Partition of Unity Methods for Heterogeneous Domain Decomposition

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1 Partition of Unity Methods for Heterogeneous Domain Decomposition Gabriele Ciaramella and Martin J. Gander 1 Heterogeneous problems and partition of unity decomposition We are interested in solving linear PDEs of the form L (u) = f in, u = g on, (1) where is a bounded domain in R d with d = 1,2, L is a linear (elliptic) differential operator, f and g are the data, and u is the solution to (1). The weak form of (1) with a Hilbert space (V,, ) of functions v : R is a(u,v) = l(v) v V 0, with u = g on, (2) where V 0 := {v V : v = 0 on }, a : V V R is the bilinear form corresponding to the operator L, and l : V R is the linear functional induced by f. We assume that (2) has a unique solution u {v V : v = g on }, and that u is heterogeneous, behaving very differently in different parts of. Typical examples are advection-diffusion problems, where there are advection dominated and diffusion dominated regions (subdomains), and the boundaries in between are not clearly defined, see [8, 10] and references therein. Apart from the χ-method [6, 1], there are no methods to determine such subdomain decompositions, and our goal is to present and study a new such method. We thus introduce (see [18, 11]) Definition 1 (Membership function). Let R d be a set. A membership function ϕ is a map ϕ : [0,1], and its support S is S := {x : ϕ(x) 0}. Given two membership functions ϕ 1,ϕ 2 : [0,1] that form a partition of unity on, ϕ 1 (x) + ϕ 2 (x) = 1 for all x, their supports provide then a domain decompo- G. Ciaramella Universität Konstanz, Germany, gabriele.ciaramella@uni-konstanz.de M. J. Gander Université de Genève, Genève, Switzerland martin.gander@unige.ch 1

2 2 Gabriele Ciaramella and Martin J. Gander sition = suppϕ 1 suppϕ 2. We introduce the approximation u dd := ϕ 1 u 1 +ϕ 1 u 2 u, where u 1 and u 2 represent two different possible behaviors of u, and we assume that u dd = g on. We proceed as follows to define the spaces that u dd, u 1 and u 2 are to be sought in: first, we introduce two approximate problems, L 1 (u 1 ) = f 1 in, B 1 (u 1, g) = g, and L 2 (u 2 ) = f 2 in, B 2 (u 2, g) = 0. (3) Here L j are approximation operators of L, f j are approximations of f, and B j are operators to define the boundary conditions of (3), see Section 3 for concrete examples. The function g represents a control and belongs to an appropriate Hilbert space W. Notice that g is different from the actual boundary data g: the latter is defined on, while we will define the former only on a subset of. We assume 1 that (3) (left) is uniquely solvable in V for any g W and (3) (right) has a unique solution u 2 V. To reformulate (3) (left), we introduce two operators A : V V and B g : W V, such that (3) (left) becomes Au 1 = B g g+ f 1. Notice that B g represents the boundary conditions of (3) (left) and takes into account also g. This problem is formally solved by u 1 = A 1 B g g + A 1 f 1, where A 1 is well defined if (3) (left) is well posed. Now, we define the spaces V 1 := {v V : v = A 1 (B g q + f 1 ), q W}, and V 2 := {u 2 }. Here V 1 represents the space of all possible solutions to the first problem in (3) generated by all the possible (control) functions in W, while V 2 is a singleton containing only the unique solution u 2 to (3) (right). Finally, we use the definition of a partition of unity method space (PUM-space [2, 13]) V PUM := ϕ 1 V 1 +ϕ 2 V 2 V, where ϕ 1,ϕ 2 are membership functions. V PUM, V 1 and V 2 are the spaces that the approximations u dd, u 1 and u 2 have to be sought in. In particular, for the approximation u dd the functions ϕ 1, ϕ 2 and g have to be computed. These are defined as solutions to optimal control problems, as described in Section 2. Here we need to remark that our approach could be computationally expensive. However, it is motivated by applications in astrophysics governed by hyperbolic equations like the Boltzmann equation. In many cases, like for supernova explosion, physical phenomena are modeled using two different (limiting) regimes. However, this would require an a-priori knowledge of the transition regime; see, e.g. [8, 3, 11] and references therein. This is exactly the role of the partition of unity functions obtained by our computational framework. In practice, one could use our computationally expensive approach to obtain the partition of unity functions for one representative case and then reuse them (as approximations) in a domain decomposition fashion to compute approximate solutions of other cases of interest. 1 This specific approximation is motivated by asymptotic expansion techniques providing in general two problems, one that is uniquely determined and a second one that is determined up to some constants for asymptotic matching [15].

