A NONLOCAL VECTOR CALCULUS WITH APPLICATION TO NONLOCAL BOUNDARY-VALUE PROBLEMS. Max Gunzburger

Transcription

1 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

2 LOCAL AND NONLOCAL DIFFUSION The elliptic equation (D(x) w(x) ) = b(x) models (steady state) diffusion in R d The nonlocal equation 2 ( u(x ) u(x) ) µ(x,x ) dx = b(x) in R d models (steady state) nonlocal diffusion

3 to see this, consider the nonlocal diffusion equation ( u t = u(x ) u(x) ) µ(x,x ) dx suppose that and µ(x,x )dx dx = 1 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 = u(x )µ(x,x ) dx u(x) µ(x,x ) dx is the rate at which u enters x less the rate at which u departs x

4 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

5 Motivations and goals ( u(x ) u(x) ) µ(x,x ) dx vs. (D(x) w(x) ) 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

6 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 u tt and not u t 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

7 A NONLOCAL GAUSS S THEOREM For any mapping r(x,x ): R d R d R, it is easily seen that r(x,x ) dx dx = r(x,x) dx dx R d If p(x,x) denotes an anti-symmetric mapping, i.e., p(x,x) = p(x,x ) for all x, x R d then p(x,x ) dx dx = 0 R d Let denote an open bounded subset of R d. Obviously, if Γ R d \ and p(x,x) is anti-symmetric, by setting = Γ, we have p(x,x ) dx dx = p(x,x )dx dx Γ

8 Let α(x,x ) : Γ 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 given by ( D(f)(x) = f(x,x ) f(x,x) ) α(x,x ) dx for x 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 N is a nonlocal normal flux operator

9 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 = Γ N(f) dx

10 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 , 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 0 q(x) = (x x)ϕ(x,x x) dx R d

11 Then, a formal application of Lemma I of Noll implies ( q(x) = f(x,x ) f(x,x) ) α(x,x )dx R d for x using the definition of the operator D( ), we then have q = Df for x Lemma II of Noll implies q(x) nda = Γ ( f(x,x ) f(x,x) ) α(x,x ) dx dx where = boundary of da = surface element on n = outward pointing unit normal vector along using the definition of the operator N( ), we then have q(x) nda = N(f) dx Γ

12 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 0 q(x) = (x x)ϕ(x,x x) dx R d with z = x x then, the nolocal Gauss s theorem for f D(f) dx = implies the classical Gauss s theorem for q q dx = Γ N(f) dx q ndx

13 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 tensorvalued

14 In the recent paper G. Gilboa and S. Osher, Nonlocal operators with applications to image processing; Multiscale Model. Simul , the nonlocal divergence theorem (Df)(x) dx = 0 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

15 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 , F. Andreu, J. Mazon, J. Rossi, and J. Toledo, A nonlocal p-laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions; SIAM J. Math. Anal , several others

16 Notational simplification In the sequel, we frequently let u = u(x) u = u(x ) v = v(x) v = v(x ) f = f(x,x ) α = α(x,x ) f = f(x,x) α = α(x,x) and similarly for functions still to be introduced

17 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)α dx dx = Γ vn(s)dx Let G denote the linear operator mapping functions v : Γ R into functions defined over Γ Γ given by G(v) = (v v)α for x,x Γ

18 Then, we obtain vd(s)dx + sg(v) dx dx = Γ vn(s) dx this is the nonlocal analog of the classical result v q dx + v q dx = vq nda the particular choice v = constant results in D(f) dx = N(f) dx i.e., the nonlocal Gauss s theorem Γ

19 NONLINEAR, NONLOCAL BOUNDARY-VALUE PROBLEMS Let U( Γ) and V ( Γ) denote Banach spaces Γ = Γ e + Γ n with Γ e Γ n = V 0 ( Γ) = {v V ( Γ) : v = 0 for x Γ e } Define the data functions b : R h e : Γ e R h n : Γ n R

20 For u U( 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( Γ) such that u = h e for x Γ e and A(u)G(v) dx dx = vbdx + Γ n vh n dx v V 0 ( Γ)

21 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 u = h e N ( A(u) ) = h n for x for x Γ e for x Γ n 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

22 if Γ e =, then - the space of test functions V 0 () is replaced by V ( Γ)/R - the compatibility condition bdx + must hold Γ h n dx = 0 - this is entirely analogous to the classical case

23 NONLOCAL LINEAR OPERATORS AND GREEN S IDENTITIES We now specialize to the case of U() = V () 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 this is the nonlocal analog of the classical Green s first identity v udx + v u dx = vn u da Γ

24 One can immediately obtain a nonlocal Green s second identity vd ( βg(u) ) dx ud ( βg(v) ) dx = Γ ( vn ( βg(u) ) un ( βg(v) )) dx this is the nonlocal analog of the classical Green s second identity v udx u v dx = vn u da un v da A nonlocal Green s third identity will come later

25 Constitutive relation The relation is a constitutive relation s = A(u) = βg(u) = (u u)αβ 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)

