A phase-based interior penalty discontinuous Galerkin method for the Helmholtz equation with spatially varying wavenumber

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1 A phase-based interior penalty discontinuous Galerkin method for the Helmholtz equation with spatially varying wavenumber Chi Yeung Lam * and Chi-Wang Shu ** * Department of Mathematics, The Chinese University of Hong Kong. cylam@math.cuhk.edu.hk ** Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. shu@dam.brown.edu July 4, 2016 Abstract This paper is concerned with an interior penalty discontinuous Galerkin (IPDG method based on a flexible type of non-polynomial local approximation space for the Helmholtz equation with varying wavenumber. The local approximation space consists of multiple polynomial-modulated phase functions which can be chosen according to the phase information of the solution. We obtain some approximation properties for this space and an a prior L 2 error estimate for the h-convergence of the IPDG method using the duality argument. We also provide numerical examples to show that, building phase information into the local spaces often gives more accurate results comparing to using the standard polynomial spaces. 1 Introduction In this paper, we consider the following Helmholtz equation on a bounded convex polygonal domain Ω R 2 with a Robin boundary condition: u κ 2 u = f in Ω (1 u n + iκu = g on Ω, (2 where κ is the wavenumber, which is a smooth real-valued function, f is the source term in L 2 (Ω, g is the boundary data in L 2 ( Ω and n is the outward normal on Ω. The Helmholtz equation with high and varying wavenumber arises from many areas, including seismology, electromagnetics, underwater acoustics [4], plasma physics [1] and medical imaging [2]. Standard finite element methods based on low-order polynomials do not perform well for the Helmholtz equation Research supported by DOE grant DE-FG02-0ER2563 and NSF grant DMS

2 at high wavenumber. On one hand, low-order polynomials do not well resolve the solution unless several grid points per wavelength is used. On the other hand, such methods suffer from the so-called pollution effect: for fix number of grid points per wavelength, the numerical error grows with κ [3, 16]. However, it was suggested in [23, 24] that the pollution effect can be suppressed by using higher order polynomial for problems with higher wavenumber. It is natural to consider oscillatory non-polynomial basis in order to overcome the shortcomings of low-order polynomials, in the hope that such basis can resolve the solution using significantly fewer degrees of freedom than polynomials. For the Helmholtz equation with homogeneous media, plane waves are extensively used as basis in the literature. Examples using such basis include the partition of unity finite element method [2, 22, 27, 31], the least-square methods [25, 30] and the discrete enrichment method [1,, 9]. A good survey of such methods can be found in [15]. Besides, Cessenat and Després proposed the ultra-weak variational formulation (UWVF for the Helmholtz equation in [7]. The UWVF approximates the exact solution of the Helmholtz equation by a linear combination of free space solutions of the homogeneous Helmholtz equations, including plane waves. Later, the UWVF was recast as a discontinuous Galerkin method in [6] and generalized into the plane wave discontinuous Galerkin (PWDG method in [12], where plane waves with uniformly-spaced directions are used as basis. The convergence of the PWDG method regarding refining the mesh, namely the h-version, is analyzed in [12] using Schatz duality argument [29]. It was shown that the h-version is still afflicted by the pollution effect. A detailed quantitative study of such effect for this method can be found in [11]. Later, it was suggested in [13] that the convergence of the PWDG method regarding using more plane waves, namely the p-version, is immune to the pollution effect. Recently, the exponential convergence of a strategy in choosing h and p locally, namely the hp-version, has been established theoretically in [14]. For problems with heterogeneous media or nonzero source function f, plane waves are no longer the free space solutions of the Helmholtz equation. In [12], the PWDG method was shown to obtain only second order accuracy for a generic source term. It was also shown numerically in [5] that using plane waves with uniformly-spaced directions for varying wavenumber did not outperform polynomials significantly in their experiments. Instead of using plane waves as basis, Imbert-Gérard and Després [1] proposed an extension of the UWVF method with high order accuracy for smooth varying wavenumber. Based on the idea of generalized plane waves proposed by Melenk [21], they considered an adapted basis of the form e P (x where P is a polynomial with complex coefficients constructed locally according to an approximation to the wavenumber. Further analysis of the adapted basis in two dimensions can be found in [17]. The above mentioned methods do not assume the knowledge of phase values of the solution. However, it has been shown that taking advantage of the knowledge of the phase values, the accuracy could be greatly enhanced with the same mesh size. For example, Betcke and Philips [5] compared uniform plane wave basis to low-order polynomial modulated plane waves with dominant directions under a DG formulation similar to the original PWDG method, and showed numerically the latter outperformed the former for a Helmholtz problem with varying wavenumber. In [26], Nguyen et al. proposed a hybridizable discontinuous Galerkin method (HDG using local approximation spaces consisting 2