3 Partition of Unity Methods for Heterogeneous Domain Decomposition 3 2 Optimal control approaches To compute ϕ 1, ϕ 2 and g, we embed the PUM formulation into an optimal control framework. We begin by inserting u dd into (2) and obtain the bounded linear functional r : V R defined by r(v) := a(ϕ 1 u 1 + ϕ 2 u 2,v) l(v), where v V. In the case that v = ϕ 1 w and v = ϕ 2 w with w V, we get the functionals r j (w) := r(ϕ j w) = a(ϕ 1 u 1 + ϕ 2 u 2,ϕ j w) l(ϕ j w) w V, for j = 1,2. Since w V r j (w), j = 1,2, are bounded linear functionals, they are elements in V, and by the Riesz representation theorem [7], there exist R 1 and R 2 in V such that R j,v = a(ϕ 1 u 1 + ϕ 2 u 2,ϕ j v) l(ϕ j v) v V 0, j = 1,2, (4) where we used V 0, since u dd is exact on and thus R 1 and R 2 must vanish there. Now, we define ϕ := ϕ 1 with ϕ 2 = 1 ϕ, and recall that r j V = R j V. Minimizing the norms of the residuals R j V leads to the optimal control problem min R 1,R 2,u 1,g,ϕ J(R 1,R 2,g,ϕ) := 1 2 R 1 2 V R 2 2 V + α 2 ϕ 2 V + β 2 g 2 W s.t. R 1,v = a(ϕu 1 + (1 ϕ)u 2,ϕv) l(ϕv) v V 0, R 2,v = a(ϕu 1 + (1 ϕ)u 2,(1 ϕ)v) l((1 ϕ)v) v V 0, Au 1 = B g g + f 1, g W, u 2 V 2, ϕ V, 0 ϕ 1 a.e. in, (5) where α,β > 0 are two regularization parameters used to tune the cost of ϕ and g, and f 1 is the same approximation to f introduced in (3). Solving (5) by an iterative procedure [5, 17] requires at each iteration to solve the two equations (4) for R 1 and R 2, and (3) for u 1. A less expensive optimal control problem is obtained by summing (4) for j = 1,2, and we obtain with R := R 1 + R 2 R,v = a(ϕu 1 + (1 ϕ)u 2,v) l(v) v V 0, (6) which is a Petrov-Galerkin type equation that we could have obtained directly applying a Petrov-Galerkin method to (2) using V PUM and V as trial and test spaces. Using (6), we get the less expensive optimal control problem min R,u 1,g,ϕ J(R,g,ϕ) := 1 2 R 2 V + α 2 ϕ 2 V + β 2 g 2 W s.t. R,v = a(ϕu 1 + (1 ϕ)u 2,v) l(v) v V 0, Au 1 = B g g + f 1, g W, u 2 V 2, ϕ V, 0 ϕ 1 a.e. in. (7)

4 4 Gabriele Ciaramella and Martin J. Gander A D M 2 1 B E C Fig. 1 Example of a boundary decomposition = Optimal control for elliptic boundary-layer problems As main test cases we consider elliptic problems of the form L (u) := µ u + a u + cu = f in, u = g on, (8) where is a bounded domain in R d, for d = 1,2, g C( ), f is sufficiently smooth, and the components of a are assumed to be strictly-positive. The assumption on a is restrictive, but it simplifies the presentation below and can be relaxed. The corresponding weak problem is to find a u { v H 1 () v = g on } such that a(u,v) := µ u v + a uv + cuvdx = f vdx =: l(v) v H0 1 (). We also assume that is such that the boundary can be decomposed into = 1 2, where the intersection 1 2 has a non-zero measure, as illustrated in Figure 1. To obtain u dd = ϕu 1 + (1 ϕ)u 2 u, we define Γ := \ 1 and introduce the operator L 1 := µ +c. Then, as in (3), for any choice of the control g H 1 0 (Γ ) the corresponding approximate problem for u 1 is µ u 1 v+cu 1 vdx = 0 v H0 1 (), { } u 1 w H 1 () w = g on 