A New Multiscale Discontinuous Galerkin Method for the One-Dimensional Stationary Schrödinger Equation

Transcription

1 J Sci Comput (06 66:3 345 DOI 0007/s A New Multiscale Discontinuous Galerkin Method for the One-Dimensional Stationary Schrödinger Equation Bo Dong Chi-Wang Shu Wei Wang 3 Received: 7 February 05 / Revised: 9 March 05 / Accepted: 6 March 05 Published online: 8 April 05 Springer Science+Business Media New York 05 Abstract In this paper, we develop and analyze a new multiscale discontinuous Galerkin (DG method for one-dimensional stationary Schrödinger equations with open boundary conditions which have highly oscillating solutions Our method uses a smaller finite element space than the WKB local DG method proposed in Wang and Shu (J Comput Phys 8:95 33, 006 while achieving the same order of accuracy with no resonance errors We prove that the DG approximation converges optimally with respect to the mesh size h in L norm without the typical constraint that h has to be smaller than the wave length Numerical experiments were carried out to verify the second order optimal convergence rate of the method and to demonstrate its ability to capture oscillating solutions on coarse meshes in the applications to Schrödinger equations Keywords Discontinuous Galerkin method Multiscale method Schrödinger equation Introduction In this paper, we propose a new multiscale discontinuous Galerkin (DG method for the following one-dimensional second order elliptic equation u f (xu = 0, ( B Bo Dong bdong@umassdedu Chi-Wang Shu shu@dambrownedu Wei Wang weiwang@fiuedu Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth, MA 0747, USA Division of Applied Mathematics, Brown University, Providence, RI 09, USA 3 Department of Mathematics & Statistics, Florida International University, Miami, FL 3399, USA

2 3 J Sci Comput (06 66:3 345 where >0isasmall parameter and f (x is a real-valued smooth function independent of The solution to this equation for positive f is a wave function with the wave number f (x One application of this type of equation is the stationary Schrödinger equation in the modeling of quantum transport in nanoscale semiconductors [5,4,8] { m h ϕ (x qv(xϕ(x = Eϕ(x on [a, b] hϕ (a + i p(aϕ p (a = i p(a, hϕ p (b = i p(bϕ(b, ( h with = and f (x = + qv(x in Eq ( Here h is the reduced Plank constant, me E m is the effective mass (assumed to be constant, q is the elementary positive charge of the electron, V (x is the total electrostatic potential in the device, E is the injection energy, p(x = m(e + qv(x is the momentum, and the solution ϕ is a wave function The quantum mechanical picture of the equation describes an electron injected from x = a with an energy E, and partially transmitted or reflected by the potential V The imposed open boundary conditions are the so-called quantum transmitting boundary conditions, see [3] In the modeling of quantum transport, the Schrödinger equation is coupled with a Poisson equation in order to compute the self-consistent potential In this paper, the total potential V (x is considered to be a known function as the coupling with the Poisson equation does not change our DG method for the Schrödinger equation When the electron has high energy E, ie is very small, the solution to the Schrödinger equation will be highly oscillating Standard numerical methods require very fine meshes to resolve such oscillations As a result, the computational cost is tremendous in order to simulate quantum transport in nano devices since millions of Schrödinger equations need to be solved The design of efficient multiscale numerical methods which produce accurate approximations of the solutions on coarse meshes is a formidable mathematical challenge Note that when the energy E is small, such that E + qv < 0 (ie f < 0, the solution is non-oscillatory Thus we mainly focus on the challenging case of f (x τ > 0and 0 < A lot of work has been dedicated to the development of efficient methods for solving stationary Schrödinger equations, such as [3 5,4 6,8] In [5], Ben Abdallah and Pinaud proposed a multiscale continuous finite element method for the one-dimensional Schrödinger equation ( Using WKB asymptotics [6], they constructed continuous WKB basis functions, which can better approximate highly oscillating wave functions than the standard polynomial interpolation functions so that the computational cost is significantly reduced This method was analyzed in [4] and second order convergence was shown independent of the wave length One limitation of the method is that it is difficult to be generalized to general two-dimensional devices due to the difficulty in extending such continuous multiscale basis functions to two dimensions Different from continuous finite element methods, DG methods [,8,9]do not enforce continuity at element interfaces, which makes them feasible to be extended to multidimensional devices Multiscale DG methods with non-polynomial basis functions have been developed and studied in the literature, including [,7,,,7 ] In [8],Wangand Shu proposeda local DG method with WKB basis (WKB-LDG for solving the one-dimensional stationary Schrödinger equation ( in the simulation of resonant tunneling diode model Numerical tests showed that the WKB-LDG method using piecewise exponential basis functions can resolve solutions on meshes that are coarser than what the standard DG with polynomial basis requires However, no convergence analysis was done for this method

