An Introduction to the Finite Element Method (FEM) for Differential Equations Part II: Problems in Rd, (D >

An Introduction to the Finite Element Method (FEM) for Differential Equations Part II: Problems in Rd, (D >

An Introduction to the Finite Element Method (FEM) for Differential Equations Part II: Problems in Rd, (d> 1). Mohammad Asadzadeh January 14, 2019 Contents 10 Approximation in several dimensions 5 10.1Introduction............................ 5 10.2 Piecewiselinearapproximationin2D . 7 10.2.1 Basis functions for the piecewise linears in 2 D . 8 10.3 Constructingfiniteelementspaces . 14 10.4 Theinterpolant.......................... 17 10.4.1 Error estimates for piecewise linear interpolation . 19 10.4.2 The L2 andRitzprojections. 23 10.5Exercises ............................. 26 11 The Poisson Equation 31 11.1 TheFundamentalSolution. 32 11.1.1 Green’sFunctions . 33 11.1.2 MethodofImages . 35 11.2Stability.............................. 36 11.3 ErrorEstimatesfortheCG(1)FEM . 36 11.3.1 ProofoftheregularityLemma . 42 11.4Exercises ............................. 43 12 The Initial Boundary Value Problems in RN 47 12.1 The heat equation in RN .................... 47 12.1.1 Thefundamentalsolution . 48 12.2Stability.............................. 49 12.3 Thefiniteelementforheatequation . 52 12.3.1 Thesemidiscreteproblem . 52 12.3.2 Afullydiscretealgorithm . 55 12.3.3 Constructing the discrete equations. 56 12.3.4 An a priori error estimate: Fully discrete problem . 57 12.4Exercises ............................. 58 12.5 The wave equation in RN .................... 61 12.5.1 Theweakformulation . 62 12.5.2 Thesemi-discreteproblem. 62 12.5.3 A priori error estimates for the semi-discrete problem 63 3 4 CONTENTS 12.5.4 Thefully-discreteproblem. 64 12.6Exercises ............................. 64 Chapter 10 Approximation in several dimensions 10.1 Introduction This chapter is, mainly, devoted to piecewise linear finite element approximation in two dimensions. To this end, we shall prove a simple version of the Green’s formula useful in weak formulations. We shall demonstrate the finite element structure through a canonical example, introduce some common general finite element spaces and finally extend the interpolation concept to higher dimen- sions. The study of one-dimensional problems in the previous chapters can be extended to higher dimensional domains in Rn, n 2. Then, the mathematical ≥ calculus, being in several variables, becomes somewhat involved. On the other hand, the two and three dimensional cases concern the most relevant models both from physical point of views as well as application aspects. In the sequel we assume that the reader is familiar with the calculus of several variables. A general problem to study is the convection-diffusion-absorption equation ∆u(x)+ β(x) u(x)+ a(x)u(x)= f(x), x Ω Rn, − ·∇ ∈ ⊂ (10.1.1) u(x) = 0, x ∂Ω, ∈ where n 2, β(x) and a(x) are convection and absorption coefficients, re- ≥ spectively and f(x) is a source term. The discretization procedure, e.g., ap- proximating with piecewise polynomials and deriving the approximation error in certain norm, would require extension of the interpolation estimates from the intervals in R (the 1d case) to higher dimensional domains in Rn, n 2. Our ≥ focus in this chapter is on the problems in two space dimensions. Then, to compute the double integrals which appear, e.g., in the weak formulations, the frequently used partial integration in the one-dimensional case is now replaced by the Green’s formula below. 5 6 CHAPTER 10. APPROXIMATION IN SEVERAL DIMENSIONS Lemma 10.1 (Green’s formula). Let Ω be an open bounded subset of R2, with a piecewise smooth boundary and u C2(Ω) and v C1(Ω), then ∈ ∈ (∆u)vdxdy = ( u n)vds u vdxdy, (10.1.2) ZΩ Z∂Ω ∇ · − ZΩ ∇ ·∇ where n := n(x,y) is the outward unit normal at the boundary point x = (x,y) ∂Ω and ds is a curve element on the boundary ∂Ω. ∈ n ∂Ω ds Ω Figure 10.1: A smooth domain Ω with an outward unit normal n Proof. Since we shall work with polygonal domains, we give a simple proof when Ω is a rectangle, see the figure below. (The classical proof for a general domain Ω can be found in any textbook in the calculus of several variables). Then, using integration by parts we have ∂2u b a ∂2u 2 v dxdy = 2 (x,y) v(x,y) dxdy ZZΩ ∂x Z0 Z0 ∂x b ∂u a a ∂u ∂v = (x,y) v(x,y) (x,y) (x,y)dx dy Z0 h∂x ix=0 − Z0 ∂x ∂x b ∂u ∂u = (a,y) v(a,y) (0,y) v(0,y) dy Z0 ∂x − ∂x ∂u ∂v (x,y) (x,y) dxdy. − ZZΩ ∂x ∂x Now identifying n on ∂Ω,viz. onΓ1 : n(a,y)=(1, 0), on Γ2 : n(x, b)=(0, 1), on Γ : n(0,y)=( 1, 0), 3 − on Γ : n(x, 0) = (0, 1), 4 − 10.2. PIECEWISE LINEAR APPROXIMATION IN 2 D 7 n(x, b)=(0, 1) Γ2 := ∂Ω2 b n(0,y)=( 1, 0) n(a,y)=(1, 0) − Ω Γ3 := ∂Ω3 Γ1 := ∂Ω1 Γ4 := ∂Ω4 a n(x, 0) = (0, 1) − Figure 10.2: A rectangular domain with outward unit normals to its sides the first integral on the right hand side above can be written as ∂u ∂u + , (x,y) n(x,y)v(x,y)ds. ZΓ1 ZΓ3 ∂x ∂y · Hence ∂2u ∂u ∂u ∂u ∂v n 2 v dxdy = , (x,y)v(x,y)ds dxdy. ZZΩ ∂x ZΓ1 Γ3 ∂x ∂y · − ZZΩ ∂x ∂x ∪ Similarly, in the y-direction we get ∂2u ∂u ∂u ∂u ∂v n 2 vdxdy = , (x,y)v(x,y)ds dxdy. ZZΩ ∂y ZΓ2 Γ4 ∂x ∂y · − ZZΩ ∂y ∂y ∪ Adding up these two relations the proof is complete. 