1, w = g + g on Γ, τ(w) C( ), where τ is the trace operator on. Notice that we have chosen f 1 = 0. As before, we introduce the operator A : H 1 () H 1 () defined as Au,v H 1,H 1 := µ u v + cuvdx for all v H1 0 (), and the operator B g : H 1 0 (Γ ) H 1 () such that v (B g g)(v) is a bounded linear functional in H 1 (). The operator B g represents the Dirichlet boundary conditions of (9). Au 1 = B g g is then equivalent to (9). The corresponding set V 1 is given by V 1 = {v H 1 () : Av = B g q for any q H 1 0 (Γ )}. Now, consider the operator L 2 := a + c and f 2 = f. The problem for u 2 is then L 2 (u 2 ) = a u 2 + cu 2 = f in, u 2 = g on 2, (10) (9)

5 Partition of Unity Methods for Heterogeneous Domain Decomposition 5 which we assume uniquely solvable in H 1 () C(). Notice that (10) is a pure advection problem and the subset 2 is given as the set of points where the characteristic curves enter the domain. This is the main assumption we make on 2 for the problem (10) to be well posed. The set V 2 contains only the solution to (10), i.e. V 2 = {u 2 }. The approximation u dd u is then obtained as u dd = ϕ 1 u 1 + ϕ 2 u 2, where the membership functions ϕ 1 = ϕ,ϕ 2 = 1 ϕ H 1 () form a partition of unity, and ϕ is such that {1} if x \ 2, ϕ(x) [0,1] if x 1 2, (11) {0} if x Γ, with τ(ϕ) C( ). Notice that this definition of ϕ makes u dd exact on the boundary, τ(u dd ) = τ(ϕ 1 u 1 + ϕ 2 u 2 ) = g. In what follows, we study the control problem (7) ((5) would have a similar structure) to optimize ϕ and g for computing the approximation u dd to the solution to (8). In particular, we first show well-posedness, and then we derive the firstorder optimality system. We consider directly a 2-dimensional problem (d = 2), since the analysis of the 1-dimensional version is simpler and relies on the same arguments. To define our optimal control problem, as in (7), we consider the cost functional J(R,g,ϕ) := 2 1 R 2 H 1 () + α 2 ϕ 2 H 1 () + β 2 g 2 H 1. Now, we introduce (Γ ) the control-to-state maps g u 1 (g) and (g,ϕ) R(u 1 (g),ϕ), where u 1 (g) and R(u 1 (g),ϕ) solve (9) and R,v H 1 () = µ u dd v + a u dd v + cu dd v f vdx v H 1 0 (). (12) Notice that the left-hand side of (12), that is R,v H 1 () = R v + Rvdx, is of a similar form to the left-hand side in (9). These maps are well defined according to the lemmas below and allow us to define the reduced cost functional J(g,ϕ) := J(R(u 1 (g),ϕ),g,ϕ) and the optimal control problem min g,ϕ J(g,ϕ) s.t. 0 ϕ(x) 1 in and (11) holds. (13) For well-posedness of this optimization problem, we need four Lemmas: Lemma 1. Let z H 1 ( ) with R 2 convex and Lipschitz. Then the problem µ u 1 v + cu 1 vdx = 0 v H0 1 () (14) with u 1 = z on is uniquely solvable by u 1 H 1 () C(), and there exists a positive constant c such that u 1 H 1 () c z H 1 ( ). Proof. To show that there exists a unique u 1 C(), we define w as the harmonic extension of z in. Recalling the embedding H 1 C for one-dimensional domains, we have that z C( ). Therefore, since is a Lipschitz domain,

