Computing eigenvalues and eigenfunctions of Schrödinger equations using a model reduction approach

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1 Computing eigenvalues and eigenfunctions of Scrödinger equations using a model reduction approac Suangping Li 1, Ziwen Zang 2 1 Program in Applied and Computational Matematics, Princeton University, New Jersey, USA Department of Matematics, University of ong Kong, Pokfulam Road, ong Kong SAR. Abstract. We present a model reduction approac to construct problem dependent basis functions and compute eigenvalues and eigenfunctions of stationary Scrödinger equations. Te basis functions are defined on coarse meses and obtained troug solving an optimization problem. We sall sow tat te basis functions span a lowdimensional generalized finite element space tat accurately preserves te lowermost eigenvalues and eigenfunctions of te stationary Scrödinger equations. Terefore, our metod avoids te application of eigenvalue solver on fine-scale discretization and offers considerable savings in solving eigenvalues and eigenfunctions of Scrödinger equations. Te construction te basis functions are independent of eac oter; tus our metod is perfectly parallel. We also provide error estimates for te eigenvalues obtained by our new metod. Numerical results are presented to demonstrate te accuracy and efficiency of te proposed metod, especially Scrödinger equations wit double well potentials are tested. AMS subject classifications: 35J10, 65F15, 65N25, 65N30 Key words: Scrödinger equation; eigenvalue problems; model reduction; two-level tecniques; problem dependent basis functions; computational cemistry. 1 Introduction In tis paper, we construct a set of problem dependent basis functions to compute eigenvalues and eigenfunctions of Scrödinger equations. To be more specific, we consider te eigenvalue problem of te stationary Scrödinger equation wit a potential V(x) of te following form u(x) := u(x)+v(x)u(x) = λu(x), x Ω R d, (1.1) u(x) =0, x Ω R d, (1.2) Corresponding autor. addresses: sl31@mat.princeton.edu (S. Li), zangzw@ku.k (Z. Zang) ttp:// Global Science Preprint

2 2 were Ω is a bounded domain in R d and V(x) : R d R is a real-valued function. λ and u(x) are te corresponding eigenvalues and eigenfunctions of te amiltonian operator = +V(x). We sould empasize tat te spectrum of te amiltonian operator can ave negative values and pysically te negative part of te spectrum corresponding to bound states and tey ave many important applications in computational cemistry [6, 17, 18, 29]. Te eigenvalue problem of (1.1) in variational form reads: find an eigenvalue λ and its associated eigenfunction u(x) W := 0 1 (Ω) suc tat ( ) a(u,v) := u(x) v(x)+v(x)u(x)v(x) dx = λ u(x)v(x)dx = λ(u,v), (1.3) Ω Ω for all v W. By using te finite element metod (FEM), we obtained te discretized problem of te eigenvalue problem (1.3): find λ and associated eigenfunctions u (x) V W suc tat a(u,v ) = λ (u,v ), for all v V, (1.4) were V is a conforming finite element space spanned by N nodal basis functions on some regular finite element mes T wit mes size. After te FEM discretization, one could apply eigenvalue algoritms, including QR-algoritm, Lanczos algoritm, and Arnoldi iteration, directly to te N -dimensional finite element matrices to obtain te eigen-pairs {λ,u }, see [10] and references terein. We remark tat it is extremely expensive to compute eigenvalues and eigenfunctions of (1.4) wen N becomes big. For example, finding all eigenvalues and eigenvectors of te matrix corresponding to te FEM discretization of (1.4) using QR-algoritm costs 6N 3+O(N2 ) flops. In practice, owever, we are mainly interested in te first few lowermost eigenvalues and eigenfunctions as tey ave important meanings in computational cemistry [19]. In addition, wen we use te FEM to approximate eigenvalues of (1.4), te number of reliable numerical eigenvalues takes up only a tiny portion of te total degrees of freedom N in te resulting discrete system. See [2, 30 32, 35] for te discussion of second-order elliptic eigenvalue problems. Tis motivates us to avoid te application of eigenvalue algoritms for te fine-scale FEM discretization (1.4) and build a low-dimensional generalized finite element space so tat we can accurately and efficiently compute te lowermost eigenvalues and eigenfunctions. Specifically, we introduce a coarse discretization of te pysical space Ω into mes T wit mes size. On te coarse mes T, we build a set of basis functions {Ψ i (x)} N i=1 tat generate a low-dimensional generalized finite element space V c. Te dimension of V c is N and it is muc smaller tan N. In te low-dimensional space V c, we compute te discretized form of te eigenvalue problem (1.3): find λ and associated eigenfunctions u (x) V c W suc tat a(u,v ) = λ (u,v ) for all v V c, (1.5)

3 3 Te basis functions {Ψ i } N i=1 ave already captured te information of te Scrödinger equation, wic enables us to accurately compute te first few important eigen-pairs of (1.1), {λ,u }. Te construction of te basis functions {Ψ i } N i=1 involves solving of N optimization problems [4]. Tese optimization problems are independent of eac oter and tus can be computed in parallel. Recall tat te size of te matrix obtained by te discretization of (1.5) is N. Tis significantly reduces te computational cost in computing te eigenvalues and eigenfunctions of Scrödinger equation (1.1). We sould point out tat te idea of using two-level tecnique or multi-level tecnique for designing algoritms for eigenvalue problem and oter problems is not new. In [33], a two-grid discretization sceme was proposed to solve eigenvalue problems, including bot partial differential equations and integral equations. In [13], ackbusc proposed a multi-grid metod to compute eigenvalues and eigenfunctions of te elliptic problem obtained by te finite element discretization. In [21], Peterseim used te numerical upscaling tecniques to compute eigenvalues for a class of linear second-order selfadjoint elliptic partial differential operators. Using similar metodology to construct lowdimensional generalized finite element spaces is pioneered by te generalized finite element metod (GFEM) [1] and te multiscale finite element metod (MsFEM) [11, 14, 15], and is pervasive in te recent developments in te numerical metods for multiscale problems and elliptic PDEs wit random coefficients, see [8, 34] and references terein. We would like to point out some similarities and differences between our approac and oter existing metods. Our construction of basis functions is inspired by te recently development in building localized basis function for multiscale elliptic PDEs and Scrödinger equations, see [16,20,21,23] and reference terein. Previous researcers ave utilized te Clément-type quasi-interpolation approac or optimization approac to construct localized basis functions tat give optimal approximation property of te elliptic operator. In te Scrödinger equation (1.1), owever, te potential V(x) is a real-valued function. Terefore, te amiltonian operator +V(x) is not necessarily positive definite, wic is quite common in Scrödinger equation models, see [12, 28]. We sall construct basis functions tat can be used to compress or upscale te amiltonian operator in Scrödinger equation (1.1) so tat we can compute te corresponding eigen-pairs in te reduced space. In addition, we sall estimate te error of te eigenvalues λ λ obtained by te FEM and our new metod. We comment tat similar ideas of computing eigenvalue problems using adaptive basis functions are considered in [24,25], toug te main point of tese papers are different and tey mainly focus on numerical investigation. Teir goals are to obtain compressed modes tat are sparse and spatially localized so tey can be used to span te low-energy eigenspace of differential operators. Te rest of te paper is organized as follows. In Section 2, we give a brief introduction of te eigenvalue problems of te Scrödinger equation and its finite element metod discretization. In section 3, we present te derivation of basis functions based on te two-scale decompositions and te approximation of eigenvalues and eigenfunctions in te reduced space. Issues regarding te practical implementation of our metod will also

