Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number
AABSOLUTELY STABLE LOCAL DISCONTINUOUS GALERKINMETHODS FOR THE HELMHOLTZ EQUATION WITH LARGEWAVE NUMBER
XIAOBING FENG AND YULONG XING
Abstract.
Two local discontinuous Galerkin (LDG) methods using some non-standard numerical fluxes are developed for the Helmholtz equation with thefirst order absorbing boundary condition in the high frequency regime. Itis shown that the proposed LDG methods are absolutely stable (hence well-posed) with respect to both the wave number and the mesh size. Optimal order(with respect to the mesh size) error estimates are proved for all wave numbersin the preasymptotic regime. To analyze the proposed LDG methods, they arerecasted and treated as (non-conforming) mixed finite element methods. Thecrux of the analysis is to establish a generalized inf-sup condition, which holdswithout any mesh constraint, for each LDG method. The generalized inf-sup conditions then easily infer the desired absolute stability of the proposed LDGmethods. In return, the stability results not only guarantee the well-posednessof the LDG methods but also play a crucial role in the derivation of the errorestimates. Numerical experiments, which confirm the theoretical results andcompare the proposed two LDG methods, are also presented in the paper. Introduction
This paper is the third installment in a series [10, 11] which devote to developingabsolutely stable discontinuous Galerkin (DG) methods for the following prototyp-ical Helmholtz problem with large wave number: − ∆ u − k u = f in Ω ⊂ R d , d = 2 , , (1.1) ∂u∂ n Ω + ı ku = g on Γ = ∂ Ω , (1.2)where ı = √− k ∈ R + is a given positive (large)number and known as the wave number. (1.2) is the so-called first order absorbingboundary condition [9].We recall that [10, 11] focused on designing and analyzing h - and hp -interiorpenalty discontinuous Galerkin (IPDG) methods which are absolutely stable (withrespect to wave number k and mesh size h ) and optimally convergent (with respectto h ). The main ideas of [10, 11] are to introduce some novel interior penalty terms Mathematics Subject Classification.
Key words and phrases.
Helmholtz equation, time harmonic waves, local discontinuousGalerkin methods, stability, error estimates .The work of the first author was partially supported by the NSF grants DMS-0710831 andDMS-1016173. The research of the second author was partially sponsored by the Office of Ad-vanced Scientific Computing Research; U.S. Department of Energy. The work of the secondauthor was performed at the ORNL, which is managed by UT-Battelle, LLC under Contract No.DE-AC05-00OR22725. a r X i v : . [ m a t h . NA ] O c t XIAOBING FENG AND YULONG XING in the sesquilinear forms of the proposed IPDG methods and to use a non-standardanalytical tool, which is based on a Rellich identity technique, to prove the desiredstability and error estimates. The numerical experiment results shown that the ab-solutely stable IPDG methods significantly outperform the standard finite elementand finite difference methods, which are known only to be stable under stringentmesh constraints hk (cid:46) hk (cid:46) −
10 grid points must be usedin a wave length). The main difficulty of analyzing the Helmholtz type problems iscaused by the strong indefiniteness of the Helmholtz equation which in turn makesit hard to establish stability estimates for its numerical approximations. The loss ofstability in the case of large wave numbers results in an additional pollution error(besides the interpolation error) in the global error bounds. Extensive research hasbeen done to address the question whether it is possible to reduce the pollutioneffect, we refer the reader to Chapter 4 of [15] and the references therein for andetailed exposition in this direction.Motivated by the success of [10, 11], the primary objective of this paper is toextend the work of [10, 11] to the local discontinuous Galerkin (LDG) formulation,which is known to be more “physical” and flexible than the IPDG formulation ondesigning DG schemes [1, 5]. As it is well-known now, the key step for constructingLDG methods is to design the numerical fluxes. As soon as the numerical fluxes areselected, for a large class of coercive elliptic and parabolic second order problems,there is a general framework for carrying out convergence analysis of LDG methods[1]. Unfortunately, this general