3 J Sci Comput (06 66: In this paper, we propose and analyze a new multiscale DG method for the stationary Schrödinger equation Our method is different from the WKB-LDG method in [8] in two aspects First, we use a smaller finite element space than that of the WKB-LDG method to achieve the same order of convergence and reduce the resonance errors Second, in the definition of the numerical traces we penalize the jumps using imaginary parameters, so our method is not an LDG method when the penalty parameters are nonzero The analysis of the multiscale DG method is challenging Due to the strong indefiniteness of the Schrödinger equation with positive f, it is hard to establish stability estimates for finite element solutions When an energy argument is used, one only gets a Gårding-type equality instead of the energy equality Moreover, there is a typical mesh constraint h for numerically resolving highly oscillating solutions using finite element methods In order to prove convergence results for both h and h cases, special projection results and inverse inequalities are needed A novel and crucial part of our analysis is Theorem 3 for the L projection onto the space of piecewise exponential functions For general H functions, the projection requires that the mesh size is small enough, at least comparable to the wave length, to have a second-order approximation But for the oscillating solutions of the model problem and the corresponding dual problem, we show that the second-order approximation can be obtained even if the mesh size h is larger than the wave length Thanks to this projection result, we are able to derive error estimates without assuming h We first use an energy argument to get a Gårding type equality The fact that the penalty parameters are imaginary allows us to estimate jumps of the errors at cell interfaces; similar idea has been used in [0] Then we apply a duality argument to estimate the error in the interior of the mesh, and the error estimate at cell interfaces allows us to control the boundary terms without using inverse inequality and losing convergence orders The paper is organized as follows In Sect, we define our multiscale DG method for the stationary Schrödinger equation In Sect 3, we state and discuss the main results The detailed proofs are given in Sect 4 We display the numerical results in Sect 5 to verify our error estimates and compare our method with the LDG method using piecewise linear and quadratic functions and the WKB-LDG method in [8] Finally, we conclude in Sect 6 Multiscale DG Method: The Methodology The DG Formulation The model problem we investigate in this paper can be written as follows: { u f (xu = 0, x [a, b], u (a + ik(a u(a = ik(a, u (b ik(b u(b = 0, ( f (x where k(x = is the wave number In order to define a DG method for the elliptic Eq (, we rewrite it into a system of first order equations with the boundary conditions q u = 0, q f (xu = 0 (a q(a + i f a u(a = i f a, q(b i f b u(b = 0, (b

4 34 J Sci Comput (06 66:3 345 where f a = f (a and f b = f (b Let I j = (x j, x j+, j =,,N, be a partition of the domain, x j = (x j + x j+, h j = x j+ x j and h = max h j Let h := {I j : j =,,N} j=,,n be the collection of all elements, h := { I j : j =,,N} be the collection of the boundaries of all elements, E h := {x j+ } N j=0 be the collection of all the cell interfaces, and Eh i := {x j+ be the collection of all interior cell interfaces The weak formulation of a standard DG method for Eq (a is to find approximate solutions u h and q h in a finite element space V h such that } N j= (q h,w h + (u h,w h û h,wn h = 0, (3a (q h,v h q h,vn h ( f (xu h,v h = 0 (3b for all test functions v h,w h V h Here, we have used the notation (ϕ, v h := N (ϕ, v I j, ψ, w n h := j= N ψ, w n I j, j= where (ϕ, v I j = ϕ(xv(x dx, I j ψ, w n I j = ψ(x j+ w(x j+ ψ(x j w(x j, where v is the complex conjugate of v and n is the unit outward normal vector For I j = (x j, x j+, we assume n(x j = andn(x j+ = The numerical traces û h and q h will be defined in the next subsection The finite element space V h contains functions which are discontinuous across cell interfaces For standard DG methods, these functions are piecewise polynomials For our proposed multiscale DG method, the basis functions are constructed to be non-polynomial functions which incorporate the small scales to better approximate the oscillating solutions, and we will give more details in Sect 3 Since the functions in V h are allowed to have discontinuities at cell interfaces x j+,weusev (x j+ and v + (x j+ to refer to the left and right limits of v at x j+, respectively The Numerical Traces The choices of numerical traces are essential for the definition of DG methods, and different numerical traces will lead to different DG methods [] In our scheme, the numerical traces û h and q h are defined in the following way At the interior cell interfaces, û h = u h i β [[q h]], q h = q h + + i α [[u h]], (4a (4b where [[ϕ]] = ϕ ϕ + represents the jump across the interface In our error analysis, we assume that the penalty parameters β and α are positive constants Our numerical experiments show that the method is also well-defined for α = β = 0, which recovers the LDG method with alternating numerical fluxes

5 J Sci Comput (06 66: At the two boundary points {a, b}, γ û h (a = ( γu h (a + i q h (a + γ, fa q h (a = γ q h (a i( γ f a u h (a + i ( γ f a, γ û h (b = ( γu h (b i q h (b, fb (5a (5b (5c q h (b = γ q h (b + i( γ f b u h (b, (5d where γ can be any real constant in (0, The numerical traces at the boundary are defined in this way to match the boundary conditions in (b It is easy to see that q h (a + i f a û h (a = i f a, q h (b i f b û h (b = 0 3 The Multiscale Approximation Space For the Schrödinger equation (, WKB asymptotic [6] states that if E + qv > 0, when h 0, ϕ(x A 4 m(e + qv(x e i S(x + B 4 m(e + qv(x e i S(x, (6 m x where A and B are constants and S(x = E + qv(sds with integration constant h x 0 x 0 Therefore, in [5], Ben Abdallah and Pinaud developed a continuous finite element method using WKB-interpolated basis function ϕ(x = A j 4 m(e + qv(x e i S(x + B j 4 m(e + qv(x e i S(x, x I j The constants A j and B j are determined by the nodal values at the cell boundaries Later, Wang and Shu in [8] developed a WKB-LDG method using the constant form of WKB basis functions Their finite element space is defined as f (x E ={v h : (v h I j span{, e i j f (x (x x j, e i j (x x j }, j =,,N}, f (x j m where = E + qv(x j h In our new proposed multiscale DG method, we use a more compact approximation space by eliminating in the previous E space The new finite element space in this paper is defined as E = { v h : (v h I j span { f (x e i j f (x (x x j, e i j (x x j }, j =,,N } Therefore, our new proposed multiscale DG method is the DG formulation (3 in the multiscale finite element space E with numerical traces defined in (4and(5 Remark WKB asymptotic (6 is valid only for E + qv > 0 and will break down close to turning points which are the points x j satisfying E + qv(x j = 0 Numerically, we require that f (x j is not zero so that the basis functions are linearly independent In implementation, we define a threshold τ>0 When f (x <τ,wesimply

