk(Ω) L2(Ω)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 52 (151) 3 Finite Element Methods for the Stokes Equations

• ﬁnite element error estimates for conforming pairs of ﬁnite element spaces (cont.) ◦ in addition: dual Stokes problem regular

h k+1 u − u ≤ Ch kukHk+1(Ω) + kpkHk(Ω) L2(Ω) h ◦ all C depend on the discrete inf-sup constant βis

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 53 (151) 3 Finite Element Methods for the Stokes Equations

• analytical example which supports the error estimates • prescribed solution 2 2 u1 ∂yψ x (1 − x) y(1 − y)(1 − 2y) u = = = 200 2 2 u2 −∂xψ −x(1 − x)(1 − 2x)y (1 − y) ! 13 13 p = 10 x − y2 + (1 − x)3 y − 2 2

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 54 (151) 3 Finite Element Methods for the Stokes Equations

• initial grids (level 0)

• red reﬁnement

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 55 (151) 3 Finite Element Methods for the Stokes Equations

h • convergence of the errors ∇(u − u ) L2(Ω) for different discretizations with different orders k

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 56 (151) 3 Finite Element Methods for the Stokes Equations

h • convergence of the errors p − p L2(Ω) for different discretizations with different orders k

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 57 (151) 3 Finite Element Methods for the Stokes Equations

h • convergence of the errors u − u L2(Ω) for different discretizations with different orders k

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 58 (151) 3 Finite Element Methods for the Stokes Equations

• implementation ◦ vector-valued velocity space

h h 3Nv V = span{φ i }i=1  h Nv  Nv  Nv   φi 0 0     h     = span  0  ∪  φi  ∪  0   0   0   h   i=1 i=1 φi i=1 ◦ pressure space h h Np Q = span{ψi }i=1 ◦ representation of unknown solution

3Nv Np h h h h h h u = ∑ u j φ j , p = ∑ p j ψ j j=1 j=1

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 59 (151) 3 Finite Element Methods for the Stokes Equations

• pressure ﬁnite element space 2 ◦ standard basis functions not in L0(Ω) ◦ it can be shown under mild assumptions that standard basis functions can be used as ansatz and test functions ◦ computed pressure with standard basis functions has to be projected 2 into L0(Ω) at the end

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 60 (151) 3 Finite Element Methods for the Stokes Equations

• linear saddle point problem

ABT u f = B 0 p 0

with

h h (A)i j = ai j = ∇φ j ,∇φ i ,i, j = 1,...,3Nv, ∑ K K∈T h h h (B)i j = bi j = − ∇ · φ j ,ψi ,i = 1,...,Np, j = 1,...,3Nv, ∑ K K∈T h h ( f )i = fi = f,φ i ,i = 1,...,3Nv ∑ K K∈T h

• dimension (3d): (3Nv + Np) × (3Nv + Np)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 61 (151) h h h h • D u ,D v instead of ∇u ,∇v ◦ equivalent only if uh weakly divergence-free ◦ generally not given for ﬁnite element velocities ◦ not longer block-diagonal matrix   A11 A12 A13 T A = A12 A22 A23 T T A13 A23 A33

3 Finite Element Methods for the Stokes Equations

• matrix A ◦ symmetric ◦ positive deﬁnite ◦ block-diagonal matrix   A11 0 0 A =  0 A11 0  0 0 A11

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 62 (151) 3 Finite Element Methods for the Stokes Equations

• matrix A ◦ symmetric ◦ positive deﬁnite ◦ block-diagonal matrix   A11 0 0 A =  0 A11 0  0 0 A11 h h h h • D u ,D v instead of ∇u ,∇v ◦ equivalent only if uh weakly divergence-free ◦ generally not given for ﬁnite element velocities ◦ not longer block-diagonal matrix   A11 A12 A13 T A = A12 A22 A23 T T A13 A23 A33

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 62 (151) 4 The Oseen Equations

• continuous equation −ν∆u +( b · ∇)u + cu + ∇p = f in Ω, ∇ · u = 0 in Ω for simplicity: homogeneous Dirichlet boundary conditions

• difﬁculties: ◦ coupling of velocity and pressure ◦ dominating convection • properties ◦ linear

Carl Wilhelm Oseen (1879 – 1944)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 63 (151) • scaling of momentum equation: one of these possibilities

◦ kbkL∞(Ω) = O (1) if ν ≤ kbkL∞(Ω) ◦ ν = O (1) if kbkL∞(Ω) ≤ ν • interesting cases ◦ ν of moderate size, c = 0 in numerical solution of steady-state Navier–Stokes equations − ◦ ν of arbitrary size, c = O (∆t) 1 in numerical solution of time-dependent Navier–Stokes equations

4 The Oseen Equations

• coefﬁcients ◦ ν > 0 ◦ b ∈ W 1,∞(Ω), ∇ · b = 0 ∞ ◦ c ∈ L (Ω), c(x) ≥ c0 ≥ 0

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 64 (151) • interesting cases ◦ ν of moderate size, c = 0 in numerical solution of steady-state Navier–Stokes equations − ◦ ν of arbitrary size, c = O (∆t) 1 in numerical solution of time-dependent Navier–Stokes equations

4 The Oseen Equations

• coefﬁcients ◦ ν > 0 ◦ b ∈ W 1,∞(Ω), ∇ · b = 0 ∞ ◦ c ∈ L (Ω), c(x) ≥ c0 ≥ 0 • scaling of momentum equation: one of these possibilities

◦ kbkL∞(Ω) = O (1) if ν ≤ kbkL∞(Ω) ◦ ν = O (1) if kbkL∞(Ω) ≤ ν

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 64 (151) 4 The Oseen Equations

• coefﬁcients ◦ ν > 0 ◦ b ∈ W 1,∞(Ω), ∇ · b = 0 ∞ ◦ c ∈ L (Ω), c(x) ≥ c0 ≥ 0 • scaling of momentum equation: one of these possibilities

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 64 (151) • existence and uniqueness of solution ◦ proof: board ◦ essential condition

((b · ∇)v,v) = 0 ∀ v ∈ V

can be proved is b is weakly divergence-free and has zero trace on Γ

4 The Oseen Equations

• weak form

ν(∇u,∇v) + ((b · ∇)u + cu,v) − (∇ · v, p) = hf,viV 0,V ∀ v ∈ V, −(∇ · u,q) = 0 ∀ q ∈ Q

• bilinear forms

a : V ×V → R, a(u,v) = ν(∇u,∇v) + ((b · ∇)u + cu,v), b : V × Q → R, b(v,q) = −(∇ · v,q)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 65 (151) 4 The Oseen Equations

• weak form

ν(∇u,∇v) + ((b · ∇)u + cu,v) − (∇ · v, p) = hf,viV 0,V ∀ v ∈ V, −(∇ · u,q) = 0 ∀ q ∈ Q

• bilinear forms

a : V ×V → R, a(u,v) = ν(∇u,∇v) + ((b · ∇)u + cu,v), b : V × Q → R, b(v,q) = −(∇ · v,q)

• existence and uniqueness of solution ◦ proof: board ◦ essential condition

((b · ∇)v,v) = 0 ∀ v ∈ V

can be proved is b is weakly divergence-free and has zero trace on Γ

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 65 (151) ◦ estimates for pressure with inf-sup condition ◦ discussion: board

4 The Oseen Equations

• stability of solution ◦ dependency of bounds on coefﬁcients is important ◦ depending on regularity of data, different estimates possible − most general