3 of p l (xe iψ l(x, where p l (x are polynomials and ψ l (x are solutions of the eikonal equation. They showed that their phase-based HDG method can obtain high-order accuracy with several orders of magnitude fewer degrees of freedom comparing to the HDG method with standard polynomial basis. More examples of using such polynomial-modulated basis can be found in [10, 19]. In this paper, we will consider a general type of local space consisting of pl (xe iq l(x where p l (x are polynomials and q l (x are real-valued functions. We have proved some approximation properties when q l are polynomials satisfying certain additional assumptions. We will use these local spaces in an interior penalty discontinuous Galerkin (IPDG framework to solve the Helmholtz equation with varying wavenumber, and give error estimates in terms of the wavenumber κ(x using Schatz s argument. We call our method a phase-based method following [26], due to its flexibility in choosing the basis related to the phase of the exact solution. This paper is organized as follows. In section 2, we give some preliminaries and then state some notations we use in this paper. In section 3, we discuss the properties of the local bases and the formulation of the IPDG method. In section 4, we use Schatz s argument to prove the convergence of the IPDG method for the Helmholtz equation with varying wavenumber. In section 5, we give numerical examples to show the h-convergence of this method, and numerically verify a few properties of the local spaces. Finally, conclusions are drawn in section 6. 2 Preliminaries We assume Ω R 2 is a convex polygonal domain. Let T h be a conforming triangulation of Ω, where the index h is a constant multiple of the maximum diameter of the triangles. For any triangle K T h, we denote the diameter of K by h K. We also define the skeleton F to be the union of boundaries of all triangles K in T h, the boundary skeleton F B := F Ω, and the interior skeleton F I := Ω \ F B. Assumption (M. We assume the triangulation T h satisfies the following properties: 1. Shape regularity. There is a positive constant α independent of h such that for any K T h, h K /ρ K α, where ρ K is the radius of the inscribed circle of K. 2. Quasi-uniformity. There is a positive constant τ independent of h such that for any K T h, h K τh. Let K ± be two triangles sharing an edge e. Let n ± be the outward normal of K ± on e and u ± be two smooth scalar functions on K ±, respectively. We define the average operator { } and jump operator by {u} := 1 2 (u+ + u and u := u + n + + u n. 3

4 Similarly, for smooth vector field σ ± defined on K ±, respectively, we define the average operator { } and jump operator by {σ} := 1 2 (σ+ + σ and σ := σ + n + + σ n. On the standard Sobolev space H s (S, we denote the Sobolev norm by s,s and the semi-norm, which is the L 2 norm for derivatives of order s, by s,s. When S = Ω, we simplify the notation by writing s and s for the Sobolev norm and semi-norm respectively. We also denote the L 2 (Ω inner product by (,. 3 The phase-based IPDG method 3.1 The local approximation spaces On each K T h, we denote the local space of polynomials with degree r by P r (K. We call a function of the form e iq(x a phase function. For a real vector-valued function q := (q 1,..., q m (H r+1 (K m, we define the local space of polynomial-modulated phase functions with degree r by { m } Q r (K, q := p l (xe iql(x : p l P r (K. We define two L 2 projection operators onto P r (K and Q r (K, q, respectively, P r : H r+1 (K P r (K, (P r u, v = (u, v for any v P r (K, Q r : H r+1 (K Q r (K, q, (Q r u, v = (u, v for any v Q r (K, q. In this section we will discuss certain special choices of q. Assumption (Q. We assume q := (q 1,... q m satisfies: 1. There is N 1 independent of h such that for any K T h and 1 l m, q l P N (K; 2. There is a constant C > 0 independent of h such that for any K T h, 1 l m, α N and x K, α q l x α (x C; 3. There is a constant C > 0 and an integer k 0 0, both independent of h, such that for any K T h, 1 l m and p l P r (K, m m p l 0,K Ch k0 p l e iq l. 0,K Remark 3.1. When m = 1, the third assumption in (Q holds with k 0 = 0, since p 1 0,K = p1 e iq1 0,K. 4

5 We will repeatedly use the following lemma in this section, which is a direct consequence of the second assumption in (Q: Lemma 3.2. There are constants C 1, C 2 > 0 independent of h and K T h such that for any 1 l m and 1 j max {r + k 0 2, r}, there are polynomials q l,j and q l,j of degree at most j such that q l q l,j C 1 h j+1 and e iq l,j q l,j C2 h j+1. Proposition 3.3 (Trace inverse inequality. There is C > 0 independent of h such that for any K T h and v Q r (K, q. v 1, K Ch 1/2 v 1,K Proof. Write v := m p le iq l and g l := ip l q l + p l for p l P r (K. Let j be an integer such that 1 j max {r + k 0 2, r}. Applying triangle inequality and Cauchy s inequality, m v 1, K g l 0, K e iq l q 0, K m l,j + g l q l,j. (3 0, K Applying trace inverse inequality for P N+r+j 1 (K, m v 1, K Ch 1/2 g l e iq l 0,K q 0, K m l,j + g l q l,j. (4 0,K Also, applying Lemma 3.2, e iq l q 0, K l e iq l e iq l,j 0, K + e iq l,j q 0, K l,j q l q l,j 0, K + e iq l,j q l,j 0, K Ch j+3/2, (5 m g l q l,j 0,K m g l e iq l 0,K m + g l ( q l,j e iq l,j 0,K m v 1,K + h j+2 g l 0,K. (6 Combining inequalities (4, (5 and (6, v 1, K Ch 1/2 ( v 1,K + (h j+3/2 + h j+2 Applying the third assumption in (Q, m g l 0,K. (7 v 1, K Ch 1/2 ( v 1,K + (h j k0+3/2 + h j k0+2 v 1,K. ( The result follows by taking j = max{k 0 1, 1}. 5