6 36 J Sci Comput (06 66:3 345 replace it by τ in the basis functions Our numerical tests show that the proposed DG method works for any smooth f W, ([a, b] In our analysis, we assume that f W, ( and f (x τ>0 forany x h (7 When f is negative, the solution is non-oscillatory and the analysis is easier 3 Error Estimates In this section, we show error estimates of our multiscale DG method for the model problem ( First, let us introduce the following L projection Foranyω H (I j where I j h, ω is a function in E I j which satisfies where (ω ω,ϕ I j =0, (3a (ω ω,ϕ I j =0, (3b ϕ = e i f j (x x j and ϕ = e i f j (x x j, (3 where f j = f (x j In the rest of the paper, we use the notation δ ω = ω ω TheH s (D-norm is denoted by s,d We drop the first subindex if s = 0, and the second if D = or h The L ( h -norm and the W, ( h semi-norm are denoted as and,, respectively By default, (ϕ, v = (ϕ, v h and ϕ,v n = ϕ,v n h We show that the projection has the following approximation properties Theorem 3 If a function ϕ is the solution to the equation ϕ f ϕ = θ, (33 where θ L ( h and f satisfies (7, and ψ = ϕ, then on each I j h we have δ ϕ I j + δ ψ I j C h ( θ I j + h f W, (I j ϕ I j and δ ϕ I j + δ ψ I j C h/ where C is independent of and h ( θ I j + h f W, (I j ϕ I j, The proofs of this Theorem and the remaining theorems in this section will be given in the next section Remark Note that Theorem 3 holds only for functions which satisfy Eq (33 For general smooth functions, the approximation result will require that h is much smaller than the parameter In our error analysis, we apply Theorem 3 to the solutions of the model problem ( and its dual problem, and it is crucial for removing the typical mesh constraint h for resolving highly oscillating solutions Applying Theorem 3 to the solution of (, we immediately get the following result

7 J Sci Comput (06 66: Lemma 3 For the exact solution u and q to the Eq (a, wehave and δ u + δ q C h f W, ( h u δ u h + δ q h C h3/ f W, ( h u where C is independent of and h Note that when f is constant, Lemma 3 implies that δ u = δ q = 0 This is consistent with the fact that the solutions u and q are in the finite element space E if f is constant To state our main results, we need the following notation Let e ω = ω ω h and η ω = ω ω h Thene ω = δ ω + η ω First we estimate the jumps of η u and η q at the cell interfaces Theorem 33 Assume that α and β are positive constants and 0 <γ < For any mesh size h > 0, wehave h 3/ [[η u ]] Eh + [[η q ]] Eh C f, ( η u + u, where C is independent of and h For the projection of error η u in the interior of the domain, we have the following result Theorem 34 Under the assumption of Theorem 33, we have ( h η u C f, + h3 u, where C is independent of and h Using the triangle inequality, Theorems 3 and 34, we now have our error estimate for the actual error e u = u u h as follows Theorem 35 Let u be the solution of the problem ( and u h E be the multiscale DG approximation defined by (3 (5 Assume that α and β are positive constants and 0 <γ < For any mesh size h > 0, wehave ( h u u h C f, + h3 u, where C is independent of and h Theorem 35 shows that when f is constant, the error of the multiscale DG method is zero When f is not constant, the method has a second order convergence rate, and the estimate holds even if h 4 Proofs In the section, we first prove the projection results in Theorem 3, which allow us to carry out error analysis without assuming that h is smaller than Thenwe usean energyargumentto get the error estimate at cell interfaces in Theorem 33 Finally, we apply a duality argument to get the main result in Theorem 34 Let us begin by introducing some preliminaries

8 38 J Sci Comput (06 66: Preliminaries Note that the exact solution to Eq ( also satisfies the weak formulation (3 Using the orthogonality property of the projection in (3, We get the following error equations (η q,w+ (η u,w ê u,wn =0, (4a (η q,v ê q,vn ( f (xδ u,v ( f (xη u,v= 0, (4b for all test functions v, w E,whereê u = u û h and ê q = q q h Before we prove our error estimates, let us gather some properties of the numerical traces It is easy to check that the following two Lemmas hold for the numerical traces defined in (4and(5 Lemma 4 Forthe numerical traces definedin (4 at the interior cell interfaces, we have ê u = (δ u + η u iβ([[δ q ]] + [[η q ]], ê q = (δ q + η q + + iα([[δ u ]]+[[η u ]] Lemma 4 For the numerical traces defined in (5 at the boundary points {a, b}, we have which implies that where n(a = and n(b = 4 Proof of Theorem 3 γ ê u = ( γe u i e q n, f ê q = γ e q + i( γ fe u n, ê q i f ê u n = 0, To prove the approximation property of the projection in Theorem 3, firstletusshow the following inverse inequality Lemma 43 For any function v E and I j h,wehave v I j Ch / j v I j, where C is a constant independent of and h j Proof Suppose that v = c ϕ + c ϕ on I j = (x j, x j+,whereϕ and ϕ are the basis functions of E I j defined in (3 Then we have v I j = v(x j + v(x j+ = ( ( c c c A c and v I j = ( c c ( (ϕ,ϕ I j (ϕ,ϕ I j (ϕ,ϕ I j (ϕ,ϕ I j ( c c = ( ( c c c B, c