4 4 be discussed. Error estimate of te eigenvalues and computational complexity analysis will be discussed in Section 4. In Section 5, we present numerical results to demonstrate te accuracy and efficiency of our metod. Concluding remarks are made in Section 6. To simplify te notation, we will write a b for two positive quantities a and b, if a Cb wit some constant C > 0 tat depends only on te size of te domain Ω, parameters in Scrödinger equation, and parameters tat measures te quality of te underlying finite element mes. We empasize tat C does not depend on te mes size and. 2 Model problem and its finite element discretization We compute te eigenpairs {λ,u(x)} of te following Scrödinger equation on te bounded domain Ω, u(x)+v(x)u(x) = λu(x), x Ω R d, (2.1) u(x) =0, x Ω R d. (2.2) Te eigenvalue problem of (2.1) in variational form reads: find an eigenvalue λ and its associated eigenfunction u(x) W := 0 1 (Ω) suc tat ( ) a(u,v) := u v+vuv dx = λ uvdx = λ(u,v), for all v W, (2.3) Ω In te finite element metod, we first partition te pysical domain Ω into a set of regular fine elements wit mes size. For example, we divide Ω into a set of non-overlapping triangles T = {τ e } N e e=1 suc tat no vertex of one triangle lies in te interior of te edge of anoter triangle, were N e is te number of finite elements. Let N f denote te set of interior vertices of T. Let N denote te number of te interior vertices, wic is also equal to te dimension of te finite element space. For every vertex x i N f, let ϕ i (x) denote te corresponding nodal basis function, i.e., ϕ i (x j ) = δ ij, x j N f. In tis paper, we assume tat all te nodal basis functions ϕ i (x) are linear functions and continuous across te boundaries of te elements, so we obtain te first-order conforming finite element space corresponding to T, V = {ϕ(x) τ e T, ϕ(x) τe is a polynomial of total degree 1} 1 0(Ω). (2.4) Ten, we apply te Galerkin metod to solve (2.3). Specifically, we find λ and associated eigenfunctions u (x) = N i=1 u iϕ i (x) V suc tat Ω a(u,v ) = λ (u,v ) for all v V. (2.5) Finally, we solve a generalized eigenvalue problem obtained from te discretization of (2.5) to obtain λ and u (x). Te amiltonian operator +V(x) is self-adjoint so te eigenvalues are real. Tey can be sorted in ascending order, λ (1) λ (2) λ (3) λ (4) (2.6)

5 5 If te potential V(x) is bounded from below, we ave te estimate for te lowermost eigenvalue λ (1). Lemma 2.1. We assume te potential V(x) V min, x Ω, ten te lowermost eigenvalue λ (1) is bounded from below. Proof. We introduce te Rayleig quotient [30] witin te finite dimension subspace V, wic is defined by R(v ) = a(v,v ) (v,v ), for all v V, (2.7) Ten, λ (1) = minr(v ), v V. Obviously, λ (1) is bounded from below if V(x) V min, x Ω. In addition, its corresponding eigenfunctions u (1) (x) is te critical point of R(v ) over te finite element space V. Remark 2.1. Te Rayleig quotient provides an alternative way to compute eigenvalues and eigenfunctions of te Scrödinger equation (1.1) and its discretized form (2.5) [30]. Te l-t eigenvalue is λ (l) =min v E l 1,v V R(v ), were E l 1 is te eigen-space spanned by eigenfunctions u 1,,ul 1 associated wit eigenvalues λ (1),,λ(l 1). In te finite element metod framework, te dimension of te discretized problem is proportional to te number of interior vertices in te fine mes T. Terefore, te finite element metod becomes expensive for 2D and 3D Scrödinger equations. It is desirable to develop model reduction metods tat can efficiently and accurately solve te eigenvalue problem of Scrödinger equation wit relatively small computational cost. 3 Construction of basis functions and teir corresponding lowdimensional space In tis section, we sall apply te two-level tecniques and decompose te finite element space V into coarse and fine parts. Te coarse part is a low-dimensional generalized finite element space tat enables us to efficiently compute te lowermost eigenvalues and eigenfunctions of Scrödinger equations. To acieve tis goal, we need to build a set of basis functions {Ψ i (x)} tat capture te information of te amiltonian operator. To construct te basis functions {Ψ i (x)}, we first partition te pysical domain Ω into a set of regular coarse elements wit mes size. Again, we divide Ω into a set of non-overlapping triangles T = {T e } suc tat no vertex of one triangle lies in te interior of te edge of anoter triangle. To facilitate te implementation, te fine mes T and coarse mes T are nested. Let N c denote te set of interior vertices of coarse mes T and N be te number of interior vertices. For every vertex x i N c, let Φ i (x) denote te corresponding FEM nodal basis functions, i.e., Φ i (x j ) = δ ij, x j N c. We also

6 6 assume tat all te nodal basis functions Φ i (x) are continuous across te boundaries of te elements, so we obtain te first-order conforming finite element space corresponding to te coarse mes T, V = {Φ(x) T e T, Φ(x) Te is a polynomial of total degree 1} 1 0(Ω). (3.1) Te dimension of te coarse finite element space V (3.1) is N, wic is far less tan tat of fine-scale FEM space V. owever, one cannot use te coarse finite element basis functions Φ i, i = 1,...,N to directly compute te eigenvalues and eigenfunctions of Scrödinger equation because tey do not capture te fine-scale information of te amiltonian operator in (1.1). Terefore, we need to construct some problem-dependent basis functions tat incorporate te fine-scale information into te coarse finite element space. In tis paper, we construct suc basis functions {Ψ i (x)} N i=1 troug an optimization approac [4]. More specifically, we compute te following constrained optimization problem to obtain Ψ i (x), ( Ψ i (x) = argmin Ψ(x) 2 +V(x) Ψ(x) 2) dx (3.2) s.t. Ω Ψ 1 0 (Ω) Ω Ψ(x)Φ j (x) = δ i,j, 1 j N, (3.3) were Φ j (x) are te nodal basis functions on te coarse FEM space V. Te objective function (3.2) contains bot te kinetic energy and te potential energy of te Scrödinger equation system. It is important to note tat te boundary condition of te Scrödinger equation as already been incorporated in te above optimization problem troug te definition of te solution space 0 1 (Ω). In general, te optimization problem cannot be solved analytically as it is an optimization problem in an infinite dimensional space. We ave to solve te optimization problem (3.2)-(3.3) using numerical metods. In tis paper, we apply te finite element metod to discretize te basis functions Ψ i (x). Specifically, we represent Ψ i (x) = N k=1 b(i) k ϕ k(x), were ϕ k (x) are te finite element basis functions defined on te fine mes T and b (i) k s are te coefficients. In tis discrete level, te optimization problem (3.2)-(3.3) is reduced to a constrained quadratic optimization problem, wic can be efficiently solved using Lagrange multiplier metods. Since te basis functions are independent of eac oter, tey can be constructed independently and te optimization problem (3.2)-(3.3) can be done perfectly in parallel. Let V c denote te conforming generalized finite element space spanned by Ψ i (x), V c = {Ψ i (x) i =1,...,N } 1 0(Ω). (3.4) Note tat te dimension of V c is equal to te coarse finite element space V. owever, te basis functions Ψ i (x) contain fine-scale information of te amiltonian operator in (1.1), wic enable us to compute te eigenvalue problem (1.3) on te coarse mes T.