6 Proposition 3.4 (Inverse estimates. For any 1 k r there is C inv,k > 0 independent of h such that for any K T h and v Q r (K, q, v k,k C inv,k h k v 0,K. (9 Proof. For v Q r (K, q, write v := m p le iq l with p l P r (K. Let j be an integer such that 1 j max{r + k 0 2, r}. Applying triangle inequality, Leibniz s rule and Cauchy s inequality, m v k,k C p l e iq l k,k q k,k m l,j + p l q l,j. (10 k,k Applying inverse inequality for P r+j (K and using Lemma 3.2, m v k,k Ch k p l e iq l 0,K q k,k m l,j + p l q l,j 0,K m Ch k h j k+2 m p l 0,K + p l q l,j. (11 0,K Also, applying triangle inequality, Cauchy s inequality and Lemma 3.2, m m p l q l,k p l e iq l m + p l e iq l 0,K q 0,K l 0,K 0,K m v 0,K + Ch j+2 p l 0,K. (12 Combining inequalities (11, (12 and taking j = max{k + k 0 2, r}, m v k,k Ch ( v k 0,K + (h k0 + h k+k0 p l 0,K. (13 The result follows from the third assumption in (Q. Theorem 3.5 (Projection error estimates. For any 0 k r, there is a constant C > 0 independent of h such that for any K T h and v H r+1 (K, v Q r v k,k Ch r+1 k v r+1,k. Proof. Let q be one of the q l. Note that by the second assumption in (Q, e iq r+1,k C. It suffices to show for any v H r+1 (K, v Q r v k,k Ch r+1 k e iq v r+1,k, (14 for the result will follow by Leibniz s rule and Cauchy s inequality. First of all we consider k = 0. We observe that e iq P r (e iq v Q r (K, q, v Q r v 0,K v e iq P r (e iq v 0,K = e iq v P r (e iq v 0,K. (15 6

7 Applying the projection error estimates for P r (K, v Q r v 0,K Ch r+1 e iq v r+1,k. (16 For the case k > 0, applying triangle inequality, v Q r v k,k v e iq P r (e iq v k,k + e iq P r (e iq v Q r v k,k. (17 Applying the projection error estimates for P r (K to the first term in (17, v e iq P r (e iq v k,k C e iq k,k e iq v P r (e iq v k,k Ch r+1 k e iq k,k e iq v r+1,k. (1 Since the second assumption in (Q implies e iq k,k C, we obtain v e iq P r (e iq v k,k Ch r+1 k e iq v r+1,k. (19 For the second term in (17, applying the inverse estimates of Q r (K in Proposition 3.4, and the projection error estimates for P r (K, e iq P r (e iq v Q r v k,k Ch k ( v e iq P r (e iq v 0,K + v Q r v 0,K Ch k v e iq P r (e iq v 0,K Ch r+1 k e iq v r+1,k. (20 The result follows from inequalities (17, (19 and (20. Corollary 3.6. For any 0 k r, there is a constant C > 0 independent of h such that for any K T h and v H r+1 (K, m v Q r v k,k Ch r+1 k min e iq l v r+1,k l. v=v v m v l H r+1 (K Proof. It follows from the proof of Theorem 3.5 and m v Q r v k,k v l Q r v l k,k. 3.2 Derivation of the discrete scheme Multiplying equation (1 by the complex conjugate of a smooth test function v on each K and applying integration by parts twice on the first term, we get u v dx + u v n ds u nv ds κ 2 uv dx = fv dx, (21 K K K where n is the outward normal on K. We approximate u, v in (21 by discrete functions u h, v h, and the boundary values of u and u by numerical fluxes û h and σ h, u h v h dx + K K û h v h n ds K K σ h nv h ds K κ 2 u h v h dx = fv h dx, (22 7 K K

8 Then we reverse the second integration by parts and get u h v h dx (u h û h v h n ds σ h nv h ds K K K κ 2 u h v h dx = fv h dx. (23 Next, we choose the numerical fluxes. On interior edges, we take On boundary edges, we take σ h = { h u h } i a h u h and û h = {u h }. σ h = (g iκu h n and û h = u h. Here h is the piecewise gradient and a is the penalty parameter to be chosen. Now, we define the finite element space V h := {v K Q r (K, q : K T h }, where q can be depending on K. Substituting the numerical fluxes into equation (23, summing over K T h and applying the magic formula : vσ n ds = v {σ} ds + {v} σ ds + vσ n ds, (24 F I F I Ω K T h K we obtain the following discrete scheme: Find u h V h such that for any v h V h, a h (u h, v h (κ 2 u h, v h = (f, v h + gv h ds, (25 K F B h K a h (u, v := h u h v dx F I h a +i u v ds + i Fh I h u { h v} ds F B h κuv ds. F I h { h u} v ds Remark 3.7. The plane wave discontinuous Galerkin method of Gittelson and Hiptmair [12] for constant wavenumber κ(x = ω is obtained by choosing the interior fluxes as σ h = { h u h } i a h u h and û h = {u h } + ibh u h, (26 and the boundary fluxes as σ h = u h (1 dωh( u h + iωu h n gn and û h = u h + idh( u h n + iωu h g, (27 in equation (23, with a a min > 0, b 0 and d > 0. Our choice of fluxes is equivalent to taking b = d = 0.