9 J Sci Comput (06 66: where ( cos A = cos ( sin and B = h j sin f j with := h j To find a constant r > 0sothat v I j r v I j for any v E,we need to find the largest eigenvalue of the generalized eigenvalue problem This can be written as Ac = λbc B Ac = λc, so we only need to find the largest eigenvalue of B A The two eigenvalues of B A are Obviously, λ = h + cos j + sin Let us estimate λ Ifh j > c 0 sin which implies that λ Ch j f j and λ = h j for any >0 cos sin for some c 0 (0, ],then>c 0Weget < sin c 0 c 0 < and sin λ Ch j If h j <,then0< < cos f j Weget sin which implies max{ λ, λ } Ch j, v I j Ch j v I j < sin c, 0 c 0 C, and then λ Ch j So we have Next, let us prove Theorem 3 by using Lemma 43 in four steps Proof It is easy to check that ( ϕ f j ϕ = 0 on each I j, j =,,N Let δ ϕ = ϕ ϕthenδ ϕ satisfies the equation δ ϕ f j δ ϕ = θ + ( f f j ϕ, which is equivalent to δ ϕ + f j δ ϕ = g(x, (4 where g(x = (θ + ( f f j ϕ/ So we can assume δ ϕ = c ϕ + c ϕ + Y (x,

10 330 J Sci Comput (06 66:3 345 where ϕ and ϕ are the basis functions of E I j defined in (3, and Y (x is a particular solution to (4 (i Let us compute and estimate Y (x Sinceϕ and ϕ are two solutions to the homogeneous equation corresponding to (4, we use variation of parameters and get where v (x = i f j Y = v (xϕ (x + v (xϕ (x, x x j ϕ gdx, and v (x = i f j x x j ϕ gdx It is easy to see that for any x I j I j, x v i (x C g dx C (h / θ I x j j + h 3/ f W, (I j ϕ I j, i =, (43 Therefore, Y I j C (h θ I j + h f W, (I j ϕ I j (44 (ii Let us compute and estimate the term c ϕ + c ϕ in δ ϕ By the definition of the projection, (3, we have (δ ϕ,ϕ I j = 0, (δ ϕ,ϕ I j = 0 After some calculations, we can write the above linear system as h j ( sin sin The solution to the linear system is ( c c = ( (Y,ϕ I j (Y,ϕ I j c = D D and c = D D 3, where ( D = h j sin, ( D = Y,ϕ sin ϕ, I j ( D 3 = Y,ϕ sin ϕ I j Note that f j c ϕ + c ϕ = D ((D + D 3 cos (x x j + i(d D 3 sin f j (x x j

11 J Sci Comput (06 66: Because ( D + D 3 = sin ( f j Y, cos (x x j I j and we have ( D D 3 = + sin ( f j Y, i sin (x x j, I j ( c ϕ + c ϕ I j D sin ( + D + sin f j Y I j cos Y I j sin (x x j I j f j (x x j I j It is easy to check that f j sin (x x j I j = ( h j sin and f j cos (x x j I j = ( h j + sin So we have c ϕ + c ϕ I j Y I j C h ( θ I j + h f W, ϕ I j (45 (iii Now we estimate δ ϕ I j and δ ψ I j From(44and(45, we immediately get δ ϕ I j 3 Y I j C h ( θ I j + h f W, ϕ I j To estimate δ ψ,welet Since v ψ E,wehave v ψ = ( ϕ i f j c ϕ δ ψ I j ψ v ψ I j So we only need to estimate ψ v ψ Notethat Hence, It is easy to check that ψ v ψ = δ ϕ + i f j c ϕ = i f j (c ϕ + c ϕ + Y (46 ψ v ψ I j f j c ϕ + c ϕ I j + Y I j Y = v ϕ + v ϕ + v ϕ + v ϕ = v ϕ + v ϕ (47

12 33 J Sci Comput (06 66:3 345 So Y I j v L (I j ϕ I j + v L (I j ϕ I j Ch / ( j v L (I j + v L (I j Using (43, we have Using the inequality above and (45, we get Y I j C h ( θ I j + h f W, (I j ϕ I j δ ψ I j ψ v ψ I j C h ( θ I j + h f W, (I j ϕ I j (48 (iv Let us estimate δ ϕ I j and δ ψ I j Because δ ϕ = c ϕ +c ϕ+y and c ϕ +c ϕ E, by Lemma 43 we have δ ϕ I j Ch / c ϕ + c ϕ I j + Y I j Using the triangle inequality and (43, we get Y I j v L ( I j ϕ I j + v L ( I j ϕ I j Using the last inequality and (45 wehave v L ( I j + v L ( I j C (h / θ I j + h 3/ f W, (I j ϕ I j δ ϕ I j C h/ ( θ I j + h f W, (I j ϕ I j Next, we estimate δ ψ I j Using the triangle inequality and the expression of ψ v ψ,(46, we get δ ψ I j ψ v ψ I j + v ψ ψ I j C c ϕ + c ϕ I j + Y I j + v ψ ψ I j Note that c ϕ + c ϕ E and v ψ ψ E By Lemma 43,weget δ ψ I j Ch / ( c ϕ + c ϕ I j + v ψ ψ I j + Y I j (49 From the Eq (47, we obtain that Then by (43, we have Note that Y I j v L ( I j ϕ I j + v L ( I j ϕ I j C( v L ( I j + v L ( I j Y I j C h/ ( θ I j + h f W, (I j ϕ I j (40 v ψ ψ I j v ψ ψ I j + ψ ψ I j v ψ ψ I j