2 1 ν 2 1/2 2 k∇ukL2(Ω) + c u ≤ kfkH−1(Ω) 2 L2(Ω) 2ν

2 − f ∈ L (Ω) and c0 > 0 1 2 1 2 1/2 2 ν k∇ukL2(Ω) + c u ≤ kfkL (Ω) 2 L2(Ω) 2c0 2

◦ proof: board

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 66 (151) 4 The Oseen Equations

• stability of solution ◦ dependency of bounds on coefﬁcients is important ◦ depending on regularity of data, different estimates possible − most general

2 1 ν 2 1/2 2 k∇ukL2(Ω) + c u ≤ kfkH−1(Ω) 2 L2(Ω) 2ν

2 − f ∈ L (Ω) and c0 > 0 1 2 1 2 1/2 2 ν k∇ukL2(Ω) + c u ≤ kfkL (Ω) 2 L2(Ω) 2c0 2

◦ proof: board ◦ estimates for pressure with inf-sup condition ◦ discussion: board

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 66 (151) 4 The Oseen Equations – Galerkin FEM

• Galerkin ﬁnite element method

auh,vh + bvh, ph = f,vh ∀ vh ∈ V h, buh,qh = 0 ∀ qh ∈ Qh

◦ homogeneous Dirichlet boundary conditions ◦ conforming, inf-sup stable ﬁnite element spaces • existence, uniqueness, stability like for continuous problem

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 67 (151) 4 The Oseen Equations – Galerkin FEM

• ﬁnite element error estimate for the L2(Ω) norm of the gradient of the velocity d ◦ Ω ⊂ R , bounded, polyhedral, Lipschitz-continuous boundary ◦ regularity of coefﬁcients like stated above

ν1/2 ∇ u − uh + c1/2 u − uh L2(Ω) L2(Ω) " ! # 1 h 1 h ≤ C 1 + Cos inf ∇(u − v ) + inf p − q , h h h L2(Ω) 1/2 h h L2(Ω) βis v ∈V ν q ∈Q where ( ) 1/2 1/2 1 1 Cos = ν + kck ∞ + kbk ∞ min , L (Ω) L (Ω) ν1/2 1/2 c0

◦ C does not depend on coefﬁcients and triangulation, but on Ω (Poincaré–Friedrichs inequality)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 68 (151) 4 The Oseen Equations – Galerkin FEM

• ﬁnite element error estimate for the L2(Ω) norm of the gradient of the velocity (cont.) ◦ proof: principally same as for Stokes equations ◦ estimates for convective term h h h (b · ∇)η,φ = − (b · ∇)φ ,η ≤ kbkL∞(Ω) ∇φ kηkL2(Ω) L2(Ω) 2 2 2 2 ν h ≤ kbkL∞(Ω) kηkL2(Ω) + ∇φ ν 8 L2(Ω)

or if c0 > 0 h h (b · ∇)η,φ ≤ kbkL∞(Ω) k∇ηkL2(Ω) φ L2(Ω) 2 2 1/2 h 2 kbkL∞(Ω) k∇ηk 2 c φ 2 ≤ L (Ω) + L (Ω) c0 4

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 69 (151) 4 The Oseen Equations – Galerkin FEM

• ﬁnite element error estimate for the L2(Ω) norm of the pressure ◦ same assumptions as for previous estimate

" ! h 1 1 2 h p − p ≤ C 1 + Cos inf ∇(u − v ) L2(Ω) h h h h L2(Ω) βis βis v ∈V ! # 1 1 C + 1 + + os inf p − qh , h h 1/2 h h L2(Ω) βis βis ν q ∈Q

◦ proof: as for Stokes equations, with discrete inf-sup condition

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 70 (151) 4 The Oseen Equations – Galerkin FEM

C 1 h k ∇(u − u ) ≤ h Cos kukHk+1(Ω) + kpkHk(Ω) , L2(Ω) ν1/2 ν1/2 C h k 2 os p − p ≤ Ch Cos kukHk+1(Ω) + 1 + kpkHk(Ω) L2(Ω) ν1/2

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 71 (151) 4 The Oseen Equations – Galerkin FEM

• Cos for kbkL∞(Ω) = 1

discussion: board • error bounds not uniform for small ν or small time steps

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 72 (151) 4 The Oseen Equations – Galerkin FEM

• b = u

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 73 (151) • Q2/Q1, convergence of errors for c = 100 and different values of ν

4 The Oseen Equations – Galerkin FEM

• Q2/Q1, convergence of errors for c = 0 and different values of ν

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 74 (151) 4 The Oseen Equations – Galerkin FEM

• Q2/Q1, convergence of errors for c = 0 and different values of ν

• Q2/Q1, convergence of errors for c = 100 and different values of ν

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 74 (151) • summary ◦ Galerkin discretization in some cases unstable

4 The Oseen Equations – Galerkin FEM

−4 • Q2/Q1, convergence of errors for ν = 10 and different values of c

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 75 (151) 4 The Oseen Equations – Galerkin FEM

−4 • Q2/Q1, convergence of errors for ν = 10 and different values of c

• summary ◦ Galerkin discretization in some cases unstable

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 75 (151) 4 The Oseen Equations – Residual-Based Stabilizations

• principal idea • given: linear partial differential equation in strong form

2 Astrustr = f , f ∈ L (Ω)

• Galerkin discretization ah uh,vh = f ,vh ∀ vh ∈ V h

h h 2 • needed: modiﬁcation of strong operator Astr : V → L (Ω) • residual h h h h 2 r u = Astru − f ∈ L (Ω)

• generally rh uh 6= 0

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 76 (151) • generalization δ(x) > 0 2 argmin δ 1/2rh uh = argmin δrh uh ,rh uh 2 uh∈V h L (Ω) uh∈V h with necessary condition

h h h h δr u ,Astrv = 0

4 The Oseen Equations – Residual-Based Stabilizations

• principal idea (cont.) • consider optimization problem 2 argmin rh uh = argmin rh uh ,rh uh 2 uh∈V h L (Ω) uh∈V h • necessary condition for solution (board)

h h h h r u ,Astrv = 0

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 77 (151) 4 The Oseen Equations – Residual-Based Stabilizations

• principal idea (cont.) • consider optimization problem 2 argmin rh uh = argmin rh uh ,rh uh 2 uh∈V h L (Ω) uh∈V h • necessary condition for solution (board)

h h h h r u ,Astrv = 0

• generalization δ(x) > 0 2 argmin δ 1/2rh uh = argmin δrh uh ,rh uh 2 uh∈V h L (Ω) uh∈V h with necessary condition

h h h h δr u ,Astrv = 0

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 77 (151) • example: Oseen equations, board

4 The Oseen Equations – Residual-Based Stabilizations

• principal idea (cont.) • minimizing residual alone: not good

◦ solid line – function with layer ◦ dashed line – optimal piecewise linear approximation

• consider combination

h h h h h h h h h h a u ,v + δr u ,Astrv = f ,v ∀ v ∈ V

optimal choice of weighting function δ(x) by numerical analysis

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 78 (151) 4 The Oseen Equations – Residual-Based Stabilizations

• principal idea (cont.) • minimizing residual alone: not good

◦ solid line – function with layer ◦ dashed line – optimal piecewise linear approximation

• consider combination

h h h h h h h h h h a u ,v + δr u ,Astrv = f ,v ∀ v ∈ V

optimal choice of weighting function δ(x) by numerical analysis • example: Oseen equations, board

• SUPG/PSPG/grad-div stabilization • ﬁnd uh, ph ∈ V h × Qh such that

h h h h h h h h h h Aspg u , p , v ,q = Lspg v ,q ∀ v ,q ∈ V × Q , ˜ ˜ with Aspg : V × Q × V × Q → R