framework does not apply to the Helmholtz typeproblems which is extremely noncoercive/indefinite for large wave number k . Nev-ertheless, when designing the numerical fluxes for our LDG methods, we borrowthe idea of [1] by establishing the connection between our LDG methods and theIPDG methods of [10, 11] although it turns out that the IPDG methods of [10, 11]do not have exactly equivalent LDG formulations due to the non-standard penaltyterms used in [10, 11]. This then leads to the construction of our first LDG method.It is proved and numerically verified that this LDG method is absolutely stable andoptimally convergent for the scalar variable. However, it is sub-optimal for thevector/flux variable. To improve the approximation accuracy for the vector/fluxvariable, we design another set of numerical fluxes which result in the constructionof our second LDG method. It is proved that the second LDG method is also ab-solutely stable and gives a better approximation for the vector/flux variable thanthe first method. On the other hand, it is computationally more expensive thanthe first LDG method, which is expected.To analyze the proposed LDG methods, we take an opposite approach to thatadvocated in [1], that is, instead of converting LDG methods to their “equivalent”IPDG methods in the primal form, we recast and treat our LDG methods as non-conforming mixed finite element methods. To avoid using the standard techniquessuch as Schatz argument (cf. [2, 8]) or Babuˇska’s inf-sup condition argument [16]to derive error estimates (and to prove stability), both approaches would certainlylead to stringent mesh constraints, our main idea is to establish a generalized inf-sup condition, which holds without any mesh constraint, for each LDG method. Thegeneralized inf-sup conditions then immediately infer the desired absolute stability DG METHODS FOR THE HELMHOLTZ EQUATION 3 of the proposed LDG methods. In return, the stability results not only guaranteethe well-posedness of the LDG methods but also play a crucial role in the derivationof the (optimal) error estimates.It should be pointed out that a lot of work has recently been done on develop-ing DG methods using piecewise plane wave functions, oppose to simpler piecewisepolynomial functions as done in this paper, for the Helmholtz type problems. How-ever, to the best of our knowledge, none of these plane wave DG method is provedto be absolutely stable with respect to wave number k and mesh size h . We referthe reader to [12, 14, 17] and the references therein for more discussions in thisdirection. We also refer to [10, 11] for more discussions and references on otherdiscretization techniques for the Helmholtz type problems.This paper consists of four additional sections. In Section 2, we introduce thenotations used in this paper and present the derivations of our two LDG methods.In Section 3, we present a detailed stability analysis for both LDG methods. Themain task of the section is to prove a generalized inf-sup condition for each proposedLDG method. Similar to [10, 11], a nonorthodox test function trick is the key toget the job done. In Section 4, a non-standard two-step error estimate procedureis used to derive error estimates for the proposed LDG methods. Once again, thestability estimates established in Section 3 play a crucial role. Finally, Section 5contains some numerical experiments which are designed to verify the theoreticalerror bounds proved in Section 4 and to compare the performance of the proposedtwo LDG methods.2. Formulation of local discontinuous Galerkin methods
The standard space, norm and inner product notation are adopted in this paper.Their definitions can be found in [1, 2, 4, 19]. In particular, ( · , · ) Q and (cid:104)· , ·(cid:105) Σ forΣ ⊂ ∂Q denote the L -inner product on complex-valued L ( Q ) and L (Σ) spaces,respectively. ( · , · ) := ( · , · ) Ω and (cid:104)· , ·(cid:105) := (cid:104)· , ·(cid:105) ∂ Ω . Throughout the paper, C is usedto denote a generic positive constant which is independent of h and k . We also usethe shorthand notation A (cid:46) B and B (cid:38) A for the inequality A ≤ CB and B ≥ CA . A (cid:39) B is a shorthand notation for the statement A (cid:46) B and B (cid:46) A .Assume that Ω ⊂ R d ( d = 2 ,
3) is a bounded and strictly star-shaped domain withrespect to a point x Ω ∈ Ω. We now recall the definition of star-shaped domains.
