A HYBRIDIZABLE WEAK GALERKIN METHOD FOR THE HELMHOLTZ EQUATION WITH LARGE WAVE NUMBER: hp ANALYSIS

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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 4, Number 4-5, Pages c 7 Institute for Scientific Comuting and Information A HYBRIDIZABLE WEAK GALERKIN METHOD FOR THE HELMHOLTZ EQUATION WITH LARGE WAVE NUMBER: h ANALYSIS JIANGXING WANG AND ZHIMIN ZHANG Abstract. In this aer, an h hybridizable weak Galerkin (h-hwg) method is introduced to solve the Helmholtz equation with large wave number in two and three dimensions. By choosing a secific arameter and using the duality argument, we rove that the roosed method is stable under certain mesh constraint. Error estimate is obtained by using the stability analysis and the duality argument. Several numerical results are rovided to confirm our theoretical results. Key words. Weak Galerkin method, hybridizable method, Helmholtz equation, large wave number, error estimates.. Introduction In this aer, we develo an h-version hybridizable weak Galerkin (h-hwg) method to solve the Helmholtz equation with Robin boundary condition: (a) (b) (c) u+κ u = f in Ω, u = u on Γ, u n +iκu = g on Γ, where Ω R d, d =,3, is a bounded convex Lischitz domain, Γ and Γ form a artition of the boundary Ω, κ > is the wave number, i = is the imaginary unit, and n denotes the unit outward normal to Ω. The condition (c) is the first order aroximation of the radiation condition for Helmholtz scattering roblem. The Helmholtz equation has imortant alications in electrodynamics, esecially in otics and acoustics involving time harmonic wave roagation. The Helmholtz system is not ositive definite. When the wave number κ, the solution is highly oscillatory. It is very challenging to design an efficient numerical method to solve the Helmholtz equation with high wave number. In the literature, there have been extensive investigations devoted to numerical aroximations for Helmholtz equations with various boundary conditions. In articular, the finite element method (FEM) has been widely used [3, 7, 7, 8,,, 35]. It has been shown that the H -errors of th order FEM solutions to the Helmholtz equation have accuracy order O(κ + h ) [,, 35, 36]. In [7], Wu et al. analyzed the reasymtotic error of high order FEM and continuous interior enalty FEM (CIP-FEM) for Helmholtz equation with large wave number. They roved that, when κ + h is sufficiently small, the ollution errors are of order k + h. Discontinuous Galerkin methods have also been used to solve Helmholtz equations [8,,, 3,, 3]. Detailed analyses have been carried out in [, ] on the discrete disersive relation by h-fem and high-order discontinuous Galerkin methods. In [8, 9], Shen and Wang used the sectral method to solve Received by the editors on December 3, 6, acceted on May 5, 7. Mathematics Subject Classification. 65N5, 65N3, 35J5. 744