Aspg ((u, p),(v,q)) = ν (∇u,∇v) + ((b · ∇)u + cu,v) − (∇ · v, p) + (∇ · u,q)

+ ∑ µK (∇ · u,∇ · v)K + ∑ δE ([|p|]E ,[|q|]E )E K∈T h E∈E h v p + ∑ −ν∆u + (b · ∇)u + cu + ∇p,δK (b · ∇)v + δK ∇q K K∈T h ˜ and Lspg : V × Q → R v p Lspg ((v,q)) = (f,v) + ∑ f,δK (b · ∇)v + δK ∇q K K∈T h

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 79 (151) • no saddle point problem because of

h h i h h i h h − δE p , q − δK ∇p ,∇q ∑ E E E ∑ K E∈E h K∈T h

◦ analysis for elliptic partial differential equations applicable ◦ inf-sup stable spaces not necessary ◦ choice of stabilization parameters affected by choice of ﬁnite element spaces

4 The Oseen Equations – Residual-Based Stabilizations

• SUPG/PSPG/grad-div stabilization (cont.) • ﬁnite element error analysis in [1] v p h • δK = δK = δK for all K ∈ T

δ = max δK, µ = max µK K∈T h K∈T h

[1] Tobiska, Verfürth, SINUM 33, 107–127, 1996

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 80 (151) 4 The Oseen Equations – Residual-Based Stabilizations

• SUPG/PSPG/grad-div stabilization (cont.) • ﬁnite element error analysis in [1] v p h • δK = δK = δK for all K ∈ T

δ = max δK, µ = max µK K∈T h K∈T h

• no saddle point problem because of

h h i h h i h h − δE p , q − δK ∇p ,∇q ∑ E E E ∑ K E∈E h K∈T h

◦ analysis for elliptic partial differential equations applicable ◦ inf-sup stable spaces not necessary ◦ choice of stabilization parameters affected by choice of ﬁnite element spaces

[1] Tobiska, Verfürth, SINUM 33, 107–127, 1996

• properties ◦ consistency

h h h h h h h h Aspg (u, p), v ,q = Lspg v ,q , ∀ v ,q ∈ V × Q

◦ Galerkin orthogonality

h h h h h h h h Aspg u − u , p − p , v ,q = 0, ∀ v ,q ∈ V × Q

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 81 (151) 4 The Oseen Equations – Residual-Based Stabilizations

• mesh-dependent norm ( 2 2 1/2 2 k(v,q)kspg = ν k∇vkL2(Ω) + c v + µK k∇ · vkL2(K) L2(Ω) ∑ K∈T h )1/2 2 2 + ∑ δE k[|q|]E kL2(E) + ∑ δK k(b · ∇)v + ∇qkL2(K) E∈E h K∈T h

◦ proof: board ◦ additional control on error of − divergence − pressure jumps − streamline derivative + gradient of pressure ◦ norm with pressure: later

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 82 (151) 4 The Oseen Equations – Residual-Based Stabilizations

• existence and uniqueness of a solution ◦ assumptions

( 2 ) hK 1 µK ≥ 0, 0 < δK ≤ min 2 , 3νCinv 3kckL∞(K)

h δE > 0 if Q 6⊂ C(Ω) ◦ proof: application of Lax–Milgram lemma − coercivity (board if time admits), ∀ vh,qh ∈ V h × Qh

2 h h h h 1 h h Aspg v ,q , v ,q ≥ v ,q 2 spg − boundedness, ∀ uh, ph,vh,qh ∈ V h × Qh

h h h h h h h h Aspg u , p , v ,q ≤ C u , p v ,q spg spg using: all norms are equivalent in ﬁnite-dimensional spaces

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 83 (151) 4 The Oseen Equations – Residual-Based Stabilizations

• stability

( 2 2 ) 2 12 kfk −1 kfk h h H (Ω) L2(Ω) 2 u , p ≤ min , + 4 ∑ δK kfkL2(K) spg 5 ν c0 K∈T h

◦ proof: as usual ◦ estimate in stronger norm than for Galerkin ﬁnite element method ◦ estimate for pressure with inf-sup condition possible

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 84 (151) • ﬁrst step: inf-sup conditions for Aspg

h h h h inf sup Aspg v ,q , w ,r ≥ βspg h h h h (v ,q )∈V ×Q (wh,rh)∈V h×Qh h h k(u ,p )k =1 vh,qh =1 spg,p k( )kspg,p

2 ◦ some conditions on stabilization parameters, e.g., δ0hK ≤ δK ◦ proof very technical ◦ βspg = O (δ0)

4 The Oseen Equations – Residual-Based Stabilizations

• norm for ﬁnite element error estimates 1/2 −2 2 k(v,q)kspg,p = k(v,q)kspg + wpres kqkL2(Ω) with n −1/2 1/2 o wpres = max 1,ν ,kckL∞(Ω) for the interesting cases of small ν and large c: small contribution of the pressure

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 85 (151) 4 The Oseen Equations – Residual-Based Stabilizations

h h h h inf sup Aspg v ,q , w ,r ≥ βspg h h h h (v ,q )∈V ×Q (wh,rh)∈V h×Qh h h k(u ,p )k =1 vh,qh =1 spg,p k( )kspg,p

2 ◦ some conditions on stabilization parameters, e.g., δ0hK ≤ δK ◦ proof very technical ◦ βspg = O (δ0)

• ﬁnite element error estimate u − uh, p − ph + ν1/2 p − ph spg L2Ω " 1/2 1/2 ! k 1/2 νδ h 1/2 µδ 1/2 ≤ C h ν + + + δ + + kck h + δ kck ∞ h kuk k+1 h δ 1/2 h L∞(Ω) L (Ω) H (Ω) ! # 1/2 −1/2 l+1 1/2 δ 1 n µ o +h ν + + max 1, kpk l+1 h ν1/2 ν H (Ω)

◦ k ≥ 1, l ≥ 0 ◦ C independent of the coefﬁcients of the problem ◦ proof: based on inf-sup condition

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 86 (151) ◦ equal-order discretizations with k = l ≥ 1 1 O (δ) = O (µ) = O (h) =⇒ order of error reduction: k + 2

• optimal asymptotics for stabilization parameters, ν ≥ h ◦ inf-sup stable discretizations with k = l + 1

2 δ = O h , µ = O (1) =⇒ order of convergence: k

◦ equal-order discretizations with k = l ≥ 1

2 δ = O h , µ arbitrary =⇒ order of convergence: k

4 The Oseen Equations – Residual-Based Stabilizations

• optimal asymptotics for stabilization parameters, ν < h (board) ◦ inf-sup stable discretizations with k = l + 1

2 δ = O h , µ = O (1) =⇒ order of error reduction: k

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 87 (151) • optimal asymptotics for stabilization parameters, ν ≥ h ◦ inf-sup stable discretizations with k = l + 1

2 δ = O h , µ = O (1) =⇒ order of convergence: k

◦ equal-order discretizations with k = l ≥ 1

2 δ = O h , µ arbitrary =⇒ order of convergence: k

4 The Oseen Equations – Residual-Based Stabilizations

• optimal asymptotics for stabilization parameters, ν < h (board) ◦ inf-sup stable discretizations with k = l + 1

2 δ = O h , µ = O (1) =⇒ order of error reduction: k

◦ equal-order discretizations with k = l ≥ 1 1 O (δ) = O (µ) = O (h) =⇒ order of error reduction: k + 2

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 87 (151) 4 The Oseen Equations – Residual-Based Stabilizations

• optimal asymptotics for stabilization parameters, ν < h (board) ◦ inf-sup stable discretizations with k = l + 1

