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A Nonlocal Vector Calculus with Application to Nonlocal Boundary-Value Problems

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A Nonlocal Vector Calculus with Application to Nonlocal Boundary-Value Problems A NONLOCAL VECTOR CALCULUS WITH APPLICATION TO NONLOCAL BOUNDARY-VALUE PROBLEMS Max Gunzburger Department of Scientific Computing Florida State University Collaboration with: Richard Lehoucq Sandia National Laboratories Last updated in 2009 LOCAL AND NONLOCAL DIFFUSION • The elliptic equation −∇ · D(x) · ∇w(x) = b(x) in Ω ⊂ Rd models (steady state) diffusion • The “nonlocal” equation 2 u(x′) − u(x) µ(x, x′) dx′ = b(x) in Ω ⊂ Rd Z Ω models (steady state) nonlocal diffusion – to see this, consider the nonlocal diffusion equation u = u(x′) − u(x) µ(x, x′) dx′ t Z Ω – suppose that µ(x, x′) dx′ dx =1 ZΩ ZΩ and µ(x, x′)= µ(x′, x) ≥ 0 so that µ(x, x′) can be interpreted as the joint probability density of moving between x and x′ – then u(x′) − u(x) µ(x, x′) dx′ Z Ω = u(x′)µ(x, x′) dx′ − u(x) µ(x, x′) dx′ ZΩ ZΩ is the rate at which u “enters” x less the rate at which u “departs” x – the nonlocal diffusion equation can be derived from a “nonlocal” random walk – any skew-symmetry of µ represents drift – the nonlocal diffusion equation is an example of a differential Chapman-Kolmogorov equation Motivations and goals u(x′) − u(x) µ(x, x′) dx′ vs. ∇· D(x) · ∇w(x) Z Ω • The nonlocal operator always contains length scales it is a multiscale operator – the local operator contains length scales only when the diffusion tensor D does • The nonlocal operator has lower regularity requirements u may be discontinuous • The nonlocal diffusion equation does not necessarily smooth discontinuous initial conditions • Extension to the vector case is (formally) straightforward – our ultimate goal – the analysis and numerical analysis of Silling’s nonlocal, continuum peridynamic model for materials – then follows – note that the peridynamic model is a mechanical model so that we have utt and not ut • Here we consider the steady-state case • Nonlocal models are able to model phenomena at smaller length and time scales at which the underlying assumption of locality associated with the classical diffusion equation or the classical balance of linear momentum lead to less accurate modeling A NONLOCAL GAUSS’S THEOREM • For any mapping r(x, x′): Rd × Rd → R, it is easily seen that r(x, x′) dx′dx = r(x′, x) dx′dx ∀ Ω ⊆ Rd ZΩ ZΩ ZΩ ZΩ b b b b b • If p(x′, x) denotes an anti-symmetric mapping, i.e., p(x′, x)= −p(x, x′) for all x, x′ ∈ Rd then p(x, x′) dx′dx =0 ∀ Ω ⊆ Rd ZΩ ZΩ b b b • Let Ω denote an open bounded subset of Rd. Obviously, if Γ ⊆ Rd \ Ω and p(x′, x) is anti-symmetric, by setting Ω=Ω ∪ Γ, we have p(x, x′) dx′dx =b− p(x, x′) dx′dx ZΩ ZΩ∪Γ ZΓ ZΩ∪Γ • Let α(x, x′):Ω ∪ Γ × Ω ∪ Γ → R denote a symmetric function ⇒ α(x′, x)= α(x, x′) • Let D denote the linear operator mapping functions f into functions defined over Ω given by D(f)(x)= f(x, x′) − f(x′, x) α(x, x′) dx′ for x ∈ Ω Z Ω∪Γ – D is a nonlocal “divergence” operator • Similarly, let N denote the linear operator mapping functions f into functions defined over Γ given by N (f)(x)= − f(x, x′) − f(x′, x) α(x, x′) dx′ for x ∈ Γ Z Ω∪Γ – N is a