2 h HWG METHOD FOR THE HELMHLOTZ EQUATION 745 the Helmholtz equation in both interior and exterior domains. Their results indicate that high-order methods are referable, if not necessary, for highly oscillatory roblems. In [4, 5, 4], hybridizable discontinuous Galerkin methods were used to solve the Helmholtz equation. The weak Galerkin (WG) method was first introduced by Wang and Ye [3] for second-order ellitic equations. It can be derived from the variational form of the continuous roblem by relacing derivatives involved by weak derivatives with some stabilizers. WG methods have been alied to solve many roblem [, 3, 4, 5, 6, 3, 3, 33, 34]. The HWG method [7] was introduced by Mu et al., which alies Lagrange multilier so that the comutational comlexity can be significantly reduced. In this aer, we will develo an h-hwg method to solve the Helmholtz equation with high wave number. The main difficulty in the analysis of the numerical method is due to the strong indefiniteness of the Helmholtz equation. As a consequence, the stability of the numerical aroximation is hard to establish. In this work, we use the duality argument to show that the roosed h-hwg method is stable under roer mesh condition. This stability result not only guarantees the existence of the HWG method but also lays an imortant role in the error analysis. In articular, we first construct an auxiliary roblem and establish its h-hwg error estimates; then we combined the estimates with the stability result to derive the error estimates of the h-hwg scheme for the original Helmholtz roblem. Notation. In this aer, standard notations for Sobolev saces (e.g., L (Ω), H k (Ω) for k N, etc.) and the associated norms and seminorms will be adoted. Plain and bold fonts are used for scalars and vectors, resectively. The rest of this aer is organized as follows. The h-hwg scheme for the Helmholtz equation is develoed in Section. Section 3 is devoted to show the stability result of the numerical scheme. In Section 4, we derive the error estimate of the numerical scheme. Numerical results are given in Section 5 to confirm the theoretical results.. Weak Divergence and the h-hwg Scheme.. Weak divergence. Let K be a subdomain in Ω. A weak vector-valued function on K refers to a vector field v = {v,v b }, where v [L (K)] d carries the information of v in K, and v b [L ( K)] d reresents artial or full information of v on K. It is imortant to oint out that v b may not necessarily be related to the trace of v on K, but shall be well-defined. Denote by V(K) the sace of all weak vector-valued functions on K; that is V(K) = {v = {v,v b } : v [L (K)] d,v b [L ( K)] d }. A weak divergence can be taken for any vector field in V(K) by following the definition [7]. Definition.. For any v V(K), the weak divergence of v, denoted by w v, is defined as a linear functional on H (K), whose action on each φ H (K) is given by () ( w v,φ) K = (v, φ) K + < v b n,φ > K, where (, ) K and <, > K stand for the inner roducts in L (K) and L ( K), resectively. Next, we introduce a discrete weak divergence oerator ( w,k ) in a olynomial subsace of the dual of H (K) [7].

3 746 J. WANG AND Z. ZHANG Definition.. For any v V(K), the discrete weak divergence of v, denoted by ( w,k v), is defined as the unique olynomial in P k (K) satisfying (3) ( w,k v,ψ) K = (v, ψ) K + < v b n,ψ > K, ψ P k (K). Here, for k, P k (K) is the set of olynomials on K with total degree u to k... Numerical scheme. Let T h be aartitionofω, with ossiblehangingnodes. For an element K T h, denote by h K = diam(k) the diameter of K. Let h = max K Th h K be the mesh size of T h. Denote by E h = K Th K the skeleton of the mesh, and set E B h = E h Ω and E I h = E h \E B h. To introduce the h-hwg method, we rewrite (a)-(c) as a first order system (4a) (4b) (4c) (4d) iκq+ u = in Ω, iκu+divq = f in Ω, u = u on Γ, q n+u = g on Γ. Multilying v [L (Ω)] d and w L (Ω) to (4a) and (4b), resectively, and using integration by arts, we get the weak formulation (5a) (5b) iκ(q,v) K ( v,u) K + < u,v n > K =, iκ(u,w) K +(divq,w) K = (f,w) K. For each element K T h, let n be the outwardnormal direction to the boundary K. Let W k (K) = P k (K) and V k (K) = {v = {v,v b } : v [P k (K)] d,v b e = v b n for v b P k (e),e K}. On the wired-basket E h, define a finite element sace Let Λ h be a subset of Λ such that Λ h = {λ : λ e P k (e), e E h }. Λ h = {λ Λ h : λ e =, e E h Γ }. Define discontinuous finite element saces V h = {v [L (Ω)] d : v K V k (K), K T h }, W h = {w L (Ω) : w K W k (K), K T h }. Further, for the mesh T h, we define (u,v) Th = (u,v) K, (u,v) Th = (u,v) K, < u,v > Th = < u,v > K,< u,v > Γ = < u,v > e. e Eh B Γ Our hybridizable weak Galerkin method is: find q h = {q,q b } V h, u h W h, λ Λ h, such that λ = Π b u on Γ and (6a) (6b) (6c) (6d) S t (q h,v)+iκ(q,v ) Th ( w,k v,u h ) Th + < λ,v b n > Th =, iκ(u h,w) Th +( w,k q h,w) Th = (f,w) Th, < q b n,φ > Th \ Ω=, < q b n+λ,φ > Γ =< g,φ > Γ,