7 7 Now we use te Galerkin metod to solve te eigenvalue problem (1.3) in te generalized finite element space V c : find λ (j) and its associated eigenfunctions u(j) c (x) V c, j =1,...,N, suc tat a(u (j) c,v) = λ (j) (u(j) c,v), for all v V c. (3.5) In general, te stiffness matrices and mass matrices corresponding to te discretization of (3.5) are not sparse. owever, te dimension of te coarse generalized finite element space V c is N N so te lack of sparsity is not an issue. Te discrete eigenvalues are ordered in ascending order, λ (1) λ(2) λ(3) λ(4) λ(n ). (3.6) Let u (j) c, j=1,2,...,n be normalized to one in L 2 (Ω), i.e., (u (j) c,u (j) c ) L2 (Ω) =1. Te discrete eigenfunctions satisfy te ortogonal constraints 4 Errors analysis a(u (j) c,u (k) c ) = (u (j) c,u (k) c ) =0, j = k. (3.7) In tis section, we present te error estimate of te approximate eigenvalues λ λ obtained by te finite element metod (FEM) and our metod. Te computational complexity analysis of te FEM and our metod can be obtained easily. 4.1 Ortogonal decomposition of te solution space in L 2 (Ω) sense We first introduce some notations tat will be used in te error estimate. Let V 0 := V(x) L (Ω) < + and W := 0 1 (Ω). We define a norm to be u(x) := V 0 u 2 L 2 (Ω) + u 2, for any u W. (4.1) L 2 (Ω) Recall tat te bilinear form a(u,v) used in te variational form corresponding to te eigenvalue problem of (1.1) is defined by a(u,v) := ( u, v)+(vu,v), for any u,v W, (4.2) were V = V(x) is te potential function and (, ) stands for te standard inner product on Ω. Under mild conditions, te second part (Vu,v) in (4.2) can be viewed as a perturbation. Our metod requires te following assumption on te resolution of te coarse mes T. Assumption 4.1. We assume tat te potential V(x) is bounded, i.e., V 0 := V(x) L (Ω)< +, and te mes size of T satisfies V 0 1.

8 8 Under tis assumption, many typical bounded potentials from Scrodinger equation (1.1) can be treated as a perturbation to te kinetic operator. Tus, tey can be computed using our metod. We also point out tat tis assumption restrains our ability to andle Scrodinger equation (1.1) wit unbounded potential, suc as Coulomb potential. We sall consider tis issue in our future work. Before we proceed te error estimate, we first study te ortogonal decomposition of te solution space W. Write V f as te subset of W defined by V f = { v(x) W Ωv(x)Φ i (x) =0, i =1,...,N }. (4.3) From te definition of V f, one can find tat V f contains functions wit fine-scale information tat cannot be captured by te coarse-scale finite element basis functions Φ i (x) defined in (3.1). Tis property is closely related to te Clément-type interpolation operator [5, 9, 27] I v(x) := x i N c α i (v)φ i (x), (4.4) were N c contains all te interior nodes of coarse mes, Φ i (x) are te nodal basis functions corresponding to x i, and te quasi-interpolation coefficient α i (v) is defined by α i (v) = Ω Φ i(x)v(x)dx Ω Φ, x i N c (4.5) i(x)dx In order to define interpolators for roug functions and to preserve piecewise polynomial boundary conditions, te approximated functions are averaged appropriately using (4.5) in order to generate nodal values for te interpolation operator. Compare te dimension of te fine-scale space, we obtain tat space V f and te kernel space of te Clément-type interpolation operator I are equal. In addition, te solution space W as te ortogonal decomposition W=kernel(I ) V =V f V in L 2 (Ω) sense. Namely, u W, we ave te decomposition u=u +u f, were u V, u f V f, and tey satisfy (u,u f ) L2 (Ω) =0. Te Clément-type interpolation operators possess te local approximation and stability properties tat are crucial in our error estimate. Tere exists a generic constant C I suc tat for all v W and for all coarse element T e T, we ave v(x) I v(x) L2 (T e ) C I v(x) L2 (S e ), (4.6) were S e := {K T K T e = } [7]. We also assume tat tere exists a constant C ol > 0 suc tat te number of elements covered by S e is uniformly bounded by C ol. Bot C I and C ol may depend on te sape regularity of te finite element mes but not on te coarse mes size.

9 9 4.2 Quasi-ortogonal decomposition of te solution space Wit tese preparations, we are ready to study te structure of te generalized finite element space V c (spanned by Ψ i (x)) and te corresponding ortogonal decomposition of te solution space W:=0 1 (Ω). First, we explore te connections between te standard 1 norm L2 (Ω), norm, and te bilinear form a(u,v). We get te following lemmas. Lemma 4.1. For any u,v in W, we ave a(u,v) u v. Proof. Using te Caucy-Scwarz inequality, we can obtain tat a(u,v) 2 ( u L2 (Ω) v L2 (Ω)+V 0 u L2 (Ω) v L2 (Ω)) ( u 2 L 2 (Ω) +V 0 u 2 L 2 (Ω) )( v 2 L 2 (Ω) +V 0 v 2 L 2 (Ω) ) u 2 v 2. Lemma 4.2. L2 (Ω) and are equivalent in V f, given V 0 1. Proof. For any w in V f, w L2 (Ω) w is obvious. For te oter direction, w 2 = (w I w,w I w)v 0 +( w, w) V 0 w I w 2 L 2 (Ω) +( w, w) = V 0 ( T T w I w 2 L 2 (Ω) )+( w, w) 2 V 0 C 2 I C ol w 2 L 2 (Ω) + w 2 L 2 (Ω) = ( 2 V 0 C 2 I C ol +1) w 2 L 2 (Ω) w 2 L 2 (Ω). Lemma 4.3. Te bilinear form a(, ) is V f -elliptic, given V 0 < 1 C I Col. Proof. For any w in V f, we ave a(w,w) V 0 w 2 L 2 (Ω) + w 2 L 2 (Ω) = V 0 w I w 2 L 2 (Ω) + w 2 L 2 (Ω) (1 2 V 0 C 2 I C ol ) w 2 L 2 (Ω) 1 2 V 0 C 2 I C ol 1+V 0 2 C 2 I C ol w 2 Ω, were te last inequality directly follows from te previous lemma 4.3.