10.2 Piecewise linear approximation in 2 D In this section we introduce the principle ideas of piecewise linear polynomial approximations of the solutions for differential equations in two dimensional polygonal domains. Hence we consider partitions (meshes) without any concern about curved boundaries. At the end of this chapter, in connection with a formal description of the finite element procedure, we shall briefly discuss some common extensions of the linear approximations to higher order polynomials both in triangular and quadrilateral (as well as tetrahedrons in the three dimensional case) elements. See Tables 9.1-9.4 8 CHAPTER 10. APPROXIMATION IN SEVERAL DIMENSIONS 10.2.1 Basis functions for the piecewise linears in 2 D Recall that in the one-dimensional case a function which is linear on a in- terval is uniquely determined by its values at two points (e.g. the endpoints of the interval), since there is only one straight line connecting two points. y x xk 1 Ik xk − Figure 10.3: A linear function on a subinterval Ik =(xk 1,xk). − Similarly a plane in R3 is uniquely determined by three points which are not lying on the same straight line. Therefore, for piecewise linear approximation of a function u defined on a two dimensional polygonal domain Ω R2, i.e. p ⊂ u : Ω R, it is natural to make partitions of Ω into triangular elements p → p and let the sides of the triangles correspond to the endpoints of the intervals in the one-dimensional case. Ω Ν Ν Ν Ν Ν 3 4 5 6 Ν 2 1 Ωp Figure 10.4: Example of triangulation of a 2D domain The Figure 10.4 illustrates a “partitioning”, i.e., triangulation of a domain Ω with curved boundary where the partitioning concerns only the polygonal domain Ω generated by Ω. Here we have 6 internal nodes N , 1 i 6 p i ≤ ≤ and Ωp is the large polygonal domain inside Ω, which is triangulated. 10.2. PIECEWISE LINEAR APPROXIMATION IN 2 D 9 z = f(x,y) (x3,y3,z3) (x1,y1,z1) (x2,y2,z2) y (x3,y3, 0) C (x1,y1, 0) A (x2,y2, 0) B x Figure 10.5: A triangle in 3D as a piecewise linear function and its pro- jection in 2D. Figure 10.5 shows a piecewise linear function on a single triangle (ele- ment) which is determined by its values, zi, i = 1, 2, 3, at the vertices of the triangle ABC. Below we define the concepts of subdivision in any spatial dimension and triangulation in the 2D case. Definition 10.1. A subdivision of a computational domain Ω is a finite collection of open sets K such that { i} a) K K = i j ∅ T ¯ ¯ b) i Ki = Ω. S Definition 10.2. A triangulation of a polygonal domain Ω R2 is a sub- ⊂ division of Ω consisting of triangles with the property that no vertex of any triangle lies in the interior of an edge of another triangle. Returning to Figure 10.4, for every linear function U on Ωp, U(x)= U1ϕ1(x)+ U1ϕ2(x)+ ... + U6ϕ6(x), (10.2.1) where Ui = U(Ni), i = 1, 2,..., 6, are numerical values (nodal values) of U at the nodes Ni. ϕi are the basis functions with ϕi(Ni)=1, ϕi(Nj)=0 for j = i and ϕ (x) is linear in x in every triangle/element. 6 i 10 CHAPTER 10. APPROXIMATION IN SEVERAL DIMENSIONS Note that with the Dirichlet boundary condition, as in the one-dimensional case, the test functions use ϕ (x) = 0for x ∂Ω . Hence, for a Dirichlet i ∈ p boundary value problem, to determine the approximate solution U, in Ωp, is reduced to find the nodal values U1,U2,...,U6, obtained from the corre- sponding discrete variational formulation. ϕ i = 1 Ni Figure 10.6: A linear basis function in a triangulation in 2D Example 10.1. Let Ω = (x,y) : 0 <x< 4, 0 <y< 3 and make a { } piecewise linear FEM discretization of the following boundary value problem: ∆u = f in Ω − (10.2.2) u = 0 on ∂Ω. Solution. Using Green’s formula, the variational formulation for the problem (10.2.2) reads as follows: Find the function u H1(Ω) such that ∈ 0 1 ( u v)dxdy = fvdxdy, v H0 (Ω).

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