Definition 2.1. Q ⊂ R d is said to be a star-shaped domain with respect to x Q ∈ Q if there exists a nonnegative constant c Q such that (2.1) ( x − x Q ) · n Q ≥ c Q ∀ x ∈ ∂Q.Q ⊂ R d is said to be strictly star-shaped if c Q is positive. Let T h be a family of partitions of Ω parameterized by h >
0. For any tri-angle/tetrahedron K ∈ T h , we define h K := diam( K ) and h := max K ∈T h h K .Similarly, for each edge/face e of K ∈ T h , define h e := diam( e ). We assume thatthe elements of T h satisfy the minimal angle condition. We also define E Ih := set of all interior edges/faces of T h , E Bh := set of all boundary edges/faces of T h on Γ = ∂ Ω , E h := E Ih ∪ E Bh . XIAOBING FENG AND YULONG XING
Let e be an interior edge shared by two elements K and K whose unit outwardnormal vectors are denoted by n and n . For a scalar function v , let v i = v | ∂K i ,and define { v } = 12 ( v + v ) , [ v ] = v K − v K (cid:48) , [[ v ]] = v n + v n on e ∈ E Ih , where K is K or K , whichever has the bigger global labeling and K (cid:48) is the other.For a vector field v , let v i = v | ∂K i and define { v } = 12 ( v + v ) , [ v ] = v K − v K (cid:48) , [[ v ]] = v · n + v · n on e ∈ E Ih . As it is well-known now (cf. [1]) that the first step for formulating an LDGmethod is to rewrite the given PDE as a first order system by introducing anauxiliary variable. For the Helmholtz problem (1.1)–(1.2) we have σ = ∇ u in Ω , (2.2) − div σ − k u = f in Ω , (2.3) ∂u∂ n Ω + ı ku = g on Γ , (2.4)Clearly, the vector-valued function (often called the flux variable) σ is the auxiliaryvariable.Then, multiplying (2.2) and (2.3) by test functions τ and v , respectively, andintegrating both equations over an element K ∈ T h yields (cid:90) K σ · τ dx = − (cid:90) K u div τ dx + (cid:90) ∂K u n K · τ ds, (2.5) (cid:90) K σ · ∇ v dx − k (cid:90) K uv dx = (cid:90) K f v dx + (cid:90) ∂K σ · n K v ds, (2.6)where n K denotes the unit outward normal vector to ∂K . The above equationsform the weak formulation one uses to define LDG methods for the Helmholtzproblem (1.1)–(1.2).Next, we define LDG spaces as follows V h := { v ∈ L (Ω); Re( v ) | K , Im( v ) ∈ P r ( K ) ∀ K ∈ T h } , Σ h := { τ ∈ ( L (Ω)) d ; Re( τ ) | K , Im( τ ) ∈ ( P (cid:96) ( K )) d ∀ K ∈ T h } , where P r ( K ) ( r ≥
1) stands for the set of all polynomials of degree less than orequal to r on K .Finally, we are ready to define the following general LDG formulation: Find( u h , σ h ) ∈ V h × Σ h such that for all K ∈ Γ h there hold (cid:90) K σ h · τ h dx = − (cid:90) K u h div τ h dx + (cid:90) ∂K ˆ u K n K · τ h ds, (2.7) (cid:90) K σ h · ∇ v h dx − k (cid:90) K u h v h dx = (cid:90) K f v h dx + (cid:90) ∂K ˆ σ K · n K v h ds (2.8)for any ( v h , τ h ) ∈ V h × Σ h . Where the quantities ˆ u K and ˆ σ K , which are callednumerical fluxes, are respectively approximations to σ = ∇ u and u on the bound-ary ∂K of K . As it is well-known now that the most important issue for all LDGmethods is how to choose the numerical fluxes. The different choices of the numer-ical fluxes obviously lead to different LDG methods. It is easy to understand that DG METHODS FOR THE HELMHOLTZ EQUATION 5 these numerical fluxes must be chosen carefully in order