10 10 In te Section 3, our new basis functions {Ψ i (x)} are constructed troug an optimization problem. For any i, let Ψ i (x) be te unique minimizer of te following problem Ψ i (x) =argmin s.t. Ω Ψ(x) W a(ψ,ψ) (4.7) Ψ(x)Φ j (x)dx = δ i,j. (4.8) Ten, te generalized finite element space V c is spanned by {Ψ i (x)} N i=1. We sall sow tat for eac i, te optimization problem (4.7)(4.8) gives rise to a minimizer Ψ i (x) given certain conditions. In addition, we sall sow tat te above optimization problem yields an ortogonal decomposition of te solution space W into te generalized finite element space V c and its quasi-ortogonal complement V f. Quasi-ortogonal decomposition means u W, we ave te decomposition u=u c +u f, were u c V c, u f V f, and tey satisfy te condition a(u c,u f )=0. We notice tat relation ( f,g)=( 1 f,g)=a( 1 f,g), f,g W and te bilinear form a(, ) is symmetric. Terefore, te constrains Ω ΨΦ j = δ i,j in te optimization problem is equivalent to a(ψ, 1 Φ j ) = δ i,j. For eac i, we define W i = {Ψ(x) W Ψ(x)Φ j (x)dx = δ i,j, j =1,...,N } Ω to be te feasible set of te optimization problem (4.7). Ten, we ave tat Lemma 4.4. Under te resolution condition V 0 < 1 C I Col, te optimization problem (4.7)(4.8) is a strictly convex optimization problem over W i, for eac i. Proof. Let us coose any two different Ψ a,ψ b W i. We write for η [0,1], f (η) = a(ψ a +η(ψ b Ψ a ),Ψ a +η(ψ b Ψ a )) = a(ψ a,ψ a )+2ηa(Ψ a,ψ b Ψ a )+η 2 a(ψ b Ψ a,ψ b Ψ a ). Use te fact tat Ψ b Ψ a V f defined in (4.3) and a(, ) is V f -elliptic by te Lemma 4.3, we get tat f (η) >0. Tus, we finis te proof. Lemma 4.5. For any 1 i N, te optimization problem (4.7)(4.8) as a minimizer Ψ i (x) if and only if { 1 Φ i } N i=1 are linearly independent. Proof. We define an N -by-n matrix Θ wit Θ i,j := ( 1 Φ i,φ j ). It is clear tat Θ is invertible if and only if { 1 Φ i } N i=1 are linearly independent. Let us define Ψ i(x) = N Θ 1 i,k 1 Φ k (x), were Θ 1 i,k is te (i,k)-t entry of Θ 1. It is easy to find tat k=1 N ( Θ 1 k=1 i,k 1 Φ k,φ j ) = N k=1 Θ 1 i,k Θ k,j = δ i,j,

11 11 wic means tat N Θ 1 i,k 1 Φ k satisfies te constrains (4.8). Tus linear independency k=1 of { 1 Φ i } N i=1 will imply existence of te minimizer. As for te oter direction, assume tat tere exists a Ψ i suc tat a(ψ i, 1 Φ j ) = δ i,j, for all j = 1,2,...,N. Suppose we ave N j=1 α j 1 Φ j = 0. Ten for eac i, we will ave tat 0 = N j=1 Tus { 1 Φ j } N j=1 are linearly independent. α j a(ψ i, 1 Φ j ) = N α j δ i,j = α i. j=1 Lemma 4.6. Te optimization problem (4.7)(4.8) yields an ortogonal decomposition of te solution space W into te generalized finite element space V c and its quasi-ortogonal complement V f. Namely, a(u c,u f ) =0, for any u c V c and u f V f. Proof. Let Ψ i be a minimizer. Ten for any w V f, we consider te objective function a(ψ i +c w,ψ i +c w), c R. As Ψ i satisfies te constrains, i.e. Ω Ψ iφ j = δ i,j, we also ave Ω (Ψ i+c w)φ j = δ i,j, since w is ortogonal to every Φ i according to (4.3). We define m(c) := a(ψ i +c w,ψ i +c w) = c 2 a(w,w)+2c a(ψ i,w)+a(ψ i,ψ i ). Since a(, ) is V f -elliptic by te Lemma 4.3, we ave tat a(w,w)>0 for nontrivial w. Recall tat Ψ i is a minimizer, we obtain tat m (c) c=0 =2a(Ψ i,w)=0, and tis sould olds for every w V f. From te definition of V c in (3.4), we get te conclusion tat a(u c,u f )=0, for any u c V c and u f V f. To see tat W = V c +V f, we firstly note tat V c is an N -dimensional subspace of W and tat V c V f = 0 by definition. For any i in {1,2,,N }, write Ψ i as v f,i +v,i, were v f,i V f and v,i V. Ten, we can sow tat {v,i } are linearly independent, because oterwise, tere exists non-trivial c i s suc tat c i v,i = 0, wic implies tat i c i Ψ i V f, a contradiction. Now, as {v,i } are linearly independent, any element in V i can be written as linear combination of Ψ i s and an element in V f. Tus, furter, we can ave W = V c +V f. Remark 4.1. From Lemma 4.5 and Lemma 4.6, we can actually sow tat wen a(, ) is a positive definite bilinear form, te minimizer as a simple form Ψ i = N See [16] for more details. k=1 Θ 1 i,k 1 Φ k (x). We also comment tat V c contains some fine-scale information of V f wic is very important in our computation of te eigenvalue problem. We sall sow tis in our error estimates and numerical experiments. Te quasi-ortogonal decomposition wit respect to a(, ) does not exactly preserve te L 2 -ortogonality. owever, we find tat te error can be controlled. Teorem 4.1. For any v c V c and v f V f, we ave (v c,v f ) L2 (Ω) 2 v c v f. (4.9)