4 h HWG METHOD FOR THE HELMHLOTZ EQUATION 747 for all v V h, w W h and φ Λ h, where S t (q h,v) = S K (q h,v), S K (q h,v) = iτ < (q q b ) n,(v v b ) n > K, is the element-wise stabilizer, τ is a arameter and Π b is the L rojection oerator which will be defined later. Notice that u h and q are resectively the aroximations of u and q in element K, λ and q b are resectively the aroximations of u and q on K. 3. Stability Analysis Lemma 3.. Let (q h,u h,λ) V h W h Λ h be the solution of (6a) (6d). If u =, then (7) (8) λ,γ f,ω u h,ω + g,γ, τ (q q b ) n, T h +κ q,ω κ u h,ω + f,ω u h,ω + g,γ. Proof. Choosing v = q h,w = u h,φ = λ in (6a) (6d), we get (9a) (9b) (9c) (9d) S t (q h,q h )+iκ(q,q ) Th ( w,k q h,u h ) Th + < λ,q b n > Th =, iκ(u h,u h ) Th +( w,k q h,u h ) Th = (f,u h ) Th, < q b n,λ > Th \ Ω=, < q b n+λ,λ > Γ =< g,λ > Γ. Summing (9a) (9d) u, and using the fact that ab = ab, we get S t (q h,q h )+iκ(q,q ) Th iκ(u h,u h ) Th + < λ,λ > Γ =< g,λ > Γ +(f,u h ) Th. Taking the real art and the imaginary art of the above equality, we arrive at λ,γ f,ω u h,ω + g,γ, τ (q q b ) n, T h +κ q,ω κ u h,ω + f,ω u h,ω + g,γ. It finishes the roof of this lemma. To estimate u h,ω, we will use a duality argument. Given u h L (Ω), we define (a) (b) (c) (d) The following result is found in [4]. iκφ+ Ψ = in Ω, Φ iκψ = u h in Ω, Ψ = on Γ, Φ n = Ψ on Γ. Lemma 3.. For Φ and Ψ defined in (), they admit the following estimate: () Ψ,Ω +κ Ψ,Ω +κ Ψ,Ω + Ψ,Γ +κ Φ,Ω C u h,ω. Let Π be the standard L rojection form L (K) onto P k (K), Π = (Π ) d and Π b be the standard L rojection form L (e) onto P k (e), where K T h, e E h. Let Π h : V(K) V k (K) be defined as Π h v = {Π v, (Π b v b )n}, v = {v,v b } V(K). The following estimates have been established in [5, 9].

5 748 J. WANG AND Z. ZHANG Lemma 3.3. For any u H s+ (K), s,s N, u Π u,k + h (u Π u),k C(h/) s+ u s+,k, u Π u, K Ch s+ K (+) (s+ ) u s+,k. Moreover, we have the following results. Lemma 3.4. () () For all w P k (K), q [H (Ω)] d, we have ( w,k (Π h q),w) K = ( q,w) K < q n Π b (q n),w > K. () For all v = {v,v b } V k (K), w H (K), we have ( w,k v,π w) K = (v, w) K + < (v v b ) n,w Π w > K (3) + < v b n,w > K. Proof. The results follow the same line as in [33]. Nowwearereadytoderivethe stabilityestimateofu h, whichlaysanimortant role in the error analysis of the Helmholtz equation. Theorem 3.. If κ h/ and u =, then there exists a constant C indeendent of κ, and h, such that u h,ω C( f,ω + g,γ ). Proof. Using (b) and Green s lemma, we get (u h,u h ) Th = (u h, Φ iκψ) Th Together with (), we have (4) = ( u h,φ) Th + < u h,φ n > Th +iκ(u h,ψ) Th. (u h,u h ) Th =( w,k (Π h Φ),u h ) Th + <u h,φ n Π b (Φ n)> K +iκ(u h,ψ) Th. The combination of (6a), (6b) and (4) leads to (u h,u h ) Th = (5) < u h,φ n Π b (Φ n) > K +(f,π Ψ) Th ( w,k q h,π Ψ) Th +S t (q h,π h Φ)+iκ(q,Π Φ) Th + < λ,π b (Φ n) > K. Using (a) and (3), we have (6) iκ(q,π Φ) Th = iκ(q,φ) Th = iκ(q, i κ Ψ) T h = (q, Ψ) Th, ( w,k q h,π Ψ) Th = (q, Ψ) Th + + < q b n,ψ > Ω. < (q q b ) n,ψ Π Ψ > K By the definition of S t (w,v), we have, for any w [H (Ω)] d, Hence, (7) S t (w,v) = v V h. S t (Π h Φ,v) = S t (Π h Φ Φ,v).