to ensure the stability andaccuracy of the resulted LDG methods.In this paper, we shall only consider the linear element case (i.e., r = (cid:96) = 1) andpropose two sets of numerical fluxes (ˆ u K , ˆ σ K ), which lead to two LDG methods.Our choices of numerical fluxes are inspired by the interior penalty discontinuousGalerkin (IPDG) methods proposed by Feng and Wu [10] and are identified with thehelp of the unified DG framework of [1] which bridges the primal DG formulations(e.g. IPDG methods) and the flux DG formulations (i.e., LDG methods).(1) LDG method : Setˆ σ K = {∇ h u h } − ı β [[ u h ]] , ˆ u K = { u h } + ı δ [[ ∇ h u h ]] on e ∈ E Ih , ˆ σ K = − ı ku h n K + g n K , ˆ u K = u h on e ∈ E Bh . (2) LDG method : Setˆ σ K = { σ h } − ı β [[ u h ]] , ˆ u K = { u h } + ı δ [[ σ h ]] on e ∈ E Ih , ˆ σ K = − ı ku h n K + g n K , ˆ u K = u h on e ∈ E Bh . Where β and δ are positive constants to be specified later and ∇ h denotes thepiecewisely defined gradient operator over T h , that is, ∇ h | K = ∇| K ∀ K ∈ T h .For the reader’s convenience, we now briefly sketch the derivation of the primalDG formulation corresponding to our LDG method K ∈ T h and using the followingintegration by parts identity( u h , div τ h ) Ω = − ( ∇ h u h , τ h ) Ω + (cid:88) e ∈E Ih (cid:16) (cid:104){ u h } , [[ τ h ]] (cid:105) e + (cid:104) [[ u h ]] , { τ h }(cid:105) e (cid:17) + (cid:104) u h , n K · τ h (cid:105) Γ , we get ( σ h , τ h ) Ω − ( ∇ h u h , τ h ) Ω (2.9) − (cid:88) e ∈E Ih (cid:16) ı δ (cid:104) [[ ∇ h u h ]] , [[ τ h ]] (cid:105) e − (cid:104) [[ u h ]] , { τ h }(cid:105) e (cid:17) = 0 , ( σ h , ∇ h v h ) Ω − k ( u h , v h ) Ω − (cid:88) e ∈E Ih (cid:104){∇ h u h } − ı β [[ u h ]] , [[ v h ]] (cid:105) e (2.10) + ı k (cid:104) u h , v h (cid:105) Γ = ( f, v h ) Ω + (cid:104) g, v h (cid:105) Γ . Setting τ = ∇ v h in (2.9) and subtracting the resulting equation from (2.10) thenleads to the following formulation:(2.11) A h ( u h , v h ) − k ( u h , v h ) Ω = F ( v h ) ∀ v h ∈ V h , XIAOBING FENG AND YULONG XING where A h ( u h , v h ) := ( ∇ h u h , ∇ h v h ) Ω + ı k (cid:104) u h , v h (cid:105) Γ + (cid:88) e ∈E Ih ı (cid:16) δ (cid:104) [[ ∇ h u h ]] , [[ ∇ h v h ]] (cid:105) e + β (cid:104) [[ u h ]] , [[ v h ]] (cid:105) e (cid:17) − (cid:88) e ∈E Ih (cid:16) (cid:104) [[ u h ]] , {∇ h v h }(cid:105) e + (cid:104){∇ h u h } , [[ v h ]] (cid:105) e (cid:17) .F ( v h ) := ( f, v h ) Ω + (cid:104) g, v h (cid:105) Γ . Hence, (2.11) is the corresponding primal (i.e., IPDG) formulation of our LDGmethod
Stability analysis
From their constructions, it is easy to see that both LDG methods proposed inthe previous section are consistent schemes for the Helmholtz problem (1.1)–(1.2).For coercive elliptic (and parabolic) problems, the stability of such a numericalscheme can be proved easily as demonstrated in [1] (the same statement is truefor their corresponding PDE stability analysis). However, the Helmholtz problem(1.1)–(1.2) is an indefinite problem and it becomes notoriously non-coercive for largewave number k . Deriving its stability estimates (i.e., a priori estimates of its PDEsolution), particularly wave-number-dependent estimates, has been proved not to bean easy job (cf. [6, 7, 13, 18] and the references therein). Numerically, such a questhas been known to be even harder because of the lower order of the smoothnessand the inflexibility of (piecewise) approximation functions (cf. [6, 16, 18] andthe references therein). The stability of the numerical methods in all the abovequoted references was proved under some very restrictive mesh constraints. Anopen question was then raised by Zienkiewicz [20] which asks whether it is possibleto construct an absolutely stable (and optimally convergent) numerical method(i.e., no restriction on the mesh size h and the wave number k ) for the Helmholtzequation. Almost a decade later Feng and Wu [10, 11] were able to design for thefirst time such numerical methods, which happen to belong to the IPDG family, forthe Helmholtz problem (1.1)–(1.2).The goal of this section is to show that in the case of the linear element (i.e., r = (cid:96) = 1) the LDG method h > k > inf-sup or Babuˇska-Brezzi condition for the (augmented) sesquilinear forms foreach method. However, we are not able to prove such an inf-sup condition for either DG METHODS FOR THE HELMHOLTZ EQUATION 7 method without imposing mesh constraints (which we believe is not possible). Toovercome the difficulty, our main idea is to prove a generalized (and weaker) inf-sup condition which holds for all h, k >
0. It turns out that this generalized inf-sup condition is sufficient for us to establish the desired absolute stability for the LDGmethod inf-sup condition, the key techniquewe use is a special test function technique which was first introduced and developedin [10].3.1.
Absolute stability of LDG method
We first recast the LDG method u h , σ h ) ∈ V h × Σ h such that(3.1) A h ( u h , σ h ; v h , τ h ) = F ( v h , τ h ) ∀ ( v h , τ h ) ∈ V h × Σ h , where A h ( w h , χ h ; v h , τ h ) = ( χ h , ∇ h v h ) Ω − k ( w h , v h ) Ω + ı k (cid:104) w h , v h (cid:105) Γ (3.2) − (cid:88) e ∈E Ih (cid:104){∇ h w h } − ı β [[ w h ]] , [[ v h ]] (cid:105) e − (cid:88) e ∈E Ih (cid:16) ı δ (cid:104) [[ ∇ h w h ]] , [[ τ h ]] (cid:105) e − (cid:104) [[ w h ]] , { τ h }(cid:105) e (cid:17) + ( χ h , τ h ) Ω − ( ∇ h w h , τ h ) Ω .F ( v h , τ h ) := ( f, v h ) Ω + (cid:104) g, v h (cid:105) Γ . (3.3)3.1.1. A generalized inf-sup condition.
The goal of this subsection is to showthat the sesquilinear form A h defined in (3.2) satisfies a generalized inf-sup con-dition, which will play a vital role for us to establish the absolute stability of theLDG method Proposition 3.1.
There exists an h - and k -independent constant c > such thatfor any ( w h , χ h ) ∈ V h × Σ h sup ( vh, τ h ) ∈ Vh × Σ hvh (cid:54) =0 Re A h ( w h , χ h ; v h , τ h ) (cid:107) v h (cid:107) DG (3.4) + sup ( vh, τ h ) ∈ Vh × Σ hvh (cid:54) =0 Im A h ( w h , χ h ; v h , τ h ) (cid:107) v h (cid:107) DG ≥ c γ (cid:107) w h (cid:107) DG , where γ := 1 + k + (cid:114) βδ + max e ∈E Ih (cid:16) k + 1 βh e + 1 h e + 1 βh e (cid:17) , (3.5) (cid:107) w h (cid:107) DG := (cid:16) k (cid:107) w h (cid:107) L (Ω) + k (cid:107) w h (cid:107) L (Γ) + c Ω (cid:107)∇ h w h (cid:107) L (Γ) + | w h | ,h (cid:17) , (3.6) | w h | ,h := (cid:16) (cid:88) K ∈T h (cid:107)∇ w h (cid:107) L ( K ) (cid:17) . Proof.