13 J Sci Comput (06 66: Using (48, we get v ψ ψ I j C h ( θ I j + h f W, (I j ϕ I j (4 Applying (40, (4and(45to(49, we get δ ψ I j C h/ ( θ I j + h f W, (I j ϕ I j 43 Proof of Theorem 33 To prove Theorem 33, first we show the following Lemma by an energy argument Lemma 44 L = B + B, where L = η q ( f, η u + iβ [[η q ]] Eh i + iα [[η u ]] Eh i + iγ, η q + i( γ η u, f, f B = (η u, f δ u, B = δu,η q n h \ + η u,δ q + n h \ iα η u, [[δ u ]] n h \ iβ [[δ q ]],η q n h \ γ + ( γδ u i δ q n,η q n + η u,γδ q n + i( γ f δ u f Proof Taking w = η q in the error Eq (4aandv = η u in the complex conjugate of (4b and adding the equations together, we get 0 = (η q,η q + (η u,η q ê u,η q n + (η u,η q η u, ê q n (η u, f η u (η u, f δ u Using integration by parts for the second term on the right hand side of the equation, we have where 0 = η q (η u, f η u (η u, f δ u +, (4 = η u,η q n ê u,η q n η u, ê q n Using Lemma 4 to rewrite ê u and ê q on interior cell interfaces, we get = 4 k= A k

14 334 J Sci Comput (06 66:3 345 where A = η u,η q n h \ ηu,η q n h \ η u,η q + n h \, A = δu,η q n h \ η u,δ q + n h \, A 3 = iβ [[δ q ]]+[[η q ]],η q n h \ + iα η u,([[δ u ]]+[[η u ]]n h \, A 4 = η u,η q n ê u,η q n η u, ê q n It is easy to see that and A = 0, A 3 = iβ [[η q ]] Eh i + iα [[η u ]] Eh i + iβ [[δ q ]],η q n h \ + iα η u, [[δ u ]] n h \ Using Lemma 4 and the fact that e ω = δ ω + η ω for ω = u, q to rewrite ê u and ê q on the domain boundary, we obtain A 4 = iγ η q,η q + i( γ η u, f η u f γ ( γδ u i δ q n,η q n η u,γδ q n + i( γ f δ u f Adding above expressions of A, A, A 3, A 4 to get and using it in (4, we get the conclusion Now let us prove Theorem 33 using Lemmas 44 and 3 Proof Taking the imaginary part of L in Lemma 44, weget β [[η q ]] Eh i + α [[η u ]] Eh i + γ f, η q + ( γ η u, f B + B Now we estimate B and B LetP 0 be the L projection onto constant functions on each element I j h Using the orthogonality property of the projection, wehave Using the Cauchy inequality, we get B = ( η u,(f P 0 f δ u Ch f, δu η u B C [[η q ]] E i h ( δ u h \ + β δ q h \ + C [[η u ]] E i ( δ q h \ + α δ u h h \ ( + C η q ( γ δ u + γ δ q + C η u ( γ δ q + ( γ δ u Because α, β are positive constant and 0 <γ <, we have B C( [[η q ]] Eh + [[η u ]] Eh ( δ u h + δ q h

15 J Sci Comput (06 66: So β [[η q ]] Eh i + α [[η u ]] Eh i + γ f, η q + ( γ f, η u ( B + B C h f, δ u η u +C ( [[η q ]] Eh + [[η u ]] Eh ( δu h + δ q h, which implies that [[η q ]] E h + [[η u ]] E h C h f, δ u η u +C ( δ u h + δ q h C h3 f, η u + C h δ u + C ( δ u h + δ q h Finally, applying the projection result in Lemma 3,weget [[η q ]] E h + [[η u ]] E h C h3 f (, ηu + u, which completes the proof 44 Proof of Theorem 34 To obtain the error estimate for u h in Theorem 34, we use a duality argument We first consider the following dual problem for any given θ L ( ψ ϕ = 0 in, (43a ψ f ϕ = θ in, (43b ψ(a i f a ϕ(a = 0, (43c ψ(b + i f b ϕ(b = 0 (43d For the dual problem, we have the following regularity result Lemma 45 Given θ L (, the solution ϕ and ψ of the dual problem (43 satisfy where C is independent of Proof From (43aand(43bweget ψ + ϕ C θ, ϕ f ϕ = θ (44 First, we multiply the Eq (44byϕ and integrate over to get (ϕ,ϕ ( f, ϕ = (θ, ϕ Using integration by parts and the boundary conditions (43cand(43d, we get ϕ ( f, ϕ + i f b ϕ(b + i f a ϕ(a = (θ, ϕ Taking the imaginary part of above equation, we get ϕ(b + ϕ(a C θ ϕ (45

16 336 J Sci Comput (06 66:3 345 which implies that ϕ (b + ϕ (a C θ ϕ (46 by using the boundary conditions (43cand(43d Next, we multiply the Eq (44byϕ and get ϕ ϕ f ϕϕ = θϕ Taking the real part of the above equation, we have which implies d dx ϕ d ( f ϕ = dx f ϕ + Re(θϕ, d dx ( ϕ + f ϕ = f ϕ Re(θϕ C ϕ + ϕ + θ Let G(x = ϕ + f ϕ Then from above we have Using Gronwall s inequality, we get G(x CG(a + C From (45and(46, we have So C( ϕ + f ϕ + θ G (x CG(x + θ(x x a θ(x dx C (G(a + θ G(a = ϕ (a + f (a ϕ(a C θ ϕ Now integrating G(x over [a, b], weobtain ϕ + τ ϕ Therefore, b a G(x C θ ϕ + θ G(xdx C θ ϕ +C θ C θ + τ ϕ ϕ + ϕ C θ Using the projection result in Theorem 3 and the regularity result in Lemma 45, we immediately get the following result