6 h HWG METHOD FOR THE HELMHLOTZ EQUATION 749 The combination of (5) (7) and the definition of Π b lead to (8) (u h,u h ) Th = (f,π Ψ) Th +S t (q h,π h Φ Φ)+ < Π b u,π b (Φ n) > Γ < (q q b ) n,ψ Π Ψ > K + < g,π Ψ > Γ. Hence, if u =, we get the following estimate u h,ω f,ω Ψ,Ω + g,γ Ψ,Γ + (q q b ) n, Th (h/) 3 Ψ,Ω +τ (q q b ) n, Th (h/) Φ,Ω. Using Lemma 3. and Lemma 3., we have u h,ω f,ω u h,ω + g,γ u h,ω +(τ κ (h/) 3 +τ κ(h/) ) ) (κ uh,ω + f,ω u h,ω + g,γ u h,ω. We choose τ = κh to get the minimum of term τ κ (h/) 3 +τ κ(h/). Finally, we obtain u h,ω C( f,ω + g,γ )+κ h u h,ω. It follows that, if κ h/ <, then 4. Error Analysis u h,ω C( f,ω + g,γ ). In this section, we will carry out the error analysis of the h-hwg method. For this urose, we first consider the following auxiliary roblem: (9a) (9b) (9c) (9d) iκq+ U = in Ω, Q iκu = f iκu in Ω, U = u on Γ, Q n+u = g on Γ, where u is the solution of (4a) (4d), and u,f, and g are the same as in (4a) (4d). The h-hwg scheme for (9a) (9d) is: find (Q h,u h,λ h ) V h W h Λ h such that λ h = Π b u on Γ and (a) (b) (c) (d) S t (Q h,v)+iκ(q,v ) Th ( w,k v,u h ) Th + < λ h,v b n > Th =, iκ(u h,w) Th +( w,k Q h,w) Th = (f iκu,w) Th, < Q b n,φ > Th \ Ω=, < Q b n+λ,φ > Γ =< g,φ > Γ, for all v V h,w W h,φ Λ h.

7 75 J. WANG AND Z. ZHANG Denote θ q = q h Q h,θ u = u h U h and θ λ = λ λ h. According to (6a) (6d) and (a) (d), we have θ λ = on Γ and (a) (b) (c) (d) S t (θ q,v)+iκ(θ q,v ) Th ( w,k v,θ u ) Th + < θ λ,v b n > Th =, iκ(θ u,w) Th +( w,k θ q,w) Th = iκ(u h u,w) Th, < θq b n,φ > Th \ Ω=, < θq b n+θ λ,φ > Γ =. By Theorem 3., in order to get the estimates of θ q,θ u and θ λ, we need to estimate u U h. 4.. Error estimates of the auxiliary roblem. Direct calculation leads to the following error equations of the h-hwg method for (9a) (9d): S t (R q,v)+iκ(r q,v ) Th ( w,k v,π u U h ) Th (a) + < (v v b ) n,u Π u > K + < v b n,u λ h > K =, (b) iκ(r u,w) Th +( w,k (Π h q Q h ),w) Th + < q n Π b (q n),w > K =, (c) < q n Q b n,φ > Th \ Ω=, (d) < q n Q b n,φ > Γ + < u λ h,φ > Γ =. In the current alication, we will emloy the following decomosition: R q = q Q h = q Π h q (Q h Π h q), R u = u U h = u Π u (U h Π u), ζ q = Q h Π h q = {ζ,ζ b }, ξ u = u Π u, η u = U h Π u. Lemma 4.. Suose that u H s+ (Ω) and q [H s+ (Ω)] d are smooth functions on Ω. Then, the following estimates hold true: (3a) <(v v b ) n,π u u> K (3b) S t (q Π h q,q h Π h q) Cτ for all v V h. ( ) s+ h ( u s+,ω (v v b ) n, K ( ) s+ h ( ) q s+,ω τ (ζ ζ b ) n, K, ),