9 4 Convergence analysis In this section, we replace the assumption (Q in the previous section by a weaker one: Assumption (Q*. We assume the space Q r (K, q satisfies: 1. Trace inverse inequality. There is C tinv > 0 independent of h such that for any K T h, v Q r (K, q, v 1, K C tinv h 1/2 v 1,K ; (2 2. Projection error estimates. For 1 k r, there is a constant C proj,k > 0 independent of h such that for any K T h, v H r+1 (K, v Q r v k,k C proj,k h r+1 k v r+1. (29 Let V H 2 (Ω to be the space of all possible solutions of the model problem (1 and (2. We extend the definition of a h to the space (V + V h (V + V h. Note that by the definition our discrete scheme (25 is consistent, meaning that for any v h V h, a h (u u h, v h (κ 2 (u u h, v h = 0, (30 where u is the analytic solution of the model problem and u h is the discrete solution of (1 and (2. Assumption (P. We assume there is a constant a min > C 2 tinv such that a a min on F I h. Proposition 4.1. The discrete problem (25 has a unique solution. Proof. Suppose a h (u h, u h = 0. The imaginary part of the equation gives u h = 0 on F B h and u h = 0 on F I h while the real part gives u h = 0 on every K T h. Next, we define an auxiliary bilinear form b h and a mesh-dependent norm DG on V + V h : b h (u, v := a h (u, v + (κ 2 u, v, v 2 DG := hv 2 a 0,Ω + 1/2 h 1/2 v 2 + a 1/2 h 1/2 { v} 2 + 0,Fh I Fh I κ 1/2 v 2 0,F B h + κv 2 0,Ω. Note that by applying Cauchy s inequality, we have a h (u, v ± (κ 2 u, v 2 u DG v DG. (31 for any u, v V + V h. 9

10 Proposition 4.2 (Coercivity of b h. There is a constant C coer > 0 independent of h such that for any v V h, b h (v, v C coer v 2 DG. Proof. Let v V h. By definition we have b h (v, v = v 2 0 2Re v { h v} ds + i a 1/2 h 1/2 v +i κ 1/2 v 2 0,F B h F I h 2 0,F I h + κv 2 0. (32 Using the Cauchy Schwarz inequality and Young s inequality, for any s > 0, 2Re v { h v} ds s a 1/2 h 1/2 v + 1 h 1/2 { h v}. a min s F I h 2 0,F I h 2 0,F I h Inserting this into equation (32, 2 bh (v, v Re b h (v, v + Im b h (v, v v 2 0 s a 1/2 h 1/2 v 2 1 h 1/2 { v} 2 a min 0,Fh I s 0,Fh I + a 1/2 h 1/2 v 2 + κ 1/2 v 2 + κv 2 0,Fh I 0,Fh B 0 ( (1 t v 2 t 0 + Ctinv 2 1 a a 1/2 s min h 1/2 { v} 2 0,Fh I ( + 1 s a 1/2 h 1/2 v 2 a min 0,Fh I + κ 1/2 v + κv 2 0 (33 2 0,F B h Picking s, t and a min such that a min > s > Ctinv 2 and 1 > t > C2 tinv /s, the result follows. We will use Schatz s duality argument to show the convergence of our method. Assumption (K. We assume the wavenumber κ(x satisfies: 1. There are constants κ and κ such that for any x Ω, 0 < κ κ(x κ < ; 2. η κ,ω := 1 min x0 Ω max x Ω κ 1 κ (x x0 T > 0. Proposition 4.3. Let u H 2 (Ω be the analytic solution to the model problem (1, (2, and u h V h be the discrete solution of (25, then there is C abs > 0 independent of h such that ( u u h DG C abs inf u v h v h V DG + κ (κ κ 1 u u h 0. h 10

11 Proof. Let v h V h. By triangle inequality, u u h DG u v h DG + v h u h DG (34 For the second term, from the coercivity of b h (Proposition 4.2, scheme consistency (30 and inequality (31, = v h u h 2 DG 1 b h (v h u h, v h u h C coer 1 b h (v h u, v h u h + 2 (κ 2 (u u h, v h u h C coer C coer 2 ( u v h C DG v h u h DG + (κ 2 (u u h, v h u h. (35 coer Using Cauchy s inequality and comparing the L 2 norm to the DG norm, (κ 2 (u u h, v h u h κ (κ κ 1 u u h 0 v h u h DG (36 Since v h is arbitrary, the result follows by combining (34, (35 and (36. Next we modify the proof of the regularity theorem given by Melenk [20] in order to provide regularity for the adjoint problem with variable wavenumber. We define the following weighted norm on H 1 (Ω : v 2 1,κ = v 2 1,Ω + κv 2 0. Theorem 4.4. Let Ω be a bounded convex domain. adjoint problem of (1 and (2: Consider the following ϕ κ 2 ϕ = w in Ω, (37 ϕ n + iκϕ = 0 on Ω, (3 where w L 2 (Ω. Let λ κ := κ 1 κ 0. Then the solution ϕ H 2 (Ω and there is a constant C 1, C 2 > 0 only depending on Ω such that ϕ 1,κ C 1 η 1 κ,ω (κ κ 1 w 0, ϕ 2 C 2 η 1 κ,ω (κ κ 1 (1 + κ + λ κ w 0. Proof. Without loss of generality, assume the domain Ω contains the origin and the origin is the maximizer of x 0 in the definition of η k,ω. Consider the weak form of the problem: B h (ϕ, ψ := ϕ ψ dx κ 2 ϕψ dx + i κϕψ ds = wψ dx. (39 Ω Ω Ω Ω Letting ψ = ϕ in (39 and considering the real part and imaginary part separately, ϕ ϕ dx κ 2 ϕϕ dx + i κϕϕ ds = wϕ dx. (40 Ω Ω Ω Ω The real part of (40 gives κ 1/2 ϕ 2 0, Ω Ω wϕ dx ε κ 1/2 ϕ εκ w 2 0 (41