12 12 Proof. For any v c V c and v f V f, we ave (v c,v f ) L2 (Ω) = (v c I v c,v f I v f ) L2 (Ω) 2 v c L2 (Ω) v f L2 (Ω) 2 v c v f were we ave used te fact (I v c,v f ) L2 (Ω) = 0, I v f = 0, and te stable estimate of te Clément-type interpolation (4.6). Remark 4.2. In previous works [16, 20, 23], te autors utilized te Clément-type interpolation or optimization approac to upscale elliptic operators. Te corresponding optimization problem is strictly convex over te solution space W. In our case, te amiltonian operator +V(x) may not be positive definite. ence, te corresponding optimization problem is not strictly convex over te solution space W. Based on our numerical experiments and analysis, we found tat under te assumption 4.1, we can prove tat our optimization problem is strictly convex over W i, wic enables us to obtain te quasi-ortogonal decomposition of W and construct te basis functions Ψ i for model reduction. 4.3 Exponential decay of te basis function Ψ i We sall sow tat te basis function Ψ i decay exponentially fast away from its associated vertex x i N c, namely te basis functions ave exponential decay property. In practice, wen we solve te optimization problem (4.7)(4.8) to construct te basis function Ψ i, we coose a localized domain S i Ω associated wit x i and impose te condition tat supp{ψ i } S i. Terefore, te exponential decay property significantly reduces our computational cost in constructing basis functions. We first define a series of nodal patces Ω l associated wit x i N c by Ω 0 := supp{φ i } = {K T x i K}, (4.10) Ω l := {K T K Ω l 1 = }, l =1,2,3,4,. (4.11) Ten, we state te main teorem as follows and put te detailed proof in te A. Teorem 4.2. Under te resolution condition V 0 < 1 2C I Col, tere exist some constants C>0 and 0< β <1 independent of, suc tat for any i =1,2,...,N. Ψ i Ω\Ωl Cβ l Ψ i Ω (4.12) Te exponential decay of te basis functions Ψ i allows us to localize te computational domain of te basis functions and to reduce te computational cost. In practice,

13 13 we modify te constrained optimization problem (3.2)(3.3) as follows, Ψ i (x) = argmin s.t. Ω Ψ 1 0 (Ω) Ω ( Ψ(x) 2 +V(x) Ψ(x) 2) dx (4.13) Ψ(x)Φ j (x) = δ i,j, 1 j N, (4.14) Ψ(x) =0, x Ω\Ω l. (4.15) were Ω l is te support set of te basis function Ψ i (x) and l depends on te decay speed of Ψ i (x). In numerical experiments, we find tat a small integer l log(l/) will generate accurate results, were L is te diameter of domain Ω. Moreover, te optimization problem (4.13)-(4.15) can be done in parallel. 4.4 Error estimate for te eigenvalues In tis subsection, we sall provide te error estimate of te eigenvalues λ λ obtained by te FEM and our new metod. Before we proceed, we add an assumption tat describes te well-posedness condition of te bilinear form a(, ). Assumption 4.2. We assume tat te bilinear form a(, ) satisfies inf sup u W\{0} v W\{0} a(u,v) u v C >0, were te positive constant C may depend on V 0 and te domain Ω. Let E l denote te eigen-space spanned by te first l eigenfunctions obtained by te finite element metod. E l := span{u (1),,u(l) }, were u(i) s are normalized to be one in L 2 (Ω) norm. Recall tat we ave λ (1) λ (2) λ (3) λ (l). Let λ =max 1 i l{ λ (i) }. Ten, we can estimate λ λ working in te eigen-space E l. Lemma 4.7. Assume te assumption (4.2) is satisfied. For u E l wit u L2 (Ω) = 1 and let u=u c +u f be te quasi-ortogonal decomposition, were u c V c El and u f V f El. Ten, we ave te following tree estimates: u c lλ, (4.16) u f (lλ )2 2, (4.17) (u c,u f ) (lλ )3 4. (4.18) Proof. Let u= l c j u (j), were c j s are te projection coefficients of u on te eigenfunctions j=1 u (j) and c j 1. According to te assumption (4.2), tere exists u 2 W\{0}, suc tat

14 14 a(u,u 2 ) u u 2. Ten we ave Tus, we ave We also ave u u 2 a(u,u 2 ) = a( = l j=1 l j=1 u f 2 a(u f,u f ) = a(u,u f ) = c j u (j),u 2) = l j=1 c j a(u (j),u 2) c j λ (j) (u(j),u 2) lλ u(j) u 2 lλ u 2. u c u lλ. l j=1 c j a(u (j),u f ) = l j=1 lλ 2 u (j) u f (lλ )2 2 u f. c j λ (j) (u(j),u f ) Te last inequality directly follows from (4.9) of Teorem 4.1 and te above two inequalities. Finally, we estimate te error for te eigenvalues λ λ obtained by te FEM and our new metod. Teorem 4.3. Wen te coarse mes is cosen small enoug suc tat <2 1 4 (lλ ) 3 4. Ten, we can get te following estimate λ(l) λ(l) λ (l) (lλ )2 4, l =1,2,... Proof. Define σ (l) := max u E l :(u,u)=1 (u f,u f ) L2 (Ω)+2(u c,u f ) L2 (Ω). Ten, we ave te following estimate for σ (l), (u f,u f )+2(u c,u f ) = (u,u f )+(u c,u f ) = (u I u,u f I u f )+(u c,u f ) 2 u L2 (Ω) u f L2 (Ω)+(lλ )3 4 2 u Ω u f Ω +(lλ )3 4 (lλ )3 4, (4.19) were we ave used te fact tat u f (lλ )2 2. Terefore, we obtain tat σ (l) (lλ )3 4. If is cosen small enoug so tat σ (l) 1/2, i.e., (lλ ) 3 4. Ten, Lemma 6.1 in [30] implies λ (l) (1 σ(l) ) 1 λ (l) (1+2σ(l) )λ(l)

15 (x) 15 After some simple calculations, one can easily obtain te final result based on te estimate for σ (l) in (4.19). 5 Numerical Experiments In tis section, we conduct numerical experiments to illustrate our analytical results. More specifically, we will consider several different models of Scrödinger equations and test te performances of our metod. Examples include Scrödinger equations of free electrons and tose wit double-well potentials. We are able to demonstrate tat te relative error of eigenvalues converges of order at least O( 4 ). Aside from te large convergence rate, using our problem dependent basis functions {Ψ i }, we are able to acieve at a numerical metod of better computational complexity tan finite element metod. Moreover, using our basis functions, one can capture te first few eigenvalue and eigenfunctions of stationary Scrödinger Equations accurately. Te potentials taken in te examples are frequently used in cemistry models, wic sows tat our metod is a very efficient model reduction metod. 5.1 amiltonian of a free electron In tis example, we consider te amiltonian of a free electron in a bounded domain Ω wit Diriclet boundary condition. In our numerical experiments, = xx and te bounded domain Ω is taken to be [0,1] for one dimensional problems. = and te bounded domain Ω is [0,1] 2 for two dimensional problems x (a) 1D basis functions (b) 2D basis function Figure 1: Profiles of te basis functions Ψ in 1D and 2D. In 1D case, we uniformly partition our interval into N =128 patces, and for eac patc, we furter uniformly partition it into N r =8 parts for numerical computation. We use te