17 J Sci Comput (06 66: Lemma 46 For the solution ϕ and ψ to the dual problem (43, we have ( h δ ϕ + δ ψ C + h θ and ( h / δ ϕ h + δ ψ h C + h3/ θ We are now ready to prove Theorem 34 Lemma 47 For any θ L ( h, we have (η u,θ h = S + S, where S = η u,δ ψ n + η q,δ ϕ n ê q,δ ϕ n + ê u,δψn, S = (( f P 0 f η u,δϕ+ (( f P 0 f δ u, ϕ Proof By the Eq (43b in the dual problem, we have (η u,θ= (η u,ψ (η u, f ϕ = (η u,δ ψ (η u,( ψ (η u, f ϕ Using integration by parts and the property of the projection (3, we have (η u,θ= η u,δ ψ n +(η u,δ ψ (η u,( ψ (η u, f ϕ = η u,δ ψ n (η u,( ψ (η u, f ϕ Using the error Eq (4a with ω = ψ, weget (η u,θ= η u,δ ψ n +(η q, ψ ê u, ψ n (η u, f ϕ For the second term on the right hand side of the equation, we use the definition of the projection and then the equation (43btoget (η q, ψ =(η q,ψ= (η q,ϕ = (η q,δ ϕ + (η q,( ϕ Using integration by parts and the definition of the projection,wehave (η q, ψ = η q,δ ϕ n +(η q,( ϕ For the second term on the right hand side of the equation, we use the error Eq (4b with v = ϕ and obtain (η q, ψ = η q,δ ϕ n + ê q, ϕ n +( f η u, ϕ + ( f δ u, ϕ So we have (η u,θ= η u,δ ψ n + η q,δ ϕ n + ê q, ϕ n ( f η u,δϕ+ ( f δ u, ϕ ê u, ψ n It is easy to check that ê q,ϕn + ê u,ψn =0 So (η u,θ= η u,δ ψ n + η q,δ ϕ n ê q,δ ϕ n + ê u,δψn ( f η u,δϕ+ ( f δ u, ϕ The Lemma follows by using the definition of the L projection

18 338 J Sci Comput (06 66:3 345 For single-valued functions v and w at Eh i, let us introduce the notation Now we prove Theorem 34 Proof Taking θ = η u,wehave v, ω E i h = N j= v(x j+ w(x j+ η u S + S Let us estimate S and S Applying Lemma 4 and Lemma 4 to Lemma 47,weget S = η u,δ ψ n + η q,δ ϕ n + (δ u + η u iβ([[δ q ]]+[[η q ]], δ ψ n h \ (δ q + η q + + iα([[δ u ]]+[[η u ]], δ ϕ n h \ γ + ( γ (δ u + η u i (δ q + η q n,δ ψ n f γ(δ q + η q + i( γ f (δ u + η u n,δ ϕ n After some algebraic manipulations, we have S = [[η u ]],δ ψ + iα[[δ ϕ]] E i + [[η q ]],δ h ϕ + iβ[[δ ϕ]] E i h η u,(γδ ψ i( γ ( γ f δ ϕ n + η q, ( γδ ϕ + i δ ψ f + δu iβ[[δ q]], [[δ ψ ]] Eh i δ q + + iα[[δ u]], [[δ ϕ ]] Eh i γ + ( γδ u i δ q n,δ ψ n γδ q + i( γ f δ u,δ ϕ n f n Because α, β, γ, γ are positive constants, using the Cauchy s inequality, Lemma 3 and Lemma 46, weget S C( [[η u ]] Eh + [[η q ]] Eh ( δ ϕ h + δ ψ h + C( δ u h + δ q h ( δ ϕ h + δ ψ h ( C (h / + h3/ h ( [[η u ]] Eh + [[η q ]] Eh θ +C Then by Theorem 33,wehave ( h ( S C + h3 h f, η u θ +C + h3 Similarly, we get + h3 f, u θ f, u θ S f P 0 f η u δϕ + f P 0 f δ u ϕ f P 0 f η u δϕ + f P 0 f δ u ϕ ( h C + h3 f, η u θ +C h3 f, u θ

19 J Sci Comput (06 66: Table Example 5: L -errors by multiscale DG with α = β =,γ = 05 N = = 0 = 0 3 = E 5 974E 4 46E 94E E 5 84E 4 E 900E E 4 39E 3 7E 3E E 4 45E 3 30E 3E 0 So ( h η u ( C + h3 h f, η u θ +C + h3 f, u θ ( If h + h3 f, is sufficiently small, we have ( h η u C f, + h3 u 5 Numerical Results by the Multiscale DG In this section, we will perform several numerical tests The first example is to show the good approximation property of the multiscale bases E The basis functions approximate the solution exactly when f (x is constant The next example is to verify the second order convergence of our multiscale DG in the error estimates for a wide range of from to 0 4 In the last two examples, we apply the proposed scheme to the application of Schrödinger equation in the modeling of resonant tunneling diode (RTD For the proposed multiscale DG scheme, we use α = β = andγ = 05 in the tests For other constants α, β and 0 <γ <, the numerical results are similar and thus we do not show them here When α = β = γ = 0, the method becomes the minimal dissipation LDG (MD-LDG method [8] with multiscale basis E Although our analysis does not cover this case, numerically we see a similar second order convergence 5 Constant f Example 5 In the first example, we consider the simple case with constant function f (x When f (x is a constant, the exact solution of ( is in the finite element space E Thus the multiscale DG method can compute the solution exactly with only round-off errors The L -errors of the multiscale DG with α = β = andγ = 05 for the case f (x = 0 are shown in Table It is clear to see the round-off errors (in double precision for ranging from to 0 4 We remark that when is small, numerically integrating the exponential functions accumulates round-off errors Thus the errors increase when becomes small 5 Accuracy Test Example 5 In this example, we consider a smooth function f (x = sin x + Table lists the L -errors and orders of accuracy by the proposed multiscale DG scheme with