The main idea of the proof is that for a fixed ( w h , χ h ) ∈ V h × Σ h we need topick up two sets of special functions ( v h , τ h ) ∈ V h × Σ h , which, as expected, mustdepend on ( w h , χ h ) ∈ V h × Σ h , such that both quotients in (3.4) can be bounded XIAOBING FENG AND YULONG XING from below by (cid:107) w h (cid:107) DG . When that is done, the inf-sup constant c /γ will berevealed in the process. Since the proof is very long, we divide it into four steps. Step 1: Taking the first test function.
We first choose the test function ( v h , τ h ) = ( w h , −∇ h w h ) to get A h ( w h , χ h ; v h , τ h ) = A h ( w h , χ h ; w h , −∇ h w h )= ( ∇ h w h , ∇ h w h ) Ω − k ( w h , w h ) Ω + ı k (cid:104) w h , w h (cid:105) Γ + (cid:88) e ∈E Ih (cid:16) ı δ (cid:104) [[ ∇ h w h ]] , [[ ∇ h w h ]] (cid:105) e − (cid:104) [[ w h ]] , {∇ h w h }(cid:105) e (cid:17) − (cid:88) e ∈E Ih (cid:104){∇ h w h } − ı β [[ w h ]] , [[ w h ]] (cid:105) e . Taking the real and imaginary parts yieldsRe A h ( w h , χ h ; w h , −∇ h w h )(3.7) = | w h | ,h − k (cid:107) w h (cid:107) L (Ω) − (cid:88) e ∈E Ih (cid:104) [[ w h ]] , {∇ h w h }(cid:105) e , Im A h ( w h , χ h ; w h , −∇ h w h )(3.8) = (cid:88) e ∈E Ih (cid:16) δ (cid:107) [[ ∇ h w h ]] (cid:107) L ( e ) + β (cid:107) [[ w h ]] (cid:107) L ( e ) (cid:17) + k (cid:107) w h (cid:107) L (Γ) . Step 2: Taking the second test function.
Inspired by the special test function technique of [10], we now choose anothertest function ( v h , τ h ) = ( α · ∇ h w h , −∇ h w h ) with α := x − x Ω (see Definition 2.1)and use the fact that ∇ h v h = ∇ h w h to get A h ( w h , χ h ; v h , τ h ) = A h ( w h , χ h ; α · ∇ h w h , −∇ h w h )= ( ∇ h w h , ∇ h w h ) Ω − k ( w h , α · ∇ h w h ) Ω + ı k (cid:104) w h , α · ∇ h w h (cid:105) Γ + (cid:88) e ∈E Ih (cid:16) ı δ (cid:104) [[ ∇ h w h ]] , [[ ∇ h w h ]] (cid:105) e − (cid:104) [[ w h ]] , {∇ h w h }(cid:105) e (cid:17) − (cid:88) e ∈E Ih (cid:104){∇ h w h } − ı β [[ w h ]] , [[ α · ∇ h w h ]] (cid:105) e . Taking the real part immediately gives (note that v h = α · ∇ h w h )Re A h ( w h , χ h ; v h , −∇ h w h )(3.9) = | w h | ,h − k ( w h , v h ) Ω − k Im (cid:104) w h , v h (cid:105) Γ − (cid:88) e ∈E Ih β Im (cid:104) [[ w h ]] , [[ v h ]] (cid:105) e − (cid:88) e ∈E Ih Re (cid:16) (cid:104) [[ w h ]] , {∇ h w h }(cid:105) e + (cid:104){∇ h w h } , [[ v h ]] (cid:105) e (cid:17) . Step 