12 and hence ( κϕ 2 ε 0, Ω κ κ 1 2 κϕ ε w 2 0. (42 The imaginary part of (40 gives ϕ 2 1 κϕ wϕ dx 2 κϕ Ω 4κ 2 w 2 0. (43 By the regularity theory, ϕ H 2 (Ω. Therefore it is valid to substitute ψ = ϕ x T in (39. We apply integration by parts and the following identities for smooth function v to the imaginary part of equation (43: Re (v v = 1 2 v 2, (44 v (x T v = 1 2 ( v 2 x T, (45 we obtain the following identity: Ω(κ 2 + κ κ x T ϕ 2 dx + 1 ϕ 2 x T n dx 1 κϕ 2 x T n dx 2 Ω 2 Ω +Re i κϕx T ϕ ds = Re wx T ϕ dx. (46 Ω For the first term in (46, we have (κ 2 + κ κ x T ϕ 2 dx = Ω Ω ( 1 + κ 1 κ x T κϕ 2 dx η κ,ω κϕ 2 0. (47 We also note that since Ω is a bounded convex domain, there is γ > 0 such that on Ω, x T n γ By combining (43, (46 and using x T n γ, there is a constant C > 0 such that η κ,ω κϕ γ 2 ϕ 2 1, Ω ( κϕ C κϕ 0, Ω ϕ 1, Ω + w 0 ϕ 1, (4 where the constant C here and hereafter only depends on Ω. Applying Cauchy s inequality to (4, η κ,ω κϕ 2 0 ( κϕ C 2 0, Ω + w 0 ϕ 1. (49 Applying (42 and (43 and Cauchy s inequality with suitable weight to this equation, κϕ 2 0 Cη 2 κ,ω (κ κ 1 2 ( 1 + κ 2 2 w 0. (50 The first estimate follows from (43 and the assumption that κ > 0. For the second estimate, using the regularity theory for, ϕ H 2 (Ω C ( ϕ 0 + n ϕ 1/2, Ω ( w C + κ 2 ϕ 0 + κϕ 1/2, Ω. (51 Ω 12

13 Also, using the trace theorem for the Sobolev space, Combining (51 and (52, κϕ 1/2, Ω C κϕ 1 C ( κϕ 0 + λ κ κϕ 0 + κ ϕ 1. (52 ϕ H 2 (Ω C( w 0 + κ κϕ 0 + ( 1 + κ 1 κ κϕ κ ϕ 1 C ( w 0 + (1 + κ + λ κ ϕ 1,κ, and the result follows from the first estimate. Lemma 4.5. There is C > 0 which only depends on the parameter a and Ω such that for any u H r+1 (Ω, u Q r u DG C h r (1 + κ h u r+1. Proof. Here we use the multiplicative trace inequality: there is C > 0 independent of h such that for any K T h, v H 1 (K, which implies, v 2 0, K C v 0,K (h 1 v 0,K + v 1,K, (53 h u Q r u 2 1, K C ( u Q r u 2 1,K + h u Q ru 1,K u Q r u 2,K, (54 h 1 u Q r u 2 0, K C (h 2 u Q r u 2 0,K + h 1 u Q r u 1,K u Q r u 0,K. (55 Also for a boundary edge e, κ 1/2 (u Q r u 2 ( (κ h h 1 u Q r u 2 0,e. (56 0,e Applying inequalities (55, (56 and the projection error estimate in Assumption (Q* to the definition of DG, and the result follows. u Q r u 2 DG Ch2r (1 + κ h + (κ h 2 u 2 r+1, (57 Lemma 4.6. Let ϕ be the solution to the adjoint problem (37, (3 with right hand side w L 2 (Ω. Then there is a constant C dual > 0 only depends on the parameter a, q and Ω such that ϕ Q 1 ϕ DG C dual η 1 κ,ω (1 + κ h(1 + κ + κ + λ κ h w 0. Proof. By Lemma 4.5 and the definition of 1,κ, ϕ Q 1 ϕ DG Ch(1 + κ h ϕ 2 Ch(1 + κ h ((1 + κ ϕ 1,κ + ϕ 2. (5 The result follows from Theorem