16 V(x) 16 Lagrange multiplier metod to solve te optimization problem. Ten, wit te computed problem dependent basis functions, we discretize te amiltonian operator onto te N dimensional space spanned by Ψ 1,,Ψ N and approximate te smallest N eigenvalues of. In 2D case, we set te coarse mes = 1 16 and partition our unit square into 256 squares (512 triangle elements). For eac element, we furter uniformly partition it into fine triangle element wit mes = Te computation metod is similar. In Figure 1, we plot te profiles of te basis functions obtained from our metod. One can see tat te basis functions decay exponentially fast, wic numerically verify our proof on te exponential decay of te basis functions Ψ i. Terefore, we can localize te computational domain of te basis functions and reduce te computational cost. Due to te exponential decay of te basis functions, we can maintain a certain level of accuracy using te localized basis functions. 100 Double Well Potential Function x Figure 2: Profile of te double well potential function 5.2 1D Scrödinger equations wit double well potential Anoter model problem we consider is Scrödinger equations wit double well potential [12,22,28], wic can be used to model te proton motion restricted to te line joining te two bridged atoms separated by a fixed distance. Te stationary Scrödinger equation in tis case can be formulated as u (ξ)+ 1 2 (E V )u(ξ) =0, (5.1) were E = 1 4 βe, V = 1 4 βv, ξ = αx, α = (µβ ) 1 2, is te reduced Planck constant, µ is te reduced mass of te bond A- B, and β is an arbitrary frequency. We suppose also tat te potential energy of te proton can be represented as a polynomial in te proton coordinate x. On te basis of bot experimental and teoretical investigations, it is generally assumed tat te potential-energy surface of many ydrogen bonds as two

17 17 l λ (l) e (l) (1/4) e (l) (1/8) e (l) (1/16) e (l) (1/32) Table 1: Relative errors e (l) = λ(l) λ(l) λ (l) for l = 1, 15, potential function being V(x), and various coices of te coarse mes size. Space means no available data. minima in te region available for protonic movement [28]. In most cases te minima are not equivalent since te pysical situation is canged wen te proton is transferred from one minimum to te oter. We consider te interval [ 4,4] and partition it into N = 4,8,16,32 patces respectively and furter partition tem suc tat te partition number of te fine mes is N = Our potential function is taken to be V(x) = 5.26x x 4 and its grap is plotted in Figure 2. In te Table 1, we compare te eigenvalues λ (l) wit te eigenvalues λ (l) obtained troug finite element metod obtained on coarse-scale approximations using problem-dependent basis functions wit different mes size. Te cart illustrates tat te convergence rate of relative error is at least of order O( 4 ), wic matces our analysis. We ave also run some tests to compare te errors of te approximated eigenfunctions. In Figure 3, one can find tat te eigenfunctions obtained from our metod can capture te reference eigenfunctions accurately. Especially, note tat te double well potentials take negative values at some x. Tis indicates te operator L is no longer positive semidefinite. Even in tis case, our metod can still accurately approximate te first few eigenfunctions of te Scrödinger equation. As in te Figure 3, qualitatively, we can see tat te grap of approximated eigenfunctions computed troug our metod overlap wit tat of te fine-scale finite element

18 u(x) u(x) u(x) u(x) Eigenvector No. 1 Finite Element Metod Our Metod 0.1 Eigenvector No. 2 Finite Element Metod Our Metod x (a) First eigenfunction Eigenvector No. 3 Finite Element Metod Our Metod x (b) Second eigenfunction Eigenvector No. 4 Finite Element Metod Our Metod x (c) Tird eigenfunction x (d) Fourt eigenfunction Figure 3: Selected examples of computed eigenfunctions using our metod as compared to eigenfunctions obtained troug finite element metod in fine scale. metod. In te Table 2, we sow te relatively error u(l) u(l) L 2 (Ω) u (l) L 2 (Ω), l = 1,2,... One can see tat te relative errors are very small, wic indicates tat our metod can accurately compute te eigenfunctions. To demonstrate te computational savings of te our metod over te finite element metod, we sow in Figure 4 te computational time of compute first 15 eigenvalues and eigenfunctions of te Scrödinger equation. Based on our previous result in Table 1, we assume tat if te coarsening ratio is 64, ten te relative error λ(l) λ(l) λ ( l) <1%. We coose N = 2 11,2 12,2 13,2 14,2 15, and N = 2 5,2 6,2 7,2 8,2 9, respectively. We record te wall time of running te eigenvalue algoritm in Matlab. From te Figure 4, one can see tat our new metod offers considerable savings over te finite element metod. Te slope of te blue line wit stars is approximate equal to 2.56, wic means te computational complexity of eigenvalue algoritm is O(N 2.56 ). Te complexity of our metod is also in te similar

19 19 order O(N α ) wit α 2.5, but N is far less tan N. Terefore, our metod can bring significant savings over te FEM. Te advantage of our metod will be more obvious in two or iger dimensional problems. We sould point out tat in tis example we coose te coarsening ratio to be 64, wic is used to demonstrate te main idea. In general, tis ratio is problem-dependent and may be different. owever, considerable savings over te finite metod can always be acieved if we only compute first few eigenvalues and eigenfunctions of te Scrödinger equation because our metod can efficiently reduce te dimension of te problem. ere we only compare te wall time of running te eigenvalue algoritm. In te FEM, one needs extra time to form te large-scale stiffness and mass matrices. In our metod, we need extra time to compute te problem dependent basis functions, wic is not a serious issue as tis can be done in parallel. l e (l) (1/4) e (l) (1/8) e (l) (1/16) e (l) (1/32) e (l) (1/64) Table 2: Errors e (l) () =: u(l) u(l) L 2 (Ω) u (l) L 2 (Ω) various coarse mes sizes. Space means no available data. for l = 1, 9, potential function being V(x) and 5.3 2D Scrödinger equations wit double well potential In 2D problems, we consider te double well potential again, wic can be used to mimic te nuclear attraction potential generated by two separate nuclei. Our computational domain is unit square Ω=[0,1] 2. We partition Ω into 2N 2 =8,32,128,512 rigt triangular elements respectively and furter partition tem suc tat te lengt of te fine-scale triangular elements remains to be 1/128. In tis setting, te fine-scale finite element space contains triangular elements. Te potential function is taken to be V(x) = e ( 100((x 3 1 )2 +(y 3 1 )2)) e ( 100((x 3 2 )2 +(y 2 3 )2))) and its profile is plotted in te Figure 5. In te Table 3, we compare te eigenvalues λ (l) coarse scale wit mes size and λ (l) obtained troug our metod on obtained troug finite element metod on fine mes. Te cart illustrates tat te convergence rate of relative error is at least of order O( 4 ), wic matces our analysis.