26 LINEAR, NONLOCAL BOUNDARY-VALUE PROBLEMS For s = A(u) = βg(u) = (u u)βα, the variational problem reduces to seek u V ( Γ) such that and u = h e for x Γ e βg(v)g(u) dx dx = vbdx + Γ n vh n dx v V 0 ( Γ) The corresponding boundary-value problem reduces to the linear problem D ( βg(u) ) = b for x u = h e N(βG(u)) = h n for x Γ e for x Γ n

27 We have the explicit relations βg(v)g(u) dx dx = D ( βg(u) ) = 2 N ( βg(u) ) = 2 (v v)(u u)α 2 β dx dx (u u)α 2 β dx for x (u u)α 2 β dx for x Γ n

28 WELL POSEDENESS OF NONLOCAL LINEAR BOUNDARY-VALUE PROBLEMS We now assume that the constitutive function β is positive K symmetric, positive definite is sufficient for β > 0 For all u,v V ( Γ), define the symmetric bilinear form B(u,v) = βg(v)g(u) dx dx = note that B(u,u) 0 (v v)(u u)βα 2 dx dx

29 Let and ((u, v)) = B(u,v) and u = ( B(u,u) ) 1/2 V ( Γ) = {u : u < } We have, for both V 0 ( Γ) and V ( Γ) \ R that ((, )) defines an inner product and defines and norm

30 We restrict attention to the linear nonlocal variational formulation with homogeneous Dirichlet boundary condition: or given b, seek u V 0 ( Γ) such that βg(v)g(u) dx dx = vbdx dx v V 0 ( Γ) given b, seek u V 0 ( Γ) such that B(u, v) = F(v) v V 0 ( Γ) where F(v) = vbdx The Lax-Milgram theorem can be applied to obtain the well-posedness of the variational problem

31 Related papers that give more extensive analyses, including making connections between the function space V () and fractional Sobolev spaces with index 0 s 1 Q. Du and K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory; submitted. and a paper in preparation (Du, G., Lehoucq, Zhou)

32 Decomposition of the solution space Let the space S( Γ) consist of functions u V ( Γ) that satisfy D ( βg(u) ) = 2 N ( βg(u) ) = 2 (u u)βα 2 dx = 0 x (u u)βα 2 dx = 0 x Γ. Then, we have that, for all u S( Γ) and v V 0 ( Γ), ((u,v)) = Thus, we conclude that G(v)G(u)β dx dx = 0 V ( Γ) = V 0 ( Γ) S( Γ)

33 thus, any function in V ( Γ) can be written as a sum of two functions that are orthogonal with respect to the inner product ((, )) - the first a function that vanishes on Γ e - the second a function belonging to S( Γ) this is entirely analogous to the decomposition of the Sobolev space H 1 () into functions belonging to H 1 0() and harmonic functions

34 NONLOCAL GREEN S FUNCTIONS AND A NONLOCAL GREEN S THIRD IDENTITY For each y R d, let g(x;y) denote a solution of D ( βg(g(x;y)) ) = δ( x y ) x R d g(x; y) is then a nonlocal fundamental solution or nonlocal free-space Green s function for the operator D(βG( )) We assume that α and β are radial functions of x and x, e.g., α(x,x ) = α(x x) g(x;y) = g(x y) α 2 β dx = 1, where the scaling 1 is without loss of generality R d

35 Let µ denote the Fourier transform of µ = βα 2 then, it can be shown that a fundamental solution is given by g(x;y) = (2π) d e ik (x y) 1 2 R d (2π) d/2 µ 1 dk note that the special choice µ(x x) = δ(x x) + d2 x) dx 2δ(x leads to the same fundamental solution as that for the Laplace operator: 1 x y for d = 1 2 (2π) d 2 R d e ik (x y) k 2 dk = where ω d denotes the volume of the unit ball in R d 1 ln x y for d = 2 2π 1 x y 2 d for d 3 2ω d 2 d

36 Nonlocal Green s third identity For any y Γ, let G(x;y) : Γ R denote any function satisfying D(βG(G(x;y))) = δ( x y ) x Then, from the nonlocal Green s second identity we obtain the nonlocal Green s third identity u(y) = G(x;y)D ( βg(u(x)) ) dx ( G(x;y)N ( βg(u(x)) ) u(x)n ( βg(g(x;y)) )) dx y Γ this the analog of the classical Green s third identity u(y) = G(x;y) u(x)dx ( G(x;y) u(x) n u(x) G(x;y) n ) da

37 Suppose that the constitutive function β = 1 so that D ( G(u) ) = 2 (u u)α 2 dx = 0 x then, the solution u(x) represents the nonlocal harmonic function ( u(y) = u(x)n ( G(G(x;y)) ) G(x;y)N ( G(u(x)) )) dx, Γ i.e., harmonic functions are determined by their boundary values on Γ Nonlocal versions of the Poisson integral formula and Gauss s law of the arithmetic mean can also be derived