8 h HWG METHOD FOR THE HELMHLOTZ EQUATION 75 Proof. By Cauchy-Schwarz inequality, we arrive at < (v v b ) n,π u u > K ( ) ( (v v b ) n, K C(h/) s+ u s+,ω ( Using the definition of S t (v,r), we have S t (q Π h q,q h Π h q) (τ q Π h q, K ) (τ Cτ (h/) s+ q s+,ω ( It comletes the roof of this lemma. Π u u, K (v v b ) n, K ). ) (ζ ζ b ) n, K ) τ (ζ ζ b ) n, K Theorem 4.. Let u H s+ (Ω), q [H s+ (Ω)] d be the solution of (4a) (4d). Let Q h, U h be the solution of (a) (d). If κ h/ <, then there exists a constant C indeendent of κ, h and such that κ ζ,ω +κ ηu,ω +τ (ζ ζ b ) n, Th ( C(+τ h ) s+ ( ) ) u s+,ω + q s+,ω. Proof. Taking v = Q h Π h q,w = U h Π u and φ = Π b u λ h in (a) (d), resectively, we get ). (4a) S t (R q,q h Π h q)+iκ(r q,ζ ) Th ( w,k (Q h Π h q),π u U h ) Th + < (ζ ζ b ) n,u Π u > K + < ζ b n,u λ h > K =, (4b) (4c) (4d) iκ(r u,u h Π u) Th +( w,k (Π h q Q h ),U h Π u) Th + < q n Π b (q n),u h Π u > K =, < q n Q b n,π b u λ h > Th \ Ω=, < q n Q b n,π b u λ h > Γ + < u λ h,π b u λ h > Γ =. Summing the conjugate of (4b) and (4a) together, we get S t (R q,q h Π h q)+iκ(r q,ζ ) Th + < (ζ ζ b ) n,u Π u > Th +iκ(r u,u h Π u) Th + < ζ b n,u λ h > K + < q n Π b (q n),u h Π u > K =. Meanwhile, summing (4c) and (4d) together, we have < q n Q b n,π b u λ h > K =< u λ h,π b u λ h > Γ, where we have used the fact that u = u,λ h = Π b u on Γ.

9 75 J. WANG AND Z. ZHANG By using the orthogonality roerties of Π,Π and Π b, we conclude from the oeratordecomositions of R q and R u that the two equations abovecan be written as S t (R q,q h Π h q) iκ(ζ,ζ ) Th (5) (6) +<(ζ ζ b ) n,u Π u> Th iκ(η u,u h Π u) Th + < ζ b n,λ h Π b u > Th + < q n Π b (q n),u h Π u > Th =, < Π b (q n) Q b n,π b u λ h > Th =< Π b u λ h,π b u λ h > Γ. Substituting (6) into (5), one has (7) iκ(ζ,ζ ) Th +iκ(η u,u h Π u) Th + < Π b u λ h,π b u λ h > Γ = S t (R q,q h Π h q)+ < (ζ ζ b ) n,u Π u > Th. Taking the imaginary art of (7), we get (8) κ ζ,ω +κ η u,ω +τ (ζ ζ b ) n, T h S t (q Π h q,q h Π h q) + < (ζ ζ b ) n,u Π u > Th. Using lemma 4. and (8), we arrive at κ ζ,ω +κ ηu,ω +τ (ζ ζ b ) n, Th ( h ) s+ C(+τ ( ) ) u s+,ω + q s+,ω. It comletes the roof of this Theorem. 4.. The estimate of η u. In this subsection, we will use the Aubin-Nitsche duality argument to get an error estimate of u U h. Consider the following dual roblem, (9a) (9b) (9c) (9d) iκφ+ Ψ = in Ω, Φ+iκΨ = η u in Ω, Ψ = on Γ, Φ n = Ψ on Γ. The following result is found in [4]. Lemma 4.. Let Ψ, Φ be the solution of (9a) (9d). Then, Φ,Ω + Ψ,Ω +κ Ψ,Ω C η u,ω.