3: Deriving an upper bound for k (cid:107) w h (cid:107) L (Ω) . To bound (3.7) from below, we need to get an upper bound for the term k (cid:107) w h (cid:107) L (Ω) on the right hand side of (3.7). This will be done by carefully and judicially com-bining (3.8) and (3.9) with some other differential identities, which we now explain. DG METHODS FOR THE HELMHOLTZ EQUATION 9
Using the integral identity(3.10) 2 (cid:107) w h (cid:107) L ( K ) = (cid:90) ∂K α · n K | w h | ds − w h , v h ) K − ( d − (cid:107) w h (cid:107) L ( K ) , and (3.7), (3.9) we get2 k (cid:107) w h (cid:107) L (Ω) = 2 Re A h ( w h , χ h , v h , −∇ h w h )(3.11) + ( d −
2) Re A h ( w h , χ h ; w h , −∇ h w h )+ 2 k Im (cid:104) w h , v h (cid:105) Γ − | w h | ,h − ( d − | w h | ,h + 2 Re (cid:88) e ∈E Ih (cid:16) (cid:104) [[ w h ]] , {∇ h w h }(cid:105) e + (cid:104){∇ w h } , [[ v h ]] (cid:105) e (cid:17) + 2( d −
2) Re (cid:88) e ∈E Ih (cid:104) [[ w h ]] , {∇ h w h }(cid:105) e + (cid:88) e ∈E Ih (cid:16) β Im (cid:104) [[ w h ]] , [[ v h ]] (cid:105) e + k (cid:88) K ∈T h (cid:90) ∂K α · n K | w h | ds (cid:17) . By the elementary identity | a | − | b | = Re( a + b )(¯ a − ¯ b ) for any two complexnumbers a and b we have(3.12) (cid:88) K ∈T h (cid:90) ∂K α · n K | w h | ds = 2 Re (cid:88) e ∈E Ih (cid:104) α · n e { w h } , [ w h ] (cid:105) e + (cid:10) α · n Ω , | w h | (cid:11) Γ . Next, using the local Rellich identity (see [10, Lemma 4.1])( d − (cid:107)∇ w h (cid:107) L ( K ) + 2 Re( ∇ w h , ∇ v h ) K = (cid:90) ∂K α · n K |∇ v h | ds and the fact that ∇ h v h = ∇ h w h , we get d | w h | ,h = ( d − | w h | ,h + 2 Re (cid:88) K ∈T h ( ∇ w h , ∇ v h ) K (3.13) = Re (cid:88) K ∈T h (cid:90) ∂K α · n K |∇ v h | ds = 2 Re (cid:88) e ∈E Ih (cid:104) α · n e {∇ h w h } , [ ∇ h w h ] (cid:105) e + (cid:88) e ∈E Bh (cid:10) α · n e , |∇ h w h | (cid:11) e . Substituting (3.12) and (3.13) into (3.11) we get2 k (cid:107) w h (cid:107) L (Ω) = 2 Re A h ( w h , χ h ; v h , −∇ h w h )(3.14) + ( d −
2) Re A h ( w h , χ h ; w h , −∇ h w h )+ 2 k Re (cid:88) e ∈E Ih (cid:104) α · n e { w h } , [ w h ] (cid:105) e + k (cid:10) α · n Ω , | w h | (cid:11) Γ + 2 k Im (cid:104) w h , v h (cid:105) Γ − (cid:88) e ∈E Bh (cid:10) α · n e , |∇ h w h | (cid:11) e − (cid:88) e ∈E Ih (cid:16) (cid:104) α · n e {∇ h w h } , [ ∇ h w h ] (cid:105) e − (cid:104){∇ h w h } , [[ v h ]] (cid:105) e (cid:17) + 2( d −