6 6 Gabriele Ciaramella and Martin J. Gander w C 2 () C(); see, e.g., [12]. Now, consider the problem µ v + cv = cw in with v = 0 on. Since is convex, Theorems in [14] ensure that this problem is uniquely solved by v H 2 () H0 1(). Since R2, the Sobolev embedding H 2 () C() [7] ensures that v C(). Noticing that the function w + v solves (14), u 1 C() and is unique by the linearity of (14). Next, we show that u 1 H 1 () with u 1 H 1 () c z H 1 ( ). Consider the trace operator τ : H 1 () H 1/2 ( ). Since is a Lipschitz domain, by [16, Theorem 3.37, page 102] this operator has a bounded right-inverse τ 1 : H 1/2 ( ) H 1 (). Now, we define w := τ 1 z and note that w H 1 (). So, if we decompose u 1 as u 1 = w + ṽ, then ṽ must solve in a weak sense the problem µ ṽ + cṽ = ( µ w + cw) in with ṽ = 0 on. By the Lax-Milgram theorem we have that the unique solution is ṽ H0 1() and there exists a constant C such that ṽ H 1 () C w H 1 (). Therefore, u 1 H 1 () and using the decomposition u 1 = w+ṽ we get u 1 H 1 () (1 +C) w H 1 () = (1 +C) τ 1 z H 1 () K z H 1 ( ), for some positive constant K, where we used the boundedness of τ 1 [16]. Lemma 2. Let ϕ H 1 () such that 0 ϕ(x) 1 a.e. in. Then for any function v H 1 () C() it holds that ϕv H 1 (). Proof. An application of Theorem 1 in [9, page 247] shows that (vϕ) = v ϕ + ϕ v. Then a simple estimate of the norm (vϕ) L 2 () allows us to obtain the result. Lemma 3. Let {z n } n be a sequence that converges weakly in H 1 ( ) to a weak limit ẑ H 1 ( ), i.e. z n ẑ in H 1 ( ). Define the sequence {u 1,n } n by u 1,n := u 1 (z n ), where u 1 (z n ) solves (14) with u 1 = z n on. Then there exists a subsequence that converges weakly in H 1 () and strongly in L 2 () to the limit û 1 = u 1 (ẑ) H 1 (), i.e., û 1 in H 1 () and û 1 in L 2 (). Proof. Since the sequence {z n } n converges weakly in H 1 ( ), it is bounded in the norm H 1 ( ). By Lemma 1, we have that u 1,n H 1 () c z n H 1 ( ) K, for some positive constant K, and the sequence u 1,n is bounded in H 1 (). Since H 1 () is reflexive, there exists a weakly convergent subsequence û 1 in H 1 (). Now, from (14), we have that for any v H0 1() µ v + c vdx µ û 1 v + cû 1 vdx. Moreover, the weak convergence z n j ẑ and the continuity of the trace operator τ : H 1 () H 1/2 ( ) [16, Theorem 3.37] implies that z n j = τ( ) τ(û 1 ) = ẑ, weakly in H 1/2 ( ). Therefore, û 1 = u 1 (ẑ). We conclude by recalling the Sobolev compact embedding H 1 () L 2 (); see, e.g., [7]. Lemma 4. Let {u 1,n } n be the sequence defined in Lemma 3 such that û 1 (weakly) in H 1 (). Consider a sequence {ϕ n } n in H 1 () such that 0 ϕ n (x) 1

7 Partition of Unity Methods for Heterogeneous Domain Decomposition 7 and ϕ n ϕ (weakly) in H 1 () with 0 ϕ(x) 1. Then there exist two subsequences {ϕ n j } j and { } j such that ϕ n j ϕ and û 1 (strongly) in L 2 (), and for any v H 1 0 () (ϕ n j ) v + ϕ n j vdx ( ϕû 1 ) v + ϕû 1 vdx. Proof. The existence of the subsequences {ϕ n j } j and { } j such that ϕ n j ϕ and û 1 (strongly) in L 2 () follows from the fact that ϕ n ϕ (weakly in H 1 ()), Lemma 3, and the Sobolev (compact) embedding H 1 () L 2 () [7]. Now, recalling Lemma 1 and according to the proof of Lemma 2 it holds that ( ϕ n j ) = ϕ n j + ϕ n j. Therefore, to treat the products of sequences û 1.n j ϕ n j, ϕ n j, and ϕ n j, we use [7, Theorem ] to obtain for any v H0 1 () that (ϕ n j ) v+ϕ n j vdx= ϕ n j v+ϕ n j v+ϕ n j vdx û 1 ϕ v+ ϕ û 1 v+û 1 ϕvdx= ( ϕû 1 ) v+ ϕû 1 v+û 1 ϕvdx. We are now ready to prove that (13) is well posed. Theorem 1. Let α,β > 0, then there exists a solution to problem (13). Proof. Consider a minimizing sequence {(R n,ϕ n,u 1,n,g n )} n, where g n is extended by zero on. Since J is coercive in ϕ and g we have the bounds ϕ n H 1 () c and g n H 1 ( ) c, for two positive constants c,c ; see, e.g., [17]. The reflexivity of H 1 () and H 1 ( ) ensures the existence of weakly convergent subsequences: ϕ n j ϕ in H 1 () and g n j ĝ in H 1 ( ). By the Sobolev (compact) embedding H 1 () L 2 () [7], the sequence {ϕ n j } j converges strongly in L 2 () to ϕ. Since the set {v L 2 () : 0 v(x) 1 a.e. in } is (weakly) closed in L 2 () [17], we have 0 ϕ(x) 1. Consider now the sequence {u 1,n } n and the corresponding subsequence = u 1 (g n j ). By Lemma 3, we have that û 1 = u 1 (ĝ) weakly in H 1 () and û 1 = u 1 (ĝ) strongly in L 2 (). Consider the sequence {R n } n. Since R n satisfies R n,v H 1 () = µ u dd,n v + a u dd,n v + cu dd,n v f vdx v H0 1 (), where u dd,n = ϕ n u 1,n + (1 ϕ n )u 2, from the Lax-Milgram theorem we have that R n H 1 () K( u 1,n H 1 (), ϕ n H 1 () ), where the constant K depends on u 1,n H 1 () and ϕ n H 1 (), which are bounded. Therefore, R n is bounded as well, and by Lemma 4, one can show that R n j R = R(û 1, ϕ) weakly in H 1 (). Now, the weaklower semi-continuity of J implies the claim [17, 4]. To obtain the first-order optimality system, we rely on the Lagrange multiplier approach and work in the reduced space of solutions of constraint and adjoint equa-

8 8 Gabriele Ciaramella and Martin J. Gander tions; see, e.g., [5, 17]. We first recall the control-to-state maps g u 1 (g) and (g,ϕ) R(u 1 (g),ϕ) and the reduced cost functional J(g,ϕ). Then we notice that its derivatives, for δg H0 1(Γ ) and δϕ H1 0 (), are D g J(g,ϕ)(δg)= βg+r g,δg H 1 (Γ ), D ϕ J(w,ϕ)(δϕ)= αϕ +R ϕ,δϕ H 1 (). (15) Here R g is the solution of the problem R g,δg H 1 0 (Γ ) = B gδg,λ H 1,H1, (16) where, H 1,H 1 : H 1 () H0 1() R denotes the duality pairing, and R ϕ is the Riesz representative of the linear functional δϕ µ [ (u 1 u 2 )δϕ ] Rdx + a [ (u 1 u 2 )δϕ ] R + c(u 1 u 2 )δϕ Rdx. In (16), λ H0 1 () is a Lagrange multiplier that solves the adjoint equation λ v + cλ vdx = µ (vϕ) R + a (vϕ)r + cvϕ Rdx, (17) for all v H0 1 (). Therefore, the first-order optimality system is given by (9), (12), (17) and (16) together with the conditions [4, 17] D g J(g,ϕ)(δg) = 0, for all δg H0 1 (Γ ), and for any arbitrary θ > 0 ϕ = P Vad (ϕ θ ( αϕ + R ϕ ) ), where P Vad is the projection onto V ad := {v H 1 () : 0 v(x) 1 a.e. in }. 4 Numerical experiments We present now numerical experiments for the one-dimensional elliptic problem µ xx u x u = 1 in (0,1), with u(0) = 0, u(1) = 0, (18) for given µ = 0.01, computing u dd = ϕ 1 u 1 + ϕ 2 u 2, with µ xx u 1 = 0 in (0,1), u 1 (0) = 0, u 1 (1) = g, and x u 2 = 1 in [0,1), u 2 (1) = 0. We solve both the PUM and Petrov-Galerkin optimality systems discretized by linear finite-elements with a projected-lbfgs method with stopping tolerance