2 δ = O h , µ = O (1) =⇒ order of error reduction: k

◦ equal-order discretizations with k = l ≥ 1 1 O (δ) = O (µ) = O (h) =⇒ order of error reduction: k + 2

• optimal asymptotics for stabilization parameters, ν ≥ h ◦ inf-sup stable discretizations with k = l + 1

2 δ = O h , µ = O (1) =⇒ order of convergence: k

◦ equal-order discretizations with k = l ≥ 1

2 δ = O h , µ arbitrary =⇒ order of convergence: k

• b = u

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 88 (151) 4 The Oseen Equations – Residual-Based Stabilizations

• Q2/Q1 ﬁnite element • stabilization parameters (based on numerical simulations from [1])

2 µK = 0.2, δK = 0.1hK

• convergence of errors for c = 0 and c = 100, different values of ν

[1] Matthies, Lube, Röhe, Comput. Methods Appl. Math. 9, 368 – 390, 2009

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 89 (151) 4 The Oseen Equations – Residual-Based Stabilizations

−4 • Q2/Q1, convergence of errors for ν = 10 and different values of c

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 90 (151) 4 The Oseen Equations – Residual-Based Stabilizations

• P1/P1 ﬁnite element • stabilization parameters ( 0.5hK if ν < hK, δ = µ = 0.5h K 2 K K 0.5hK else,

• convergence of errors for c = 0 and c = 100, different values of ν

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 91 (151) 4 The Oseen Equations – Residual-Based Stabilizations

−4 • P1/P1, convergence of errors for ν = 10 and different values of c

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 92 (151) 4 The Oseen Equations – Residual-Based Stabilizations

• implementation: same approach as for Stokes equations • grad-div term leads to matrix block     A11 A12 A13 A11 0 0 T A12 A22 A23 instead of  0 A11 0  T T A13 A23 A33 0 0 A11

• PSPG term introduces pressure-pressure couplings • SUPG term inﬂuences velocity-velocity coupling and the pressure (ansatz) - velocity (test) coupling • ﬁnal system ! AD u f = BC p fp

much more matrix blocks to store than for Galerkin FEM

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 93 (151) 4 The Oseen Equations – Residual-Based Stabilizations

• Summary and remarks h h ◦ errors (u, p) − (u , p ) spg independent of ν ◦ versions without pressure couplings available − only for inf-sup stable pairs of ﬁnite elements − easier to implement than SUPG/PSPG/grad-div stabilization ◦ numerical analysis in [1,2,3]

[1] Tobiska, Verfürth, SINUM 33, 107–127, 1996 [2] Lube, Rapin, M3AS 16, 949–966, 2006 [3] Matthies, Lube, Röhe, Comput. Methods Appl. Math. 9, 368–390, 2009

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 94 (151) 5 The Stationary Navier–Stokes Equations

• continuous equation

−ν∆u +( u · ∇)u + ∇p = f in Ω, ∇ · u = 0 in Ω

for simplicity: homogeneous Dirichlet boundary conditions • difﬁculties: ◦ coupling of velocity and pressure ◦ dominating convection ◦ nonlinear

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 95 (151) 5 The Stationary Navier–Stokes Equations

• different forms of the convective term

(u · ∇)u : convective form, ∇ · uuT : divergence form, (∇ × u) × u : rotational form

◦ convective form and divergence form equivalent if ∇ · u = 0 (board, if time permits) ◦ convective form and rotational form 1 (∇ × u) × u + ∇uT u = (u · ∇)u 2 deﬁnition of new pressure (Bernoulli pressure) in rotational form 1 p = p + uT u Bern 2

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 96 (151) 5 The Stationary Navier–Stokes Equations

• variational form of the steady-state Navier–Stokes equations: Find (u, p) ∈ V × Q such that

(ν∇u,∇v) + ((u · ∇)u,v) − (∇ · v, p) = (f,v) −(∇ · u,q) = 0

for all (v,q) ∈ V × Q • equivalent: Find (u, p) ∈ V × Q such that

(ν∇u,∇v) + ((u · ∇)u,v) − (∇ · v, p) + (∇ · u,q) = (f,v)

for all (v,q) ∈ V × Q

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 97 (151) 5 The Stationary Navier–Stokes Equations

• properties of convective term ◦ linear in each component (trilinear) ◦ u,v,w ∈ H1(Ω), product rule

((u · ∇)v,w) = ∇ · vuT ,w − ((∇ · u)v,w)

◦ u,v,w ∈ H1(Ω), product rule

((u · ∇)v,w) = (u,∇(v · w)) − ((u · ∇)w,v)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 98 (151) 5 The Stationary Navier–Stokes Equations

• convective terms in the variational formulation ◦ convective form nconv(u,v,w) = ((u · ∇)v,w) ◦ divergence form

ndiv(u,v,w) = nconv(u,v,w) + ((∇ · u)v,w)

◦ rotational form nrot(u,v,w) = ((∇ × u) × v,w) with momentum equation

(ν∇u,∇v) + nrot(u,u,v) − (∇ · v, pBern) = (f,v) ∀ v ∈ V,

◦ skew-symmetric form (for u weakly divergence-free, u · n = 0 on Γ, board) 1 n (u,v,w) = (n (u,v,w) − n (u,w,v)) skew 2 conv conv

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 99 (151) • estimates: u,v,w ∈ H1(Ω)

|nconv(u,v,w)| ≤ C kukH1(Ω) k∇vkL2(Ω) kwkH1(Ω) ,

|nskew(u,v,w)| ≤ C kukH1(Ω) kvkH1(Ω) kwkH1(Ω)

◦ proof: board

5 The Stationary Navier–Stokes Equations

• further properties of convective term • vanishing ◦ rotational and skew-symmetric form

nrot(u,v,v) = nskew(u,v,v) = 0

◦ convective and divergence form: if u weakly divergence-free and u · n = 0 on Γ nconv(u,v,v) = ndiv(u,v,v) = 0

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 100 (151) 5 The Stationary Navier–Stokes Equations

• further properties of convective term • vanishing ◦ rotational and skew-symmetric form

nrot(u,v,v) = nskew(u,v,v) = 0

◦ convective and divergence form: if u weakly divergence-free and u · n = 0 on Γ nconv(u,v,v) = ndiv(u,v,v) = 0 • estimates: u,v,w ∈ H1(Ω)

|nconv(u,v,w)| ≤ C kukH1(Ω) k∇vkL2(Ω) kwkH1(Ω) ,

|nskew(u,v,w)| ≤ C kukH1(Ω) kvkH1(Ω) kwkH1(Ω)

◦ proof: board

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 100 (151) 5 The Stationary Navier–Stokes Equations

• existence and uniqueness of a solution d ◦ Ω ⊂ R , d ∈ {2,3}, bounded domain with Lipschitz boundary ◦ f ∈ H−1(Ω) ◦ then: existence • main ideas of the proof ◦ equivalent problem in the divergence-free subspace, only velocity ◦ consider problem in finite dimensional spaces (Galerkin method) ◦ fixed point equation, existence of a solution of the finite dimensional problems: fixed point theorem of Brouwer ◦ dimension of the spaces → ∞: show subsequence of the solutions tends to a solution of the problem in the divergence-free subspace ◦ existence of the pressure: inf-sup condition

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 101 (151) • numerical simulations ◦ case of unique solution is of interest ◦ steady-state solutions unstable in non-unique case, solve time-dependent solution

5 The Stationary Navier–Stokes Equations

• existence and uniqueness of a solution (cont.) ◦ ν sufﬁciently large, i.e.