nonlocal “normal flux” operator • Note that the operators D and N differ only in their domains and in their signs • Then, setting p(x, x′)= f(x, x′) − f(x′, x) α(x, x′) results in the nonlocal Gauss’s theorem D(f) dx = N (f) dx ZΩ ZΓ Relation to the classical Gauss’s theorem • We apply two remarkable Lemmas due to Walter Noll W. Noll, Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistischen Mechanik; Indiana Univ. Math. J. 4 1955, 627–646. Originally published in J. Rational Mech. Anal. See also W. Noll, Derivation of the fundamental equations of continuum thermodynamics from statistical mechanics; Translation with corrections by R. Lehoucq and O. von Lilienfeld, to appear in J. Elasticity, 2009 • Given a mapping f(x, x′), let p(x, x′)= f(x, x′) − f(x′, x) α(x, x′) and, with z = x′ − x 1 ϕ(x, z)= p x + λz, x − (1 − λ)z dλ Z 0 q(x)= − (x′ − x)ϕ(x, x′ − x) dx′ ZRd • Then, a formal application of Lemma I of Noll implies ∇· q(x)= f(x, x′) − f(x′, x) α(x, x′) dx′ for x ∈ Ω Z d R – using the definition of the operator D(·), we then have ∇· q = Df for x ∈ Ω • Lemma II of Noll implies q(x) · n dA = − f(x, x′) − f(x′, x) α(x, x′) dx′dx Z Z Z ∂Ω Γ Ω∪Γ where ∂Ω= boundary of Ω dA = surface element on ∂Ω n = outward pointing unit normal vector along ∂Ω – using the definition of the operator N (·), we then have q(x) · n dA = N (f) dx Z∂Ω ZΓ • Summarizing, given a function f(·, ·), if q(·) is determined from f(·, ·) according to p(x, x′)= f(x, x′) − f(x′, x) α(x, x′) 1 ϕ(x, z)= p x + λz, x − (1 − λ)z dλ with z = x′ − x Z 0 q(x)= − (x′ − x)ϕ(x, x′ − x) dx′ ZRd then, the nolocal Gauss’s theorem for f D(f) dx = N (f) dx ZΩ ZΓ implies the classical Gauss’s theorem for q ∇· q dx = q · n dx ZΩ Z∂Ω • Evidently, Gausss theorem can be given a meaning without the notions of a divergence operator, a unit normal vector, or a surface • Lehoucq and Silling have shown that the vector-valued function q satisfies a constrained energy principle • Noll’s Lemma also holds for vector-valued functions in which case q is tensor- valued • In the recent paper G. Gilboa and S. Osher, Nonlocal operators with applications to image processing; Multiscale Model. Simul. 7 2008, 1005-1028. the nonlocal divergence theorem (Df)(x) dx =0 ZΩ is given – this is a the special case of our nonlocal Gauss’s theorem for Γ= ∅ – Gilboa and Osher also provide the germs of a nonlocal calculus but do not provide Green’s identities or consider nonlocal BVPs • Important related papers by Rossi and co-workers that deal with nonlocal difussion problems F. Andreu, J. Mazon, J. Rossi, and J. Toledo, A nonlocal p-Laplacian evolution equation with Neumann boundary conditions; J. Math. Pures Appl. 40 2008, 201–227. F. Andreu, J. Mazon, J. Rossi, and J. Toledo, A nonlocal p-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions; SIAM J. Math. Anal. 40 2009, 1815–1851. + several others Notational simplification • In the sequel, we frequently let u = u(x) u′ = u(x′) v = v(x) v′ = v(x′) f = f(x, x′) f ′ = f(x′, x) α = α(x, x′) α′ = α(x′, x) and similarly for functions still to be introduced An application of the nonlocal Gauss’s theorem • For functions v(x) and s(x, x′), set f = sv so that f − f ′ = sv − s′v′ =(s − s′)v + s′(v − v′) • Using the definitions of the operators D and N , one obtains