10 h HWG METHOD FOR THE HELMHLOTZ EQUATION 753 Lemma 4.3. Let u H s+ (Ω) and q [H s+ (Ω)] d be the solution of (4a) (4d). Then, we have the following estimates: (3a) S t (q Π h q,π h Φ) Cτ( h )s+ q s+,ω Φ,Ω, (3b) ( S t (Π h q Q h,π h Φ) Cτ h ) s+ ( ) (+τ ) u s+,ω + q s+,ω Φ,Ω, (3c) < Π (Φ n) Π b (Φ n),π u u > K C( h )s+ u s+,ω Φ,Ω, (3d) < q n Π b (q n),π Ψ Ψ > K C( h )s+ q s+,ω Ψ,Ω. Proof. By the definition of S t (, ), S t (q Π h q,π h Φ) = S t (q Π h q,π h Φ) ( ) ( Cτ q Π h q,e e E h On the other hand, Cτ( h )s+ q s+,ω Φ,Ω. S t (Π h q Q h,π h Φ) = S t (Π h q Q h,π h Φ Φ) ( ) ( Cτ (ζ ζ b ) n,e e E h Meanwhile, e E h Π h Φ,e ) e E h Π h Φ Φ,e ) ( Cτ h ) s+ ) (+τ ) ( u s+,ω + q s+,ω Φ,Ω. < Π (Φ n) Π b (Φ n),π u u > K = < Π (Φ n) Φ n+φ n Π b (Φ n),π u u > K C( h )s+ u s+,ω Φ,Ω. Then, we turn to the estimate of (3d), < q n Π b (q n),π Ψ Ψ > K ( ) ( q n Π b (q n), K C( h )s+ q s+,ω Ψ,Ω. Π Ψ Ψ, K ) Hence, we finish the roof of this lemma.

11 754 J. WANG AND Z. ZHANG Theorem 4.. Let Q h,u h be the solution of (a) (d). Let u H s+ (Ω),q [H s+ (Ω)] d be the exact solution of (4a) (4d). If κ h/ <, then there exists a constant C indeendent of κ, h and, such that ( ) s+ h u U h,ω C ( u s+,ω + q s+,ω). Proof. Using (9b) and (), we get (η u,η u ) Th = (η u, Φ+iκΨ) Th Using (a), we have = ( w,k (Π h Φ),η u ) Th + < η u,φ n Π b (Φ n) > Th iκ(η u,ψ) Th. ( w,k (Π h Φ),η u ) Th =S t (R q,π h Φ)+iκ(R q,π Φ) Th + + < Π (Φ n) Π b (Φ n),u Π u > K. By the orthogonal roerty of Π and (9a), one has < Π b (Φ n),u λ h > K iκ(r q,π Φ) Th = iκ(ζ,φ) Th = iκ(ζ, i κ Ψ) T h = (ζ, Ψ) Th. Using (3), (b) and the orthogonal roerties of Π, we obtain i(η u,ψ) Th =i(η u,π Ψ) Th and = ( w,k (Π h q Q h ),Π Ψ) Th < q n Π b (q n),π Ψ > K, ( w,k (Π h q Q h ),Π Ψ) Th = (ζ, Ψ) Th + < ζ b n,ψ > Ω + < (ζ ζ b ) n,ψ Π Ψ > K. The combination of the above equations leads to (η u,η u ) Th =S t (R q,π h Φ)+iκ(R q,π Φ) Th + + < Π (Φ n) Π b (Φ n),u Π u > K < Π b (Φ n),u λ h > K (3) + < η u,φ n Π b (Φ n) > Th iκ(η u,ψ) Th =S t (R q,π h Φ) (ζ, Ψ) Th + + < Π b (Φ n),u λ h > K < Π (Φ n) Π b (Φ n),u Π u > K + < η u,φ n Π b (Φ n) > Th + < q n Π b (q n),π Ψ > K +(ζ, Ψ) Th < ζ b n,ψ > Ω < (ζ ζ b ) n,ψ Π Ψ > K.