38 Nonlocal Green s functions Let g(x;y) denote a nonlocal fundamental solution For each y, define the nonlocal Green s function G(x;y) : R as G(x;y) = g(x;y) g(x;y) where g( ; ) is a solution of D ( βg( g) ) = 0 for x N ( βg( g) ) = N ( βg(g) ) for x Γ n g(x;y) = g(x;y) for x Γ e then, G( ; ) satisfies the homogeneous boundary conditions G(x;y) = 0 for x Γ e and N ( βg(g) ) = 0 for x Γ n

39 Then, from the nonlocal Green s third identity, we have that the solution of the nonlocal boundary-value problem is given by u(y) = G(x; y)b(x) dx + h e (x)n ( βg(g(x;y)) ) dx G(x;y)h n (x) dx Γ e Γ n y Γ n the classical analog is u(y) = G(x;y)b(x) dx + Γ e h e G n da Γ n h n G da y

40 LOCAL SMOOTH LIMITS We now see what happens are a result of the assumptions: 1. solutions of the nonlocal boundary-value -problems are smooth 2. the nonlocal operators are asymptotically local we emphasize that these assumptions are made only to make the connection to classical problems for partial differential equations and are not required for the well posedness of the nonlocal boundary-value -problems in addition, the nonlocal boundary-value -problems admit solutions that are not solutions, even in the usual sense of weak solutions, of the partial differential equations thus, one can view solutions of the nonlocal boundary-value -problems as further generalizations of solutions of the partial differential equations, generalized in two ways: they are nonlocal and they lack the smoothness needed for them to be standard weak solutions

41 Smoothness assumption: assume that the solution u(x) of the nonlocal boundary-value -problem satisfies u(x ) = u(x) + u(x) (x x) + o(ε 2 ) if x x ε where ε 2 o(ε 2 ) 0 as ε 0 actually, we only need this to hold weakly Locality assumption: assume that the constitutive function β(x,x ) is integrable and is a positive, symmetric function such that β(x,x ) = 0 whenever x x > ε

42 Next, set Γ = Γ(ε) = x supp(β) \ e = Γ e and n = Γ n and assume that e and n ε Γ Γ

43 Then as ε 0 Γ = O(ε) Γ e e Γ n n We also set α(x,x ) = 1 x x and assume a scaling such that, for some positive constants β and β, β < β(x,x )dx < β uniformly for x, S ε (x) where S ε (x) := {x R d x x < ε}

44 Then, using the assumptions we have made, we are led to where B(u,v) = (ε) v ( ) D ε + o(ε 0 ) u dx D ε (x) = so that S ε (x) ((ε)) (x x) (x x)k(x x) (x x) x x 2 dx lim B(u,v) = v ε 0 (D u ) dx where D(x) = lim ε 0 D ε (x)

45 Then, using the local and nonlocal Green s identities, it can be shown that, as ε 0, the nonlocal variational problem seek u V ( Γ) such that u = h e for x Γ e and βg(v)g(u) dx dx = vbdx + Γ n vh n dx v V 0 ( Γ) reduces to the classical local variational problem v (D u ) = vbdx + v h n dx n u = h e in on e h e and h n denote traces of the nonlocal data h e and h n, respectively

46 Also, as ε 0, the corresponding nonlocal boundary-value problem D ( βg(u) ) = b for x u = h e N(βG(u)) = h n reduces to classical boundary-value problem (D u ) = b for x Γ e for x Γ n in u = h e on e ( D u ) n = hn on n

47 NONLOCAL LINEAR CONVECTION-DIFFUSION-REACTION PROBLEMS Let σ(x,x ) and ω(x,x ) denote anti-symmetric and symmetric functions, respectively Consider the nonlocal variational principle seek u V ( Γ) such that u = h e for x Γ and βg(v)g(u) dx dx + + v ω(u + u) dx dx = v σg(u) dx dx vbdx v V 0 ( Γ) here, we only examine nonlocal Dirichlet problems; clearly, their Neumann counterparts can be defined in a similar manner

48 The corresponding nonlocal Dirichlet boundary-value problem is given by D(βG(u))+σG(u) + ω(u + u) = b for x u = h e for x Γ General problems may be defined by choosing the constitutive relations β = (x x) K (x x) σ = a (x x) ω = r where a(x,x ) denotes a symmetric vector-valued function r(x,x ) denotes a symmetric function

49 Then, as before, we now have that the general nonlocal Dirichlet boundaryvalue problem reduces to the general linear convection-diffusion-reaction problem (D u ) +w u + cu = b along with a Dirichlet boundary condition, where w and c are related to a and r in a manner similar to the relation between D and K

50 CURRENT AND FUTURE WORK With Du, Lehoucq, and Zhou fusing the nonlocal calculus to the connections made by Du and Zhou to Sobolev spaces extension to the vector-valued case extension to material interface problems application to the peridynamic model of materials extension to nonlinear problems With M. Parks and P. Seleson extension to material interface problems