((u · ∇)v,w) 2 kfkH−1(Ω) sup < ν u,v,w∈V k∇ukL2(Ω) k∇vkL2(Ω) k∇wkL2(Ω)

◦ then: uniqueness • main idea of the proof ◦ construct a contraction, apply Banach’s ﬁxed point theorem

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 102 (151) 5 The Stationary Navier–Stokes Equations

• existence and uniqueness of a solution (cont.) ◦ ν sufﬁciently large, i.e.

((u · ∇)v,w) 2 kfkH−1(Ω) sup < ν u,v,w∈V k∇ukL2(Ω) k∇vkL2(Ω) k∇wkL2(Ω)

◦ then: uniqueness • main idea of the proof ◦ construct a contraction, apply Banach’s ﬁxed point theorem • numerical simulations ◦ case of unique solution is of interest ◦ steady-state solutions unstable in non-unique case, solve time-dependent solution

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 102 (151) 5 The Stationary Navier–Stokes Equations

• stability 1 k∇uk 2 ≤ kfk −1 , L (Ω) ν H (Ω) 1 C 2 kpkL2(Ω) ≤ 2kfkH−1(Ω) + 2 kfkH−1(Ω) βis ν

◦ proof: as usual, using n(u,u,u) = 0

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 103 (151) • ﬁnite element error analysis for nskew(·,·,·) 1 n uh,vh,vh = n uh,vh,vh − n uh,vh,vh = 0 skew 2 conv conv h note that in general u 6∈ Vdiv =⇒

h h h h h h nconv u ,v ,v 6= 0, ndiv u ,v ,v 6= 0

• same as for continuous problem: ◦ existence, uniqueness ◦ stability

5 The Stationary Navier–Stokes Equations – Galerkin FEM

• Galerkin ﬁnite element method

ν ∇uh,∇vh + nuh,uh,vh − ∇ · vh, ph = f,vh ∀ vh ∈ V h, −∇ · uh,qh = 0 ∀ qh ∈ Qh,

• inf-sup stable pair of ﬁnite element spaces

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 104 (151) • same as for continuous problem: ◦ existence, uniqueness ◦ stability

5 The Stationary Navier–Stokes Equations – Galerkin FEM

• Galerkin ﬁnite element method

ν ∇uh,∇vh + nuh,uh,vh − ∇ · vh, ph = f,vh ∀ vh ∈ V h, −∇ · uh,qh = 0 ∀ qh ∈ Qh,

• inf-sup stable pair of ﬁnite element spaces

• ﬁnite element error analysis for nskew(·,·,·) 1 n uh,vh,vh = n uh,vh,vh − n uh,vh,vh = 0 skew 2 conv conv h note that in general u 6∈ Vdiv =⇒

h h h h h h nconv u ,v ,v 6= 0, ndiv u ,v ,v 6= 0

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 104 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• Galerkin ﬁnite element method

ν ∇uh,∇vh + nuh,uh,vh − ∇ · vh, ph = f,vh ∀ vh ∈ V h, −∇ · uh,qh = 0 ∀ qh ∈ Qh,

• inf-sup stable pair of ﬁnite element spaces

• ﬁnite element error analysis for nskew(·,·,·) 1 n uh,vh,vh = n uh,vh,vh − n uh,vh,vh = 0 skew 2 conv conv h note that in general u 6∈ Vdiv =⇒

h h h h h h nconv u ,v ,v 6= 0, ndiv u ,v ,v 6= 0

• same as for continuous problem: ◦ existence, uniqueness ◦ stability

• Finite element error estimate for the L2(Ω) norm of the gradient of the velocity d ◦ Ω ⊂ R bounded domain with polyhedral boundary −2 ◦ ν kfkH−1(Ω) be sufﬁciently small such that unique solution ◦ inf-sup stable ﬁnite element spaces V h × Qh

∇(u − uh) L2(Ω) ! 1 1 h ≤ C 1 + kfkH−1(Ω) 1 + inf ∇ u − v 2 h h h L2(Ω) ν βis v ∈V 1 + inf p − qh ν qh∈Qh L2(Ω)

◦ proof: main ideas and treatment of nonlinear term: board

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 105 (151) • analytical results can be supported numerically by analytical test examples

5 The Stationary Navier–Stokes Equations – Galerkin FEM

• Finite element error estimate for the L2(Ω) norm of the pressure

p − ph L2(Ω) ! 1 2 1 ν h ≤ C 1 + kfkH−1(Ω) 1 + inf ∇ u − v h 2 h h h L2(Ω) βis ν βis v ∈V ! 1 ν h +C 1 + kfkH−1(Ω) inf p − q h 2 h h L2(Ω) βis ν q ∈Q

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 106 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• Finite element error estimate for the L2(Ω) norm of the pressure

p − ph L2(Ω) ! 1 2 1 ν h ≤ C 1 + kfkH−1(Ω) 1 + inf ∇ u − v h 2 h h h L2(Ω) βis ν βis v ∈V ! 1 ν h +C 1 + kfkH−1(Ω) inf p − q h 2 h h L2(Ω) βis ν q ∈Q

• analytical results can be supported numerically by analytical test examples

• Example: steady-state ﬂow around a cylinder at Re = 20 ◦ domain

◦ velocity

◦ pressure

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 107 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• Example: steady-state ﬂow around a cylinder at Re = 20 ◦ at the cylinder

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 108 (151) • reformulation with volume integrals possible, long but elementary derivation, e.g.

2 2U cdrag = − 2 (ν∇u,∇wd) + n(u,u,wd) − (∇ · wd, p) − (f,wd) dUmean

1 T for any function wd ∈ H (Ω) with wd = 0 on Γ \ Γcyl and wd|Γcyl = (1,0)

5 The Stationary Navier–Stokes Equations – Galerkin FEM

• important: drag and lift coefﬁcient at the cylinder 2 Z ∂vt cdrag = 2 µ ny − Pnx ds, ρdUmean Γcyl ∂n 2 Z ∂vt clift = − 2 µ nx + Pny ds ρdUmean Γcyl ∂n

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 109 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• important: drag and lift coefﬁcient at the cylinder 2 Z ∂vt cdrag = 2 µ ny − Pnx ds, ρdUmean Γcyl ∂n 2 Z ∂vt clift = − 2 µ nx + Pny ds ρdUmean Γcyl ∂n

• reformulation with volume integrals possible, long but elementary derivation, e.g.

2 2U cdrag = − 2 (ν∇u,∇wd) + n(u,u,wd) − (∇ · wd, p) − (f,wd) dUmean

1 T for any function wd ∈ H (Ω) with wd = 0 on Γ \ Γcyl and wd|Γcyl = (1,0)

• reference values ◦ [1] : compiled from simulations of different groups

cdrag,ref ∈ [5.57,5.59], clift,ref ∈ [0.104,0.110]

◦ [2] : do-nothing conditions at outlet

cdrag,ref = 5.57953523384, clift,ref = 0.010618948146

◦ [3] : Dirichlet conditions at outlet

cdrag,ref = 5.57953523384, clift,ref = 0.010618937712

[1] Schäfer, Turek, Notes on Numerical Fluid Mechanics 52, 547–566, 1996 [2] Nabh, PhD thesis, Heidelberg, 1998 [3] J., Matthies, IJNMF 37, 885–903, 2001

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 110 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• initial grids

• patch for test function in computation of coefﬁcients, Q2

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 111 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• convective form of convective term • do-nothing boundary conditions • convergence of drag coefﬁcient

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 112 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• convergence of lift coefﬁcient

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 113 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• preliminary results: different forms of the convective term, P2/P1

• rotational form ◦ reconstructed pressure has boundary layers, inaccurate results