14 Theorem 4.7. Let u H r+1 (Ω be the analytic solution of (1, (2. u h V h be the the discrete solution of the DG method (25. Let Let θ κ := η 1 κ,ω κ (κ κ 1 (1 + κ + κ + λ κ. Provided that the following threshold condition holds: 2C abs C dual θ κ (1 + κ hh < 1, then there is a constant C only depending on Ω, α, τ, a, and q such that u u h DG Ch r u r+1, u u h 0 Cθ κ (1 + κ hh r+1 u r+1. Proof. Let ϕ be the solution to the adjoint problem (37 and (3 with right hand side w L 2 (Ω. By consistency of the adjoint problem, (u u h, w h = a h (u u h, ϕ Q 1 ϕ (κ 2 (u u h, ϕ Q 1 ϕ. (59 Hence by the inequality (31, Applying Lemma 4.6, (u u h, w h 2 u u h DG ϕ Q 1 ϕ DG. (60 u u h 0 2 C dual η 1 κ,ω (1 + κ h(1 + κ + κ + λ κ h u u h DG. (61 Applying Proposition 4.3, ( u u h DG C abs inf u v h v h V DG + h The threshold condition implies 2C dual θ κ (1 + κ hh u u h DG. (62 u u h DG C inf v h V h u v h DG. (63 The error estimate in DG norm follows from taking v h = Q r u in equation (63 and applying Lemma 4.5, Then the error estimate in L 2 norm follows from equation (61. 5 Numerical experiments In the following examples, we consider square domains. To form the triangulation T h, we subdivide Ω into n n squares and further subdivide each square into two triangles by the diagonal. The mesh parameter h is taken as 1/n. The penalty parameter is taken as a = 10 for all of our examples, which may not satisfy the assumption (P but we have observed expected h-convergence with this choice of parameter in all of our test cases. Note that for the first two examples, the phase functions we use in the local spaces are not polynomials, but numerical results in section 5.5 suggest these spaces satisfy the assumption (Q*. The plots of the real part of the solutions are shown in Figure 1. 14

15 5.1 Example 1 In this example, we consider the domain Ω = [0, 1] 2 with wavenumber κ(x = ω for ω = 1, 10 and 100, respectively. Also, we consider the following solution of problem (1-(2: u 0,ω (x := H (1 0 (ω x y 0, where y 0 = ( 1, 1 and H (1 0 is the zeroth order Hankel function of the first kind. We will consider the following local space: Q r 0,ω(K := Q r (K, ω x y 0 and compare the relative errors of using these spaces to local polynomials spaces P r in our scheme. The relative errors are shown in Table 1, which suggest that the numerical solutions are second and third order accurate in h, for r = 1, 2, respectively. 5.2 Example 2 In this example, we consider the domain Ω = [0, 1] 2 with wavenumber κ(x = 100. We define the points y 1, y 2, y 3 to be (0.3, 0.1, (0.7, 0.1, (0.5, 1.1, respectively. Also, we define cylindrical waves centered at y k by H k,100 (x := H (1 0 (100 x y k. We consider the following solutions of the Helmholtz problem (1 and (2 with f = 0 and corresponding g: u 1,100 (x = H 1,100 (x + H 2,100 (x, u 2,100 (x = H 1,100 (x + H 2,100 (x + H 3,100 (x, and using our scheme to solve for these solution separately. For k = 1, 2, we solve for u k,100 (x using local spaces Q 1 k,100 defined as follows: Q 1 1,100(K := Q 1 ( K, (100 x y 1, 100 x y 2 Q 1 2,100(K := Q 1 ( K, (100 x y 1, 100 x y 2, 100 x y 3. We compare the relative errors of using the local approximation space Q 1 1,100(K with P 2 (K where the number of degrees of freedom per element are both 6 for these spaces. Also, we compare the relative error using the local approximation space Q 1 2,100(K with P 3 (K where the number of degrees of freedom per element for these spaces are 9 and 10 respectively. The relative errors are shown in Table 2, which suggest that the numerical solutions for u 1,100 (x and u 2,100 (x are second and third order accurate in h, respectively. 5.3 Example 3 We consider Ω = [0.5, 1.5] 2 with κ(x = 2ωx 1, and impose the solution u 3,ω (x = e x2 e iωx2 1, 15

16 where x = (x 1, x 2. The source term is f(x = (1 + 2iωe x2 e iωx2 1 and the function g is taken accordingly. Note that we have max x Ω κ 1 κ (x (1/2, 1/2 T = 2/3, which shows the second assumption in (K is satisfied. We use the following local space to solve for u 3 : Q r 3,ω(K := Q r (K, ωx 2 1. The relative errors are shown in Table 3, which suggest that the numerical solutions are second and third order accurate in h, for r = 1, 2, respectively. 5.4 Example 4 With the same Ω and κ in Example 3, we impose the solution u 4 (x = e x2 e iωx2 1 + e iωx 2, where x = (x 1, x 2. The source term f and the function g are taken accordingly. We use the following local space to solve for u 4 : Q 1 4,ω(K := Q 1 (K, (ωx 2 1, ωx 2. The relative errors are shown in Table 4, which suggest that the numerical solutions are second order accurate in h. 5.5 A study of some constants In this section we compute the constants in the trace inverse inequality (2 and the inverse estimates in Proposition 3.4. We will also compute the constant in the third assumption of (Q for a given k 0 in order to justify that the spaces in our examples satisfy the assumption. For the trace inverse inequality, we consider the stiffness matrix S and the matrix S given by (S ij := v i v j dx. We solve the following generalized eigenvalue problem: K (hs x = λ 1 Sx, and take C tinv,h as the maximum of λ 1. We note that the stiffness matrix S is not necessarily invertible, e.g. the stiffness matrices for P r (K. For local spaces with single phase, the stiffness matrix is singular only if e iq1 is a polynomial, which is not the case of our local spaces. For our choices of local space with multiple phases, it can be checked that all the stiffness matrices are invertible. Similarly, for the inverse estimates we consider the stiffness matrix S, the matrix S 2 corresponding to the 2 2, and the mass matrix M. We solve the following generalized eigenvalue problems: (h 2 Sx = λ 2 Mx, 16