9 Partition of Unity Methods for Heterogeneous Domain Decomposition Petrov PUM Petrov PUM Exact Petrov PUM ϕ and 1 ϕ u and udd J(φ n,g n ) x x n (iteration) Fig. 2 Comparison of the Petrov and PUM approaches: Left: partition of unity functions ϕ and 1 ϕ. Middle: exact solution and approximations. Right: Decay of the cost functional. on the (relative) residual norm. The regularization parameters are α = β = In Figure 2 (left) we see that the ϕ and 1 ϕ obtained by the two approaches are very similar, and catch well the boundary layer on the left. The small bumps in the right part (close to x = 1) are due numerical effects and we checked that they disappear for smaller tolerances. In Figure 2 (middle) the exact solution is compared with the two approximations u dd, and we see good agreement. In Figure 2 (right), we show the decay of the cost functional with respect to the number of iterations, and we see that the Petrov-Galerkin approach converges a bit faster. References 1. Y. Achdou and O. Pironneau. The χ-method for the Navier-Stokes equations. IMA journal of numerical analysis, 13(4): , I. Babuska and J. M. Melenk. The partition of unity method. International Journal of Numerical Methods in Engineering, 40: , H. Berninger, E. Frnod, M. Gander, M. Liebendrfer, and J. Michaud. Derivation of the isotropic diffusion source approximation (idsa) for supernova neutrino transport by asymptotic expansions. SIAM Journal on Mathematical Analysis, 45(6): , A. Borzì, G. Ciaramella, and M. Sprengel. Formulation and Numerical Solution of Quantum Control Problems. Society for Industrial and Applied Mathematics, Philadelphia, PA, A. Borzì and V. Schulz. Computational Optimization of Systems Governed by Partial Differential Equations. SIAM, Philadelphia, F. Brezzi, C. Canuto, and A. Russo. A self-adaptive formulation for the Euler-Navier Stokes coupling. Comput. Methods Appl. Mech. Eng., 73: , P. G. Ciarlet. Linear and Nonlinear Functional Analysis with Applications. SIAM, Philadelphia, P. Degond and S. Jin. A smooth transition model between kinetic and diffusion equations. SIAM J. Numerical Analysis, 42(6): , L. C. Evans. Partial differential equations. Graduate studies in mathematics. American Mathematical Society, Providence (R.I.), M. J. Gander, L. Halpern, and V. Martin. A new algorithm based on factorization for heterogeneous domain decomposition. Numerical Algorithms, 73(1): , M.J. Gander and J. Michaud. Fuzzy domain decomposition: a new perspective on heterogeneous DD methods. In Domain Decomposition Methods in Science and Engineering XXI, pages Springer, D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, New York, 1983.

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