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 114 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• schemes for solving the nonlinearity • ﬁxed point iteration

(m+1) (m) (m) (m) (m)!! u u −1 (f,v) − N u ;u , p (m+1) = (m) − ϑNlin p p 0

with a(u,v) + n(w,u,v) + b(v, p) N(w;u, p) = b(u,q)

Nlin – linear operator ϑ ∈ (0,1] – damping factor

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 115 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• ﬁxed point iteration ◦ linear system to be solved !! δu(m+1) (f,v) − N u(m);u(m), p(m) Nlin (m+1) = δ p 0

◦ setting δu(m+1) u˜ (m+1) − u(m) = , δ p(m+1) p˜(m+1) − p(m) then !! u˜ (m+1) (f,v) − N u(m);u(m), p(m) u(m) Nlin (m+1) = + Nlin (m) p˜ 0 p

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 116 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• iteration with Stokes equations

(m+1) (m+1) Nlin = N 0;u˜ , p˜

• then

a(u˜ (m+1),v) + b(v, p˜(m+1)) b(u˜ (m+1),q) (f,v) − a(u(m),v) − n(u(m),u(m),v) − b(v, p˜(m)) = −b(u˜ (m),q) a(u(m),v) + b(v, p˜(m)) (f,v) − n(u(m),u(m),v) + = b(u˜ (m),q) 0

◦ converges only if ν is sufﬁciently large ◦ not recommended

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 117 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• iteration with Oseen-type equations, Picard iteration

(m) (m+1) (m+1) Nlin = N u ;u˜ , p˜

• then

a(u˜ (m+1),v) + n(u(m),u˜ (m+1),v) + b(v, p˜(m+1)) b(u˜ (m+1),q) (f,v) − a(u(m),v) − n(u(m),u(m),v) − b(v, p˜(m)) = −b(u˜ (m),q) a(u(m),v) + n(u(m),u(m),v) + b(v, p˜(m)) + b(u˜ (m),q) (f,v) = 0

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 118 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• iteration with Oseen-type equations, Picard iteration (cont.) • different forms of nonlinear term

(m) (m+1) (m) (m+1) nconv u ,u˜ ,v = u · ∇ u˜ ,v ,

(m) (m+1) (m) (m+1) (m) (m+1) ndiv u ,u˜ ,v = u · ∇ u˜ ,v + ∇ · u u˜ ,v ,

(m) (m+1) (m) (m+1) nrot u ,u˜ ,v = ∇ × u × u˜ ,v 1h n u(m),u˜ (m+1),v = u(m) · ∇ u˜ (m+1),v skew 2 i − u(m) · ∇ v,u˜ (m+1)

◦ discussion: board • widely used

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 119 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• Newton’s method • linear operator is derivative of the nonlinear operator at the current position u(m) N = DN lin p(m) ◦ with Gâteaux derivative at (u, p)T u N(u + εφ;u + εφ, p + εψ) − N(u;u, p) DN = lim p ε→0 ε = N(φ;u, p) + N(u;φ, p) + N(u,u,ψ) ◦ inserting and collecting terms (board) a(u˜ (m+1),v) + n(u(m),u˜ (m+1),v) + n(u˜ (m+1),u(m),v) + b(v, p˜(m+1)) b(u˜ (m+1),q) (f,v) + n(u(m),u(m),v) = 0

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 120 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• Newton’s method (cont.) ◦ order of convergence of Newton’s method expected to be better than of the Picard iteration if − the solution (u, p) is sufﬁciently smooth − the linear systems are solved sufﬁciently accurately ◦ properties of term n(u˜ (m+1),u(m),v) not clear ◦ sometimes used in practice

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 121 (151) ◦ convective form of convective term − Picard iteration   A11 0 0 A =  0 A11 0  0 0 A11 − Newton iteration   A11 A12 A13 A =  A21 A22 A23  A31 A32 A33

5 The Stationary Navier–Stokes Equations – Galerkin FEM

• implementation ◦ same principal approach as for Stokes and Oseen equations ◦ inf-sup stable ﬁnite elements lead to linear saddle point problems in ﬁxed point iteration ABT u f = B 0 p 0

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 122 (151) 5 The Stationary Navier–Stokes Equations – Galerkin FEM

• implementation ◦ same principal approach as for Stokes and Oseen equations ◦ inf-sup stable ﬁnite elements lead to linear saddle point problems in ﬁxed point iteration ABT u f = B 0 p 0 ◦ convective form of convective term − Picard iteration   A11 0 0 A =  0 A11 0  0 0 A11 − Newton iteration   A11 A12 A13 A =  A21 A22 A23  A31 A32 A33

• residual-based (and other) stabilizations possible ◦ better: solve time-dependent problem

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 123 (151) 6 The Time-Dependent NSE – Laminar Flows

• continuous equation

∂t u − ν∆u + (u · ∇)u + ∇p = f in (0,T] × Ω, ∇ · u = 0 in [0,T] × Ω, u(0,·) = u0 in Ω,

with u = 0 in [0,T] × Γ

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 124 (151) • properties ◦ no time derivative with respect to u ◦ no second order space derivative with respect to u ◦ the pressure vanished because Z Z ∇p · ϕ dx = (∇p,ϕ) = p(s)ϕ(s)·n(s) ds − (p,∇ · ϕ) = 0 Ω Γ |{z} |{z} =0 =0

6 The Time-Dependent NSE – Laminar Flows

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 125 (151) 6 The Time-Dependent NSE – Laminar Flows

• weak or variational formulation obtained by ◦ multiply Navier–Stokes equations with a suitable test function ϕ ◦ integrate on (0,T) × Ω ◦ apply integration by parts 1 • weak or variational formulation: ﬁnd u : (0,T] → H0 (Ω) such that Z T h −1 i − (u,∂t ϕ) + Re (∇u,∇ϕ) + ((u · ∇)u,ϕ) (τ) dτ 0 Z T = (f,ϕ)(τ) dτ + (u0,ϕ (0,·)) 0 • properties ◦ no time derivative with respect to u ◦ no second order space derivative with respect to u ◦ the pressure vanished because Z Z ∇p · ϕ dx = (∇p,ϕ) = p(s)ϕ(s)·n(s) ds − (p,∇ · ϕ) = 0 Ω Γ |{z} |{z} =0 =0

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 125 (151) 6 The Time-Dependent NSE – Laminar Flows

• mathematical analysis ◦ 2d: existence and uniqueness of weak solution, Leary (1933), Hopf (1951) ◦ 3d: existence of weak solution, Leary (1933), Hopf (1951) • Jean Leray (1906 – 1998) Eberhard Hopf (1902 – 1983)

Uniqueness of weak solution of 3d Navier–Stokes equations is open problem !