17 u0,1 u0,10 u0,100 u3,1 u3,10 u3,100 u1,100 u2,100 u4,100 Figure 1: Plots for the real part of the exact solution in the examples. (h4 S2 x = λ3 M x, and take Cinv,1,h and Cinv,2,h as the maximum of λ2 and λ3, respectively. Also, in order to show the third assumption in (Q is satisfied for the local spaces with multiple phases, that is Q11,100, Q12,100 and Q14,100, we consider P x = λ4 M x, and take CQ,h as the maximum of λ4. Here P is the matrix associated with Pm 2 1 `=1 kp` k0,k, p` P (K. The results are shown in Tables 5, 6, 7,. 6 Conclusion In this paper, we prove some properties of a type of local approximation space consisting of polynomial-modulated phase functions and derive an a priori estimate for an IPDG method incorporating these spaces. We provide numerical results to demonstrate the efficiency of the local spaces of polynomial-modulated phase function with the knowledge of the phase values. The error levels obtained 17

18 with such local spaces are usually several orders of magnitude lower than those with standard polynomial spaces. In the future, we would like to further investigate the method by looking for more concrete classes of local approximation spaces satisfying our assumptions. References [1] M. Amara, R. Djellouli, and C. Farhat. Convergence analysis of a discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of Helmholtz problems. SIAM Journal on Numerical Analysis, 47(2: , [2] I. Babuška and J. M. Melenk. The partition of unity method. International Journal for Numerical Methods in Engineering, 40(4:727 75, [3] I. M. Babuška and S. A. Sauter. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Journal on Numerical Analysis, 34(6: , [4] A. Bayliss, C. I. Goldstein, and E. Turkel. The numerical solution of the Helmholtz equation for wave propagation problems in underwater acoustics. Computers & Mathematics with Applications, 11(7: , 195. [5] T. Betcke and J. Phillips. Approximation by dominant wave directions in plane wave methods Preprint, available at ac.uk/id/eprint/ [6] A. Buffa and P. Monk. Error estimates for the ultra weak variational formulation of the Helmholtz equation. Modélisation Mathématique et Analyse Numérique, 42(6: , 200. [7] O. Cessenat and B. Despres. Application of an ultra weak variational formulation of elliptic pdes to the two-dimensional Helmholtz problem. SIAM Journal on Numerical Analysis, 35(1: , 199. [] C. Farhat, I. Harari, and L. P. Franca. The discontinuous enrichment method. Computer Methods in Applied Mechanics and Engineering, 190(4: , [9] C. Farhat, R. Tezaur, and P. Weidemann-Goiran. Higher-order extensions of a discontinuous Galerkin method for mid-frequency Helmholtz problems. International Journal for Numerical Methods in Engineering, 61(11: , [10] E. Giladi and J. B. Keller. A hybrid numerical asymptotic method for scattering problems. Journal of Computational Physics, 174(1: , [11] C. J. Gittelson and R. Hiptmair. Dispersion analysis of plane wave discontinuous Galerkin methods. International Journal for Numerical Methods in Engineering, 9(5: ,

19 [12] C. J. Gittelson, R. Hiptmair, and I. Perugia. Plane wave discontinuous Galerkin methods: analysis of the h-version. ESAIM: Mathematical Modelling and Numerical Analysis, 43(02: , [13] R. Hiptmair, A. Moiola, and I. Perugia. Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM Journal on Numerical Analysis, 49(1:264 24, [14] R. Hiptmair, A. Moiola, and I. Perugia. Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version. Foundations of Computational Mathematics, pages 1 39, [15] R. Hiptmair, A. Moiola, and I. Perugia. A survey of Trefftz methods for the Helmholtz equation. arxiv preprint arxiv: , [16] F. Ihlenburg. Finite element analysis of acoustic scattering, volume 132. Springer Science & Business Media, [17] L.-M. Imbert-Gérard. Interpolation properties of generalized plane waves. Numerische Mathematik, 131(4:63 711, [1] L.-M. Imbert-Gérard and B. Després. A generalized plane-wave numerical method for smooth nonconstant coefficients. IMA Journal of Numerical Analysis, 34(3: , [19] S. Langdon and S. N. Chandler-Wilde. A wavenumber independent boundary element method for an acoustic scattering problem. SIAM Journal on Numerical Analysis, 43(6: , [20] J. M. Melenk. On generalized finite element methods. PhD thesis, The University of Maryland, [21] J. M. Melenk. Operator adapted spectral element methods i: harmonic and generalized harmonic polynomials. Numerische Mathematik, 4(1:35 69, [22] J. M. Melenk and I. Babuška. The partition of unity finite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering, 139(1:29 314, [23] J. M. Melenk and S. Sauter. Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions. Mathematics of Computation, 79(272: , [24] J. M. Melenk and S. Sauter. Wavenumber explicit convergence analysis for Galerkin discretizations of the helmholtz equation. SIAM Journal on Numerical Analysis, 49(3: , [25] P. Monk and D. Q. Wang. A least-squares method for the Helmholtz equation. Computer Methods in Applied Mechanics and Engineering, 175(1: , [26] N. C. Nguyen, J. Peraire, F. Reitich, and B. Cockburn. A phase-based hybridizable discontinuous Galerkin method for the numerical solution of the Helmholtz equation. Journal of Computational Physics, 290:31 335,