vD(s) dx + s(v′ − v)αdx′dx = vN (s)dx ZΩ ZΩ∪Γ ZΩ∪Γ ZΓ • Let G denote the linear operator mapping functions v : Ω ∪ Γ → R into functions defined over Ω ∪ Γ × Ω ∪ Γ given by G(v)=(v′ − v)α for x, x′ ∈ Ω ∪ Γ • Then, we obtain vD(s) dx + sG(v) dx′dx = vN (s) dx ZΩ ZΩ∪Γ ZΩ∪Γ ZΓ – this is the nonlocal analog of the classical result v∇· q dx + ∇v · q dx = vq · n dA ZΩ ZΩ Z∂Ω – the particular choice v = constant results in D(f) dx = N (f) dx ZΩ ZΓ i.e., the nonlocal Gauss’s theorem NONLINEAR, NONLOCAL BOUNDARY-VALUE PROBLEMS • Let U(Ω ∪ Γ) and V (Ω ∪ Γ) denote Banach spaces Γ = Γe + Γn with Γe ∩ Γn = ∅ V0(Ω ∪ Γ) = {v ∈ V (Ω ∪ Γ) : v =0 for x ∈ Γe} • Define the data functions b :Ω → R he : Γe → R hn : Γn → R • For u ∈ U(Ω ∪ Γ), let s(x, x′)= A(u) for a possibly nonlinear operator A which also may depend explicitly on x and x′ • Consider the variational problem seek u ∈ U(Ω ∪ Γ) such that u = he for x ∈ Γe and ′ A(u)G(v) dx dx = vbdx + vhn dx ∀ v ∈ V0(Ω ∪ Γ) ZΩ∪Γ ZΩ∪Γ ZΩ ZΓn • The nonlocal Green’s formula shows that the variational formulation can be viewed as a weak formulation of the nonlocal “boundary-value” problem −D A(u) = b for x ∈ Ω u = he for x ∈ Γe N A(u) = hn for x ∈ Γn – the second equation is a “Dirichlet boundary” conditions that is essential for the variational principle – the third equation is a “Neumann boundary” condition that is natural for the variational principle – if Γe = ∅, then - the space of test functions V0(Ω) is replaced by V (Ω ∪ Γ)/R - the compatibility condition bdx + hn dx =0 ZΩ ZΓ must hold - this is entirely analogous to the classical case NONLOCAL LINEAR OPERATORS AND GREEN’S IDENTITIES • We now specialize to the case of U(Ω∪Γ) = V (Ω∪Γ) and to linear operators • For a mapping β(x, x′), let s = A(u)= βG(u) =(u′ − u)αβ – this is a constitutive relation • From the nonlocal Gauss’s theorem, we then obtain the nonlocal Green’s first identity vD βG(u) dx + βG(v)G(u) dx′dx = vN βG(u) dx Z Z Z Z Ω Ω∪Γ Ω∪Γ Γ – this is the nonlocal analog of the classical Green’s first identity v∆udx + ∇v · ∇udx = vn · ∇u dA ZΩ ZΩ Z∂Ω • One can immediately obtain a nonlocal Green’s second identity vD βG(u) dx − uD βG(v) dx = vN βG(u) − uN βG(v) dx Z Z Z Ω Ω Γ – this is the nonlocal analog of the classical Green’s second identity v∆udx − u∆vdx = vn · ∇u dA − un · ∇v dA ZΩ ZΩ Z∂Ω Z∂Ω • A nonlocal Green’s third identity will come later Constitutive relation • The relation s = A(u)= βG(u)=(u′ − u)αβ is a “constitutive” relation • We need to say something about β – we want something analogous to the diffusion tensor D appearing in second-order elliptic partial differential equations • Let K(x, x′) denote a tensor – then, a general nonlocal constitutive function β is given by β =(x′ − x) · K · (x′ − x) LINEAR, NONLOCAL BOUNDARY-VALUE PROBLEMS • For s = A(u)= βG(u)=(u′ − u)βα, the variational problem reduces to seek u ∈ V (Ω ∪ Γ) such that u = he for x ∈ Γe and ′ βG(v)G(u) dx dx = vbdx + vhn dx ∀ v ∈ V0(Ω ∪ Γ) ZΩ∪Γ ZΩ∪Γ ZΩ ZΓn • The corresponding “boundary-value” problem reduces to the linear problem −D βG(u) = b for x ∈ Ω u = he for x ∈ Γe N (βG(u)) = hn for x ∈ Γn • We have the explicit relations βG(v)G(u) dx′dx = (v′ − v)(u′ − u)α2βdx′dx ZΩ∪Γ ZΩ∪Γ ZΩ∪Γ ZΩ∪Γ D βG(u) =2 (u′
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