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 126 (151) • stability of velocity (board)

2 2 2 1 2 ku(T)k 2 + ν k∇uk ≤ ku(0)k 2 + kfk L (Ω) L2(0,T;L2(Ω)) L (Ω) ν L2(0,T;H−1(Ω))

6 The Time-Dependent NSE – Laminar Flows

• different form of the variational formulation

(∂t u,v) + ν (∇u,∇v) + n(u,u,v) − (∇ · v, p) = (f,v) ∀ v ∈ V, −(∇ · u,q) = 0 ∀ q ∈ Q,

and u(0,x) = u0(x)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 127 (151) 6 The Time-Dependent NSE – Laminar Flows

• different form of the variational formulation

(∂t u,v) + ν (∇u,∇v) + n(u,u,v) − (∇ · v, p) = (f,v) ∀ v ∈ V, −(∇ · u,q) = 0 ∀ q ∈ Q,

and u(0,x) = u0(x) • stability of velocity (board)

2 2 2 1 2 ku(T)k 2 + ν k∇uk ≤ ku(0)k 2 + kfk L (Ω) L2(0,T;L2(Ω)) L (Ω) ν L2(0,T;H−1(Ω))

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 127 (151) • one-step θ-schemes: n = k θ1 θ2 θ3 θ4 tk tk+1 ∆tk+1 order forward Euler scheme 0 1 1 0 tn tn+1 ∆tn+1 backward Euler scheme (BWE) 1 0 0 1 tn tn+1 ∆tn+1 1 Crank–Nicolson scheme (CN) 0.5 0.5 0.5 0.5 tn tn+1 ∆tn+1 2

6 The Time-Dependent NSE – Laminar Flows

• implicit θ-schemes as semi discretization in time

◦ ∆tn+1 = tn+1 −tn ◦ subscript k for quantities at time level k

uk+1 + θ1∆tn+1[−ν∆uk+1 + (uk+1 · ∇)uk+1] + ∆tk+1∇pk+1

= uk − θ2∆tn+1[−ν∇ · ∆uk + (uk · ∇)uk] + θ3∆tn+1fk

+θ4∆tn+1fk+1,

∇ · uk+1 = 0,

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 128 (151) 6 The Time-Dependent NSE – Laminar Flows

• implicit θ-schemes as semi discretization in time

◦ ∆tn+1 = tn+1 −tn ◦ subscript k for quantities at time level k

uk+1 + θ1∆tn+1[−ν∆uk+1 + (uk+1 · ∇)uk+1] + ∆tk+1∇pk+1

= uk − θ2∆tn+1[−ν∇ · ∆uk + (uk · ∇)uk] + θ3∆tn+1fk

+θ4∆tn+1fk+1,

∇ · uk+1 = 0,

• one-step θ-schemes: n = k θ1 θ2 θ3 θ4 tk tk+1 ∆tk+1 order forward Euler scheme 0 1 1 0 tn tn+1 ∆tn+1 backward Euler scheme (BWE) 1 0 0 1 tn tn+1 ∆tn+1 1 Crank–Nicolson scheme (CN) 0.5 0.5 0.5 0.5 tn tn+1 ∆tn+1 2

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 128 (151) 6 The Time-Dependent NSE – Laminar Flows

• fractional-step θ-scheme [1] ◦ three-step scheme ◦ two variants √ 2 θ˜ θ = 1 − , θ˜ = 1 − 2θ, τ = , η = 1 − τ 2 1 − θ

θ1 θ2 θ3 θ4 tk tk+1 ∆tk+1 order FS0 τθ ηθ ηθ τθ tn tn + θ∆tn+1 θ∆tn+1 ηθ˜ τθ˜ τθ˜ ηθ˜ tn + θ∆tn+1 tn+1 − θ∆tn+1 θ˜∆tn+1 2 τθ ηθ ηθ τθ tn+1 − θ∆tn+1 tn+1 θ∆tn+1 FS1 τθ ηθ θ 0 tn tn + θ∆tn+1 θ∆tn+1 ηθ˜ τθ˜ 0 θ˜ tn + θ∆tn+1 tn+1 − θ∆tn+1 θ˜∆tn+1 2 τθ ηθ θ 0 tn+1 − θ∆tn+1 tn+1 θ∆tn+1

[1] Bristeau, Glowinski, Periaux: Finite elements in physics, North-Holland, 73–187, 1986

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 129 (151) • number of papers with ﬁnite element error estimates available ◦ proofs become long ◦ same techniques as for steady-state problems + Gronwall’s lemma

6 The Time-Dependent NSE – Laminar Flows

• popular approaches: BWE, CN • stability ◦ BWE, FS0, FS1: strongly A-stable ◦ CN: A-stable • FS1 less expensive than FS0 if computation of right hand side costly

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 130 (151) 6 The Time-Dependent NSE – Laminar Flows

• popular approaches: BWE, CN • stability ◦ BWE, FS0, FS1: strongly A-stable ◦ CN: A-stable • FS1 less expensive than FS0 if computation of right hand side costly • number of papers with ﬁnite element error estimates available ◦ proofs become long ◦ same techniques as for steady-state problems + Gronwall’s lemma

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 130 (151) 6 The Time-Dependent NSE – Laminar Flows

• flow around a cylinder ◦ reference curves for drag and lift [1]

[1] J., Rang, CMAME 199, 514–524, 2010

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 131 (151) 6 The Time-Dependent NSE – Laminar Flows

disc • reﬁnement in space with Q2/P1

disc P2/P1 Q2/P1 level velocity pressure all velocity pressure all 3 25 408 3248 28 656 27 232 9984 37 216 4 100 480 12 704 113 184 107 712 39 936 147 648 5 399 616 50 240 449 856 428 416 159 744 588 160 • reﬁnement in time: ∆t ∈ {0.02,0.01,0.005}

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 132 (151) 6 The Time-Dependent NSE – Laminar Flows

• error to the reference curve for the drag coefﬁcient

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 133 (151) 6 The Time-Dependent NSE – Laminar Flows

• error to the reference curve for the drag coefﬁcient

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 134 (151) 6 The Time-Dependent NSE – Laminar Flows

• error to the reference curve for the lift coefﬁcient

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 135 (151) 6 The Time-Dependent NSE – Laminar Flows

• error to the reference curve for the lift coefﬁcient

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 136 (151) • BWE much to inaccurate (dissipative)

6 The Time-Dependent NSE – Laminar Flows

• temporal evolution of lift coefﬁcient

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 137 (151) 6 The Time-Dependent NSE – Laminar Flows

• temporal evolution of lift coefﬁcient

• BWE much to inaccurate (dissipative)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 137 (151) 6 The Time-Dependent NSE – Laminar Flows

• projection method ◦ motivation: schemes without need to solve (nonlinear) saddle point problems ◦ survey in [1]

[1] Guermond, Minev, Shen, CMAME 195, 6011–6045, 2006

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 138 (151) 6 The Time-Dependent NSE – Laminar Flows

• idea: decoupled NSE to obtain separate equations for velocity and pressure ◦ approximation of time derivative given (q-step scheme) ! 1 q−1 q ∂t u(tn+1) ≈ τqun+1 + ∑ τ jun− j , ∑ τ j = 0 ∆t i=0 i=0 ◦ equation for intermediate velocity: given pˆ or ∇pˆ ! 1 q−1 τqu˜ n+1 + ∑ τ jun− j − ν∆u˜ + (u˜ · ∇)u˜ = f − ∇pˆ in (0,T] × Ω ∆t i=0 ◦ correction step for divergence-free velocity 1 (τ u − τ u˜ ) + ∇ϕ (u˜) + ∇p = ∇pˆ in (0,T] × Ω, ∆t q n+1 q n+1 ∇ · u = 0 in [0,T] × Ω ϕ(·) – given function

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 139 (151) 6 The Time-Dependent NSE – Laminar Flows

2 • velocity computed in projection step is L (Ω) projection of u˜ n+1 into

2 2 Hdiv(Ω) = v ∈ L (Ω), ∇ · v ∈ L (Ω) : ∇ · v = 0 and v · n = 0 on Γ

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 140 (151) 6 The Time-Dependent NSE – Laminar Flows