12 h HWG METHOD FOR THE HELMHLOTZ EQUATION 755 (3) By definitions of Π b and λ h, we have < Π b (Φ n),u λ h > K Using (d) and (9c), it follows (33) = < Π b (Φ n),u λ h > Γ = < Π b (Φ n),π b u λ h > Γ. < ζ b n,ψ > Ω = < ζ b n,ψ > Γ, < Π b (Φ n),π b u λ h > Γ = < Π b (Φ n),q n Q b n > Γ The combination of (3) and (33) leads to (34) = < Π b (Φ n),ζ b n > Γ. < Π b (Φ n),u λ h > K < ζ b n,ψ > Ω = < Π b Ψ Ψ,ζ b n > Γ. Substituting (34) into (3), we get from the roerty of Π b that (η u,η u ) Th = S t (R q,π h Φ)+ Together with lemma 4.3, we get ( h η u,ω Cτ < Π (Φ n) Π b (Φ n),u Π u > K < (ζ ζ b ) n,ψ Π Ψ > K. ) s+ q s+,ω Φ,Ω +Cτ (+τ ) ( h +C ) s+ ( u s+,ω + q s+,ω ) Φ,Ω ( ) s+ h u s+,ω Φ,Ω +C ( ) s+ h q s+,ω Ψ,Ω. Recall that τ = κh/ and κ h/ <. Using lemma 4., we obtain ( ) s+ h η u,ω C ( u s+,ω + q s+,ω). The desired result follows from an alication of the triangle inequality. Theorem 4.3. Let q [H s+ (Ω)] d, u H s+ (Ω) be the solution of (4a) (4d), and q h, u h be the solution of (6a) (6d). If κ h/ <, then there exists a constant C indeendent of κ, h and, such that u u h,ω C(+κ) q q h,ω C Proof. Direct calculation leads to ( h ( κ + h + κh ) s+ ( u s+,ω+ q s+,ω), ) (h ) s ( u s+,ω + q s+,ω ). u u h = u Π u+(π u U h )+(U h u h ), q q h = q Π q+(π q Q h )+(Q h q h ). Then, the combination of Theorem 3., Theorem 4. and the triangle inequality comletes the roof of theorem.

13 756 J. WANG AND Z. ZHANG Table. Examle Numerical errors and their convergence rates when κ = and τ =. N Error ur order Error ui order Error qr order Error qi order e e e e e e e e e e e e e e e e e e e e e- 5.99e e e e e e e e e e e e e e e e e e e e-.4473e-.886e-.393e e e e e e e e e e e e e e e e e e-.989e-.865e-.859e e e e e e e e e e e e e e e e e Table. Examle Numerical errors and their convergence rates when κ = and τ =. N Error ur order Error ui order Error qr order Error qi order e-.6886e-.6667e 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 e e Table 3. Examle Numerical errors and their convergence rates when κh =. and τ =. κ N Error ur Error ui Error qr Error qi 5 4.8e-.98e-.33e-.e e-.833e-.36e-.48e e-.86e-.39e-.85e e-.389e-.597e-.85e e-.477e-.89e-.55e e-.5654e-.643e-.87e e e e- 6.76e- 5. Numerical Results We shall rovide several numerical examles to suort the theoretical analysis of the HWG scheme (6a) (6d). The exact solution can be written as u = ur+i ui.