20 340 J Sci Comput (06 66:3 345 Table Example 5: L -errors and orders of accuracy by multiscale DG with α = β =,γ = 05 N = = 0 = 0 3 = 0 4 Error Order Error Order Error Order Error Order 0 6E 04 56E 0 47E E E E E E E E E E E E E E E E E E E E E E E E E E E E Table 3 Example 5: L -errors and orders of accuracy by multiscale DG with α = β = 0,γ = 0 N = = 0 = 0 3 = 0 4 Error Order Error Order Error Order Error Order 0 0E 05 54E 0 5E 0 58E E E E E E E E E E E E E E E E E E E E E E α = β =,γ = 05 for from to 0 4 The reference solutions are computed by polynomial-based MD-LDG P 3 method with N = 50,000 cells for = and with N = 500,000 for = 0, 0 3, 0 4 Note that we stop refining the mesh when the errors are smaller than 0 8 We can see a clean second order convergence for both h > and h <Forthesameh, when decreases by a factor of ten, the magnitude of the error increases by ten times This verifies the convergence order is of O(h / in Theorem 34 We also show the results by the multiscale MD-LDG method with E space, ie the proposed multiscale DG method with α = β = γ = 0, in Table 3 Although our analysis is inconclusive in this case, we observe a second order convergence when h >and a third order convergence when h < Next, we compare the results with the WKB-LDG method in [8] intable4 Note that WKB-LDG method is the multiscale MD-LDG method with E space, which has one more basis function than the E space Similar to the MD-LDG method with E space, MD-LDG method with E space also has a second order convergence when h >and a third order convergence when h < However, we observe a resonance error around h = whereas our new proposed multiscale scheme does not have it In Table 5, we show the condition numbers of the global matrices for the WKB-LDG method with E space and our proposed DG method with E space Note that the condition numbers by using E space become very large when N = 0 for = 0 and when N = 60

21 J Sci Comput (06 66: Table 4 Example 5: L -errors and orders of accuracy by WKB-LDG in [8] N = = 0 = 0 3 = 0 4 Error Order Error Order Error Order Error Order 0 40E 06 54E 5E 0 58E E E E E E E E E E E E E E E E E E E E Table 5 Example 5: Condition numbers of the global matrices (in L norm by the proposed scheme with E space and WKB-LDG with E space N = 0 = 0 3 E E E E 0 654E+0 980E+0 605E E E+0 39E+04 67E E E+0 E E E E+03 63E E+03 0E E+03 44E+04 75E+03 E E+03 E E+03 96E E+04 34E+07 9E+04 40E+04 for = 0 3 Those are exactly where the resonance errors are observed in Table 4 In contrast, the condition numbers by using E space only change slightly This shows why removing the basis function from the finite element basis will reduce the resonance errors At the end of this section, we compare our multiscale DG with the MD-LDG using polynomials P and P in Table 6 Standard DG methods using polynomials do not have any order of convergence until the mesh is refined to h < For example, for = 0, MD-LDG P shows a second order starting from N = 60 and MD-LDG P shows a third order starting from N = 80 When becomes even smaller to 0 3, for both P and P, there is no order of convergence till N = 640 The multiscale DG method is convergent when h is much larger than,andwhenh <, it also approximates the solution much more accurately than the standard DG For example, when = 0 and N = 640, the error is O(0 3 by using P, O(0 5 by P,andO(0 7 by E Therefore, the multiscale DG is more efficient and accurate than the standard DG methods for solving the stationary Schrödinger equation 53 Applications to Schrödinger Equation In this section, we apply our proposed multiscale DG method to solve the Schrödinger equation in the simulation of the resonant tunneling diode (RTD model RTD model is used to collect the electrons which have an energy extremely close to the resonant energy

22 34 J Sci Comput (06 66:3 345 Table 6 Example 5: L -errors and orders of accuracy by standard MD-LDG P and P N P P = 0 = 0 3 = 0 = 0 3 Error Order Error Order Error Order Error Order 0 953E 0 95E 0 947E 0 95E E E E E E E E E E E E E E E E E E E E E E E E E 0 4 Fig External voltage with double barriers of height 03V located at [60, 65] and [70, 75] and an applied bias of 08V V x We consider the RTD model (see [5] on the interval [0, 35 nm] Its conduction band profile consists of two barriers of height 03 V located at [60, 65] and [70, 75] Abias energy v is applied between the source and the collector regions The wave function of the electrons injected at x = 0 with an energy E > 0 satisfies the stationary Schrödinger equation with open boundary conditions, (, with m = 0067 m e In our numerical simulations, we consider the total potential to be the external potential only Figure shows the external potential with the double barriers and an applied bias These numerical tests were also performed in [5,8] Example 53 In the first application example, we consider the energy E = ev which is very close to the double-barrier resonant energy For simplicity, no bias is applied to the external potential and the potential function V (x is a piecewise constant function 0, x < 60 03, 60 < x < 65 V (x = 0, 65 < x < 70 03, 70 < x < 75 0, x > 75