• non-incremental pressure-correction scheme ◦ pˆ = 0, ϕ (·) = 0 ◦ proposed in [1,2] ◦ with backward Euler • intermediate velocity

u˜ n+1 + ∆tn+1 (−ν∆u˜ n+1 + (u˜ n · ∇)u˜ n+1) = un + ∆tn+1fn+1 in Ω

with u˜ n+1 = 0 on Γ • projection step

un+1 + ∆tn+1∇pn+1 = u˜ n+1 in Ω, ∇ · un+1 = 0 in Ω, un+1 · n = 0 on Γ

[1] Chorin, Math. Comp. 22, 745–762, 1968 [2] Temam, Arch. Rational Mech. Anal. 33, 377–385, 1969

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 141 (151) • error estimates: (u, p) result of projection step

1/2 kp − pkl∞(0,T;L2(Ω)) + ku − u˜kl∞(0,T;H1(Ω)) ≤ C (u, p,T)∆t if in addition domain has regularity property

ku − ukl∞(0,T;L2(Ω)) + ku − u˜kl∞(0,T;L2(Ω)) ≤ C (u, p,T)∆t

6 The Time-Dependent NSE – Laminar Flows

• non-incremental pressure-correction scheme (cont.) • taking divergence of projection step 1 ∇ · ∇pn+1 = ∆pn+1 = ∇ · u˜ n+1 ∆tn+1 ◦ Poisson equation for the pressure ◦ boundary condition 1 ∇pn+1 · n = − (un+1 − u˜ n+1) · n = 0 ∆tn+1

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 142 (151) 6 The Time-Dependent NSE – Laminar Flows

1/2 kp − pkl∞(0,T;L2(Ω)) + ku − u˜kl∞(0,T;H1(Ω)) ≤ C (u, p,T)∆t if in addition domain has regularity property

ku − ukl∞(0,T;L2(Ω)) + ku − u˜kl∞(0,T;L2(Ω)) ≤ C (u, p,T)∆t

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 142 (151) 6 The Time-Dependent NSE – Laminar Flows

• non-incremental pressure-correction scheme (cont.) ◦ inf-sup stable finite elements not necessary ◦ however, spurious oscillations may appear if the time step becomes too small ◦ low orders of convergence ◦ splitting error is O (∆t) =⇒ first order time stepping scheme sufficient ◦ artificial Neumann boundary condition for the pressure induces a numerical boundary layer

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 143 (151) 6 The Time-Dependent NSE – Laminar Flows

• standard incremental pressure-correction scheme

◦ pˆ = pn, ϕ (·) = 0 ◦ with BDF2 • intermediate velocity

3u˜ n+1 + 2∆t (−ν∆u˜ n+1 + (u˜ n · ∇)u˜ n+1)

= 4un − un−1 + 2∆t (fn+1 − ∇pn) in Ω,

with u˜ n+1 = 0 on Γ • projection step

3un+1 + 2∆t∇(pn+1 − pn) = 3u˜ n+1 in Ω, ∇ · un+1 = 0 in Ω, un+1 · n = 0 on Γ

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 144 (151) • error estimates, with appropriate initial step, (u, p) result of projection step

kp − pkl∞(0,T;L2(Ω)) + ku − u˜kl∞(0,T;H1(Ω)) ≤ C (u, p,T)∆t if in addition domain has regularity property 2 ku − ukl∞(0,T;L2(Ω)) + ku − u˜kl2(0,T;L2(Ω)) ≤ C (u, p,T)∆t

6 The Time-Dependent NSE – Laminar Flows

• standard incremental pressure-correction scheme (cont.) • taking divergence of projection step 3 ∆(p − p ) = ∇ · u˜ in Ω n+1 n 2∆t n+1 ◦ Poisson equation for the pressure update ◦ boundary condition

∇(pn+1 − pn) · n = 0 on Γ

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 145 (151) 6 The Time-Dependent NSE – Laminar Flows

∇(pn+1 − pn) · n = 0 on Γ • error estimates, with appropriate initial step, (u, p) result of projection step

kp − pkl∞(0,T;L2(Ω)) + ku − u˜kl∞(0,T;H1(Ω)) ≤ C (u, p,T)∆t if in addition domain has regularity property 2 ku − ukl∞(0,T;L2(Ω)) + ku − u˜kl2(0,T;L2(Ω)) ≤ C (u, p,T)∆t

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 145 (151) 6 The Time-Dependent NSE – Laminar Flows

• standard incremental pressure-correction scheme (cont.) ◦ similar estimates for Crank–Nicolson scheme ◦ splitting error is O ∆t2 =⇒ second order time stepping scheme sufﬁcient ◦ artiﬁcial Neumann boundary condition for the pressure induces a numerical boundary layer

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 146 (151) 6 The Time-Dependent NSE – Laminar Flows

• rotational incremental pressure-correction scheme

◦ pˆ = pn, ϕ (u˜) = ν∇ · u˜ n+1 ◦ with BDF2 • intermediate velocity

3u˜ n+1 + 2∆t (−ν∆u˜ n+1 + (u˜ n · ∇)u˜ n+1)

= 4un − un−1 + 2∆t (fn+1 − ∇pn) in Ω,

with u˜ n+1 = 0 on Γ • projection step

3un+1 + 2∆t∇(pn+1 − pn) = 3u˜ n+1 − 2ν∆t∇(∇ · u˜ n+1) in Ω, ∇ · un+1 = 0 in Ω, un+1 · n = 0 on Γ

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 147 (151) • error estimates, with appropriate initial step, (u, p) result of projection step 3/2 kp − pkl2(0,T;L2(Ω)) +ku − u˜kl2(0,T;H1(Ω)) +ku − ukl2(0,T;H1(Ω)) ≤C (u, p,T)∆t if in addition domain has regularity property 2 ku − ukl2(0,T;L2(Ω)) + ku − u˜kl2(0,T;L2(Ω)) ≤ C (u, p,T)∆t

6 The Time-Dependent NSE – Laminar Flows

• rotational incremental pressure-correction scheme (cont.) • taking divergence of projection step 3 ∆p˜ = ∇ · u˜ with p˜ = p − p + ν∇ · u˜ n 2∆t n+1 n n+1 n n+1 ◦ Poisson equation for the modiﬁed pressure ◦ boundary condition

∇pn+1 · n = (fn+1 − ν∇ × ∇ × un+1) · n on Γ

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 148 (151) 6 The Time-Dependent NSE – Laminar Flows

∇pn+1 · n = (fn+1 − ν∇ × ∇ × un+1) · n on Γ • error estimates, with appropriate initial step, (u, p) result of projection step 3/2 kp − pkl2(0,T;L2(Ω)) +ku − u˜kl2(0,T;H1(Ω)) +ku − ukl2(0,T;H1(Ω)) ≤C (u, p,T)∆t if in addition domain has regularity property 2 ku − ukl2(0,T;L2(Ω)) + ku − u˜kl2(0,T;L2(Ω)) ≤ C (u, p,T)∆t

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 148 (151) 6 The Time-Dependent NSE – Laminar Flows

• rotational incremental pressure-correction scheme (cont.) ◦ equivalent formulation of velocity step

3un+1 + 2∆t (ν∇ × ∇ × un+1 + (u˜ n · ∇)u˜ n+1 + ∇pn+1) = 4un − un−1 + 2∆tfn+1 in Ω, ∇ · un+1 = 0 in Ω

◦ boundary condition for the pressure is consistent, can be derived from the Navier–Stokes equations

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 149 (151) 6 The Time-Dependent NSE – Laminar Flows

• only u˜ n+1 needed in implementation • ﬁrst experience with non-incremental and standard incremental scheme: very inaccurate at boundaries (bad drag and lift coefﬁcients)

Finite Element Methods for the Simulation of Incompressible Flows · Course at Universidad Autonoma de Madrid, 27.02. – 02.03.2012 · Page 150 (151) Thank you for your attention !

http://www.wias-berlin.de/people/john/

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