20 [27] P. Ortiz and E. Sanchez. An improved partition of unity finite element model for diffraction problems. International Journal for Numerical Methods in Engineering, 50(12: , [2] W. Rundell, A. K. Louis, and H. Engl. Inverse Problems in Medical Imaging and Nondestructive Testing. Springer-Verlag New York, Inc., [29] A. H. Schatz. An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Mathematics of Computation, 2(12: , [30] M. Stojek. Least-squares Trefftz-type elements for the Helmholtz equation. International Journal for Numerical Methods in Engineering, 41(5:31 49, 199. [31] T. Strouboulis, I. Babuška, and R. Hidajat. The generalized finite element method for Helmholtz equation: Theory, computation, and open problems. Computer Methods in Applied Mechanics and Engineering, 195(37 40: , John H. Argyris Memorial Issue. Part I. 20

21 1/h Local ω = 1 ω = 10 ω = 100 space Error Order Error Order Error Order 4.353e e e e e e e e e Q 1 0,ω 7.015e e e e e e e e e e e e e e e e e e e e e P e e e e e e e e e e e e e e e e e e e e e Q 2 0,ω 1.641e e e e e e e e e e e e e e e e e e e e e P e e e e e e e e e e e e Table 1: In Example 1, the relative error of the numerical solution of u 0,ω using local spaces Q r 0,ω and P r with r = 1, 2 and different ω. 21

22 Solve for u 1,100 Solve for u 2,100 1/h Local Local Error Order space space Error Order 7.44e e e e e e Q 1 1, e Q 1 2, e e e e e e e e e e e e e P e P e e e e e e e Table 2: In Example 2, the relative error of the numerical solution for u 1,100 using spaces Q 1 1,100 and P 2, and u 2,100 using spaces Q 1 2,100 and P 3 with different meshes size h. 22

23 1/h Local ω = 1 ω = 10 ω = 100 space Error Order Error Order Error Order 1.040e e e e e e e e e Q 1 3,ω 1.20e e e e e e e e e e e e e e e e e e e e e P 1.16e e e e e e e e e e e e e e e e e e e e e Q 2 3,ω 4.636e e e e e e e e e e e e e e e e e e e e e P e e e e e e e e e e e e Table 3: In Example 3, the relative error of the numerical solution of u 3,ω using spaces Q r 3,ω with r = 1, 2 and different ω. We compare the errors to those of using polynomial local space P r. 23

24 1/h Local space Error Order 6.744e e e Q 1 4, e e e e e e e P e e e e Table 4: In Example 4, the relative error of the numerical solution for u 4,100 using spaces Q 1 4,100 and P 2 with different mesh size h. 1/h Local ω = 1 ω = 10 ω = 100 space C tinv,h C inv,1,h C tinv,h C inv,1,h C tinv,h C inv,1,h Q ,ω Q ,ω Local ω = 1 ω = 10 ω = 100 1/h space C inv,2,h C inv,2,h C inv,2,h Q ,ω Table 5: In Example 1, different numerical values for the constants in the trace inverse inequality and the inverse inequality for the local space Q 1 0,ω and Q 2 0,ω with different mesh size h and ω. 24

25 Local 1/h C tinv,h C inv,1,h h 3 Local C Q,h C tinv,h C inv,1,h h 4 C space space Q,h e Q ,100 Q e-02 2, e e-02 Table 6: In Example 2, different numerical values for the constants in the trace inverse inequality, the inverse inequality and the third assumption in (Q for the local space Q 1 1,100 and Q 1 2,100 with different mesh size h 1/h Local ω = 1 ω = 10 ω = 100 space C tinv,h C inv,1,h C tinv,h C inv,1,h C tinv,h C inv,1,h Q ,ω Q ,ω Local ω = 1 ω = 10 ω = 100 1/h space C inv,2,h C inv,2,h C inv,2,h Q ,ω Table 7: In Example 3, different numerical values for the constants in the trace inverse inequality and the inverse inequality for the local space Q 1 0,ω and Q 2 0,ω with different mesh size h and ω. 1/h Local space C tinv,h C inv,1,h h 3 C Q,h e Q e-03 4, e e-03 Table : In Example 4, different numerical values for the constants in the trace inverse inequality, the inverse inequality and the third assumption in (Q for the local space Q 1 1,100 and Q 1 2,100 with different mesh size h. 25