23 J Sci Comput (06 66: Fig Wavefunction modulus by multiscale DG for the resonance energy E = ev 3 cells are used in the multiscale DG method In each cell, the solution is plotted as a function using 7 points wavefunction modulus reference multiscale DG x Fig 3 Wavefunction modulus by the multiscale DG for a high energy E = ev 3 cells are used in the multiscale DG method In each cell, the solution is plotted as a function using 7 points wavefunction modulus 5 05 reference multiscale DG x In this case, our proposed multiscale DG method can approximate the solution exactly with round off errors We only use 3 cells for the proposed multiscale DG method with 6 cells each in [0, 50], [85, 35], cells each in [50, 60], [60, 65], [70, 75] and [75, 85], and 3 cells in [65, 70] Figure shows the wave function modulus for the case of the resonance energy E = ev We see a big spike in the double barriers area [60, 75] The proposed multiscale DG method is able to capture the resonance spike in only 3 cells The standard polynomialbased DG methods will need a much more refined mesh to capture the resonance Example 54 In the next example, a bias of 008 V is added at the edges of the device We compute in the case of a very high energy E = ev and the solution will be highly oscillating The reference solution was obtained by the polynomial-based MD-LDG P 3 method of 3,500 cells Figure 3 shows the wave function modulus computed by the multiscale DG with 3 cells In each cell, the solution is plotted as a function using 7 points We can see although the solution is highly oscillating, the proposed scheme matches the reference solution very well The L -error is The standard polynomial-based DG methods cannot well approximate the solution unless the mesh is very refined

24 344 J Sci Comput (06 66: Concluding Remarks In this paper, we have developed a new multiscale discontinuous Galerkin method for a special second order elliptic equation with applications to one-dimensional stationary Schrödinger equations The basis functions are constructed based on the WKB asymptotic and thus have good approximations to the solutions The error estimate shows a second order convergent rate for both h and h Numerical experiments confirmed the convergent rate and also demonstrated excellent accuracy on very coarse meshes when applied to Schrödinger equations Compared with the continuous finite element based WKB method in [5], the multiscale DG method allows the full usage of the potential of this methodology in easy h- p adaptivity and feasibility for the extension to two-dimensional case Compared with the WKB-LDG method in [8], the new multiscale DG method uses a smaller finite element space and more importantly, it dose not have resonance errors In future work, we will generalize our multiscale DG method to two-dimensional Schrödinger equations Acknowledgments The authors consent to comply with all the Publication Ethical Standards The research of the first author is supported by NSF Grant DMS-4909 The research of the second author is supported by DOE Grant DE-FG0-08ER5863 and NSF Grant DMS The research of the third author is supported by NSF Grant DMS References Aarnes, J, Heimsund, B-O: Multiscale discontinuous Galerkin methods for elliptic problems with multiple scales In: Multiscale Methods in Science and Engineering, Lecture Notes in Computer Science Engineering, vol 44, pp 0 Springer, Berlin (005 Arnold, DN, Brezzi, F, Cockburn, B, Marini, LD: Unified analysis of discontinuous Galerkin methods for elliptic problems SIAM J Numer Anal 39, (00 3 Arnold, A, Ben Abdallah, N, Negulescu, C: WKB-based schemes for the oscillatory D Schrödinger equation in the semiclassical limit SIAM J Numer Anal 49, (0 4 Ben Abdallah, N, Mouis, M, Negulescu, C: An accelerated algorithm for D simulations of the quantum ballistic transport in nanoscale MOSFETs J Comput Phys 5, (007 5 Ben Abdallah, N, Pinaud, O: Multiscale simulation of transport in an open quantum system: resonances and WKB interpolation J Comput Phys 3, (006 6 Bohm, D: Quantum Theory Dover, New York (989 7 Buffa, A, Monk, P: Error estimates for the ultra weak variational formulation of the helmholtz equation ESAIM MAN Math Model Numer Anal 4, (008 8 Cockburn, B, Dong, B: An analysis of the minimal dissipation local discontinuous Galerkin method for convection diffusion problems J Sci Comput 3, 33 6 (007 9 Cockburn, B, Shu, C-W: Runge Kutta discontinuous Galerkin methods for convection-dominated problems J Sci Comput 6, 73 6 (00 0 Feng, X, Wu, H: Discontinuous Galerkin methods for the Helmholtz equation with large wave number SIAM J Numer Anal 47, (009 Gabard, G: Discontinuous Galerkin methods with plane waves for time-harmonic problems J Comput Phys 5, (007 Gittelson, C, Hiptmair, R, Perugia, I: Plane wave discontinuous Galerkin methods: analysis of the h-version ESAIM MAN Math Model Numer Anal 43, (009 3 Lent, CS, Kirkner, DJ: The quantum transmitting boundary method J Appl Phys 67, (990 4 Negulescu, C: Numerical analysis of a multiscale finite element scheme for the resolution of the stationary Schrödinger equation Numer Math 08, (008 5 Negulescu, C, Ben Abdallah, N, Polizzi, E, Mouis, M: Simulation schemes in D nanoscale MOSFETs: a WKB based method J Comput Electron 3, (004 6 Polizzi, E, Ben Abdallah, N: Subband decomposition approach for the simulation of quantum electron transport in nanostructures J Comput Phys 0, (005

25 J Sci Comput (06 66: Wang, W, Guzmán, J, Shu, C-W: The multiscale discontinuous Galerkin method for solving a class of second order elliptic problems with rough coefficients Int J Numer Anal Model 8, 8 47 (0 8 Wang, W, Shu, C-W: The WKB local discontinuous Galerkin method for the simulation of Schrödinger equation in a resonant tunneling diode J Sci Comput 40, (009 9 Yuan, L, Shu, C-W: Discontinuous Galerkin method based on non-polynomial approximation spaces J Comput Phys 8, (006 0 Yuan, L, Shu, C-W: Discontinuous Galerkin method for a class of elliptic multi-scale problems Int J Numer Methods Fluids 56, (008 Zhang, Y, Wang, W, Guzmán, J, Shu, C-W: Multi-scale discontinuous Galerkin method for solving elliptic problems with curvilinear unidirectional rough coefficients J Sci Comput 6, 4 60 (04