20 Time Finite Element Metod 10 1 Our Metod N Figure 4: Computational time comparison. Te slope of te blue line wit stars is approximately equal to l λ (l) e (l) ( 1 2 ) e(l) ( 1 4 ) e(l) ( 1 8 ) e(l) ( 1 16 ) Table 3: Relative errors of eigenvalues e (l) () =: λ(l) λ(l) for l =1, 15, potential function λ (l) being V(x) and various coices of te coarse mes size. Space means no available data. We ave also run some tests on approximated eigenfunctions. In Figure 6, we see tat our metod can approximate te first few eigenfunctions of te Scrödinger equation

21 21 Figure 5: Profile of te 2D double well potential function accurately. Especially, note tat te double well potentials take negative values at some x. Tis indicates te operator L is no longer positive semidefinite. Even in tis case, our metod can still approximate te first few eigenfunctions of te Scrödinger equation accurately. As sown in te figure 6, qualitatively, we can see tat te grap of approximated eigenfunctions computed troug our metod overlap wit tat of te fine-scale finite element metod. ere, te domain is taken to be [0,1] 2 and we partition te unit square into 2m 2 = 512 rigt triangular patces respectively and furter partition tem into 16 parts. Quantitatively, te relatively errors u(l) u(l) L 2 (Ω) u (l) L 2 (Ω), l = 1,2,..., are kept at a very low level, indicating te accuracy of our metod in capturing te first few eigenfunctions, were u (l) stand for eigenfunctions computed troug our metod and u(l) stand for eigenfunctions computed troug finite element metod. e (1) e (4) e (7) Table 4: Relative errors of eigenfunctions e (l) =: u(l) u(l) L 2 (Ω) u (l) L 2 (Ω) for l=1,4,7 potential function being V(x), te coarse mes size is 1/16, and te fine mes size =1/128.

22 A simple study of coosing coarse mes size We sall investigate ow te course mes size is scaled wit te fine mes size in order to keep te error rate at a similar level. We consider te one dimensional free electron model as discussed in Section 5.1. Since true eigenvalues λ (l) TRUE of tis model problem can be computed analytically, we compare te relative errors of e (l) and e (l) = λ(l) λ(l) TRUE λ (l) TRUE, were λ (l) = λ(l) λ(l) TRUE λ (l) TRUE are computed using our new basis functions and λ(l) are computed using finite element basis function wit mes size indicated in te bracket. In table 5, we compared te differences between two types of relative errors obtained troug our metod and finite element metod respectively. For eac pair of relative errors, we fix te mes size to be = 1/2 and consider te errors incurred wen computing te first few eigenvalues. We can see from te table tat relatively errors are at te same level. Tis provides us wit an empirical guidance on ow to coosing coarse mes size. More teoretical and numerical investigations of our metod will be considered in our subsequent researc. l e (l) (1/16) e(l) (1/162 ) e (l) (1/32) e(l) (1/322 ) e (l) (1/64) e(l) (1/642 ) l e (l) (1/128) e(l) (1/1282 ) e (l) (1/256) e(l) (1/2562 ) Table 5: Relative errors of e (l) = λ(l) λ(l) TRUE λ (l) TRUE and e (l) = λ(l) λ(l) TRUE λ (l) TRUE, l = 1, 7 for te free electron model wit various coices of te coarse mes size and fine mes size. Space means no available data.

23 23 6 Conclusions In tis paper, we propose a model reduction metod to construct problem dependent basis functions and compute eigenvalues and eigenfunctions of stationary Scrödinger equations. Te basis function are obtained troug solving an optimization problem. Under mild conditions, we prove tat te generalized finite element space spanned by our basis functions can accurately compute te first few eigenvalues and eigenfunctions of te stationary Scrödinger equations. In addition, our new metod can significantly reduce te computational cost in eigenvalue decomposition problems compared wit te standard finite element metod on fine mes. We demonstrate troug numerical experiments to sow tat our metod works well for Scrödinger equations wit double well potentials, in wic case te differential operators are no longer positive semidefinite. Tere are several directions we want to explore in our future work. Firstly, we would like to construct problem dependent basis functions to compute eigenvalues and eigenfunctions of Scrödinger equations wit unbounded potential, suc as Coulomb potential. Ten, we would like to employ our new basis functions to compute time-evolutionary Scrödinger equations. In addition, we sall construct problem dependent basis functions using optimization approac to solve problems arising from uncertainty quantification, suc as multiscale elliptic PDEs wit random coefficients. 7 Acknowledgements Te researc of Z. Zang is supported by te ong Kong RGC grants ( , ), National Natural Science Foundation of Cina (Project ), Seed Funding Programme for Basic Researc (KU), and te ung ing Ying Pysical Sciences Researc Fund (KU). We would like to tank Professor Tomas ou for several stimulating discussions. A Exponential decay of te basis function Ψ i In tis appendix, we provide a detailed derivation of te Teorem 4.2. For te ease of reading, we state te teorem again as follows, Teorem A.1. Under te resolution condition V 0 < 1 2C I Col, tere exist some constant C>0 and 0< β <1 independent of, suc tat Ψ i Ω\Ωl Cβ l Ψ i Ω (A.1) for any i =1,2,...,N. Proof. To facilitate te proof, we make use of a few properties of te Clément-type interpolation operator I. For more detailed arguments, we refer te interested readers to