14 h HWG METHOD FOR THE HELMHLOTZ EQUATION 757 Table 4. Examle Numerical errors and their convergence rates when κh =.5 and τ =. κ N Error ur Error ui Error qr Error qi e-.783e-.79e-.737e e-.868e-.7346e-.77e e-.94e-.896e-.7e e-.975e-.97e-.78e e-.377e-.89e-.855e e-.37e-.3368e-.9e e e e e- In this section, the L -errors are comuted as follows: ( Error ur = ur ur h dx ( Error ui = K K ui ui h dx Examle. Let Ω = [,]. We firstlyconsiderauredirichlet boundary value roblem (i.e. Γ = ). The exact solution is taken as u = ex( iκx). We choose the arameter τ = O(). Tables and show the errors and their convergent rates of the numerical solutions by the HWG scheme for κ = and κ =, resectively. It is noticed that some numerical convergence rates are suerior than the theoretical rates. On the other hand, in Tables 3 and 4, we rovide the data when κh/ is fixed as. and.5, resectively. it is observed that the errors u u h and q q h are not imroved when κh/ is fixed. Table 5. Examle Numerical errors for different κ values. ) )., κ Error ur Error ui Error qr Error qi e-6.57e e-4.e e e-8.548e e-7 3.7e e e e-.59e-.533e-.734e-.645e e e-4.496e e e-7.58e-7.436e-7.98e e e e- 7.98e e-3.78e-3.687e e-3.98e e e-6.76e-6 Examle. In this examle, we choose Ω = B \ S, where B = {(x,y) : x +y } and S = [.5,.5]. The exact solution is taken as u = ex( iκx). For simlicity, only Dirichlet boundary condition is alied. Table 5 resents the errors u h u and q h q for various κ values. Figures and are the surface lots of the HWG solutions u h with = 3, where κ = for the exact solution u, and four elements of Ω artitioned by y = ±x are used. It is evident that high order method is very effective for solving Helmholtz equations. Examle 3. Let Ω = [,], which is artitioned by a uniform mesh. We choose the inhomogeneous boundary conditions in such a way that the analytical solutions are the circular waves given, in olar coordinates x = (rcosθ,rsinθ), by u(x) = J ξ (κr)cos(ξθ), ξ,

15 758 J. WANG AND Z. ZHANG u Y u Y X.5 X Figure. Examle The numerical real art and exact real art for κ =, = u Y Y.5 u X.5 X Figure. Examle The numerical imaginary art and exact imaginary art for κ =, = 3. Table 6. Examle 3 Numerical errors and their convergence rates when κ = and ξ = N Errorur.633e e e e e e-8.94e e-7.38e e-8 order Errorqi.84e e e e e e e e e e-7 order where, Jξ denotes the Bessel function of the first kind of order ξ. If ξ N, u can be analytically extended to a Helmholtz solution in R. If ξ / N, its derivatives have a singularity at the origin. Then u H ξ+ ǫ (Ω) can be extended to a Helmholtz solution in R rovided ǫ >, but u / H ξ+ (Ω) [6].

16 h HWG METHOD FOR THE HELMHLOTZ EQUATION 759 u Y u Y X.6.8 X Figure 3. Examle 3 The numerical solution (Left) and the exact solution (Right) of u for κ =, ξ = 3, = 4, N = 3(error =.77 4 ). u Y.6.4. u Y X X Figure 4. Examle 3 The numerical solution of u for κ =, ξ = 3, =, N = 64(Left, error = ), =, N = 64(Right,error =.59 ). We comare the HWG solutions with the exact solutions for various κ and ξ on uniform meshes. Table 6 documents the errors u uh and q qh for κ = and ξ = 3. Figures 3 and 4 resent the surface lots of the HWG solutions uh and the exact solutions u for various κ and ξ. It is observed that high order methods are very effective. Acknowledgments The authors would like to thank the anonymous referees for the valuable comments and constructive suggestions to imrove this aer. This work was suorted in art by the following grants: NSFC 473 and 9436, NASF U534, NSF DMS-494, and a Tianhe JK comuting time award at Beijing Comutational Science Research Center.

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