24 24 Section 6 of [7]. Note tat tere exists a constant C I suc tat, for all v V and x i N c, 1 v I v L2 (Ω 0 )+ (v I v) L2 (Ω 0 ) C I v L2 (Ω 1 ). (A.2) Te constant C I depends on te sape regularity parameter γ but does not depend on te local mes size. Notice tat I V is a local operator, by wic we mean it gives rise to a sparse matrix, but its inverse (I V ) 1 is not. Neverteless, tere exists some bounded rigt inverse : V V of I tat is local. Tat is, tere exists a constant C I depending only on I 1,loc γ, suc tat for all v V, I (I 1,loc v ) = v, I 1,loc v Ω C I v Ω, Supp(I 1,loc v ) {T e T e T : T e Supp(v ) = }. (A.3) Te tird condition simply means tat Supp(I 1,loc v ) is included Supp(v ) union anoter layer of coarse elements. More detailed results can be found in [21, 26]. Now we are in te position to prove te decay property of te basis function Ψ i. Te main idea of te proof is based on some iterative Caccioppoli-type argument tat as been used in [16, 21, 26]. We define a projection operator P : W V f suc tat a(pv,w) = a(v,w) for any v W and w V f. More superficially, Pv= T T P T (v T ) and P T (v T ) solves te equation a(p T (v T ),w) = a T (v,w) for all w V f, were a T (, ) means te restriction of te weak form a(, ) on te element T. Now for any x i N c, we define P i := T Ωi,0 P T (Φ i T ) were Ω i,0 = Supp(Φ i ). And we sall prove te P i as exponential decay property. To simplify notation, we omit te dependence on i and use Ω 0 to denote Ω i,0. We coose an integer l wit l 7 and dist(x,ω define a cutoff function as η(x)= l 4 ), x Ω. It is easy to ceck tat te dist(x,ω l 4 )+dist(x,ω\ω l 3 ) cutoff function η(x) as te following properties: (1) η = 0 in Ω l 4, (2) η = 1 in Ω\Ω l 3, (3) 0 η 1, and η is Lipscitz continuous wit η L (Ω) 1 γ, were γ depends on sape regularity parameter γ of te finite element triangles T. Ten we ave te estimate P i Ω\Ωl 3 = ( P i, P i ) Ω\Ωl 3 ( P i,η P i ) Ω = ( P i, (ηp i )) Ω ( P i,( η)p i ) Ω ( P i, (ηp i I 1,loc (I (ηp i )))) Ω + ( P i, (I 1,loc (I (ηp i )))) Ω + ( P i,( η)p i ) Ω. (A.4) To simplify notations, we define M 1 := ( P i, (ηp i I 1,loc (I (ηp i )))) Ω, M 2 := ( P i, (I 1,loc (I (ηp i )))) Ω, M 3 := ( P i,( η)p i ) Ω.

25 25 From te definition of I in (4.4), we know tat I (ηp i I 1,loc (I (ηp i ))) Ω =0. Tis implies tat ηp i I 1,loc (I (ηp i )) V f wit support in Ω\Ω l 6. Tus, ηp i I 1,loc (I (ηp i )) vanises on Ω 0 as long as l 6 and we ave a(v,ηp i I 1,loc (I (ηp i ))) Ω0 =0 for any v in W. Ten, we ave M 1 = ( P i, (ηp i I 1,loc (I (ηp i )))) Ω = (VP i,ηp i I 1,loc (I (ηp i ))) Ω V 0 (P i,ηp i I 1,loc (I (ηp i ))) Ω. Note tat ηp i I 1,loc (I (ηp i )) is supported in Ω\Ω l 6, so wen l 6, it vanises on Ω 0. Using te properties of (A.2)(A.3), arguments used in Lemma 4.2, and te resolution condition, we ave M 1 V 0 (P i,ηp i I 1,loc (I (ηp i ))) Ω C 2 I C ol 2 V 0 P i 2 Ω\Ω l 6 +C 3 I C I C ol 2 V 0 P i 2 Ω l \Ω l P i 2 Ω\Ω l (1+C I C I ) P i 2 Ω l \Ω l 7. (A.5) Similar tecniques and te Lipscitz bound lead to upper bounds of M 2 and M 3, M 2 C I C I (ηp i ) Ωl 1 \Ω l 6 P i Ωl 1 \Ω l 6 C I ) C I (C I Col η L (Ω)+1 P i 2 Ω l \Ω l 7 and M 3 C I Col η L (Ω) P i 2 Ω l 2 \Ω l 5. Te combination of estimates (A.5)-(A.7) yields (A.6) (A.7) 1 2 P i 2 Ω\Ω l C 1 P i 2 Ω l \Ω l 7, were C 1 := C I C I +(C I C I +1)C I Col γ depends only on te sape regularity γ of te finite element triangles T. Since P i 2 Ω l \Ω l 7 = P i 2 P Ω\Ω l 7 i 2, we Ω\Ω l ave te contraction P i 2 Ω\Ω l C 1 C P i 2 Ω\Ω l 7. 2 Finally, some algebraic calculations yield te exponential decay of te P i s, P i 2 Ω\Ω l ( C 1 C ) l 7 P i 2 Ω. (A.8) After proving tat P i s decay exponentially, we sall sow tat Ψ i s ave te some property. Notice tat by definition, P i Φ i V c and moreover, tey span te space V c. Tus

26 26 eac Ψ i can be written as a linear combination of P i Φ i s, namely, Ψ i = j some coefficients a (i) j. And from te condition tat (Ψ i,φ k )=δ i,k, we ave tat ( Φ j ),Φ k ) = δ i,k. Tus j a (i) j (P j Φ j ), for j a (i) j (P j a (i) j (Φ j,φ k ) = δ i,k. Let M be an N N mass matrix wit entry (Φ i,φ k ) and write a (i) =(a (i) 1,,a(i) N ) T. Ten Ma (i) = e i, were e i is a column vector wit i-t entry equals to one and oter entries equal 0, and tus a (i) = M 1 e i. If we number te finite element basis functions Φ i in a proper way, so tat te mass M is a banded matrix wit bandwidt at most p. Ten we know tat te entries of M 1 as te decay property (M 1 ) ij 2ρ 2 i j /p M 1 2, (A.9) were ρ = ( cond 2 (M) 1)/( cond 2 (M)+1). See Teorem 4.8 in [3] for more details. Tus, a (i) j decays exponentially away from a (i) i. Now as eac P j Φ j decays exponentially, teir exponentially-decay linear combination also decays exponentially. Recall tat Ψ i = a (i) j (P j Φ j ), now take β = max{ρ 2/p, j property for Ψ i, namely, ( C 1 C Ψ i 2 Ω\Ω l Cβ l Ψ i 2 Ω. ) 1 7 }, ten we get te exponential decay We remark tat wen Φ i s are taken to be piecewise constant basis functions as proposed in [16], teir correspond mass matrix M reduces to a diagonal matrix and tus a (i) j =δ ij. In tis case, Ψ i =a (i) i (P i Φ i ) and its exponential decay property follows trivially from te decay property of P i. References [1] I. Babuska, G. Caloz, and E. Osborn. Special finite element metods for a class of second order elliptic problems wit roug coefficients. SIAM J. Numer. Anal., 31: , [2] I. Babuška and J. Osborn. Eigenvalue problems. andbook of numerical analysis, 2: , [3] M. Bebendorf. ierarcical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems, volume 212. Springer Berlin eidelberg, [4] S. Boyd and L. Vandenberge. Convex optimization. Cambridge university press, [5] S. Brenner and R. Scott. Te matematical teory of finite element metods, volume 15. Springer Science & Business Media, [6] Laurie J Butler. Cemical reaction dynamics beyond te Born-Oppeneimer approximation. Annual review of pysical cemistry, 49(1): , [7] C. Carstensen and R. Verfürt. Edge residuals dominate a posteriori error estimates for low order finite element metods. SIAM journal on numerical analysis, 36(5): , 1999.

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29 Figure 6: Selected examples of computed eigenfunctions as compared to eigenfunctions obtained troug finite element metod in fine scale. 29