1) Re (cid:88) e ∈E Ih (cid:104) [[ w h ]] , {∇ h w h }(cid:105) e + 2 Im (cid:88) e ∈E Ih β (cid:104) [[ w h ]] , [[ v h ]] (cid:105) e . To get an upper bound for k (cid:107) w h (cid:107) L (Ω) , we need to bound the terms on theright-hand side of (3.14), which we now bound as follows.2 k Re (cid:88) e ∈E Ih (cid:104) α · n e { w h } , [ w h ] (cid:105) e ≤ Ck (cid:88) e ∈E Ih h − e (cid:107) w h (cid:107) L ( K e ∪ K (cid:48) e ) (cid:107) [ w h ] (cid:107) L ( e ) (3.15) ≤ k (cid:107) w h (cid:107) L (Ω) + C (cid:88) e ∈E Ih k βh e β (cid:107) [ w h ] (cid:107) L ( e ) .k (cid:10) α · n Ω , | w h | (cid:11) Γ ≤ Ck (cid:107) w h (cid:107) L (Γ) . (3.16)It follows from the star-shaped assumption on Ω that2 k Im (cid:104) w h , v h (cid:105) Γ − (cid:88) e ∈E Bh (cid:10) α · n e , |∇ h w h | (cid:11) e (3.17) ≤ Ck (cid:88) e ∈E Bh (cid:107) w h (cid:107) L ( e ) (cid:107)∇ h w h (cid:107) L ( e ) − c Ω (cid:88) e ∈E Bh (cid:107)∇ h w h (cid:107) L ( e ) ≤ Ck (cid:107) w h (cid:107) L (Γ) − c Ω (cid:107)∇ h w h (cid:107) L (Γ) . By the trace inequality [2], we also have2 d Re (cid:88) e ∈E Ih (cid:104) [[ w h ]] , {∇ h w h }(cid:105) e (3.18) (cid:46) d (cid:88) e ∈E Ih h − e (cid:88) K = K e ,K (cid:48) e (cid:107)∇ h w h (cid:107) L ( K ) (cid:107) [ w h ] (cid:107) L ( e ) ≤ | w h | ,h + C (cid:88) e ∈E Ih βh e β (cid:107) [ w h ] (cid:107) L ( e ) . DG METHODS FOR THE HELMHOLTZ EQUATION 11 − (cid:88) e ∈E Ih (cid:16) (cid:104) α · n e {∇ h w h } , [ ∇ h w h ] (cid:105) e − (cid:104){∇ h w h } , [[ v h ]] (cid:105) e (cid:17) (3.19) = 2 Re (cid:88) e ∈E Ih d − (cid:88) j =1 (cid:90) e (cid:16) ( α · τ je ) {∇ h w h · n e }− ( α · n e ) {∇ h w h · τ je } (cid:17) ∇ h [ w h ] · τ je (cid:105) (cid:46) (cid:88) e ∈E Ih d − (cid:88) j =1 h − e (cid:88) K = K e ,K (cid:48) e (cid:107)∇ h w h (cid:107) L ( K ) (cid:107) [ ∇ h w h · τ je ] (cid:107) L ( e ) ≤ | w h | ,h + C (cid:88) e ∈E Ih βh e d − (cid:88) j =1 β (cid:107) [ ∇ h w h · τ je ] (cid:107) L ( e ) . By the definition of v h := α · ∇ h w h , we get2 Im (cid:88) e ∈E Ih β (cid:104) [[ w h ]] , [[ v h ]] (cid:105) e = 2 Im (cid:88) e ∈E Ih β (cid:104) [ w h ] , [ v h ] (cid:105) e (3.20) = 2 Im (cid:88) e ∈E Ih β (cid:42) [ w h ] , (cid:104) ( α · n e ) ∇ h w h · n e + d − (cid:88) j =1 ( α · τ je ) ∇ h w h · τ je (cid:105)(cid:43) e ≤ C (cid:88) e ∈E Ih β (cid:107) [ w h ] (cid:107) L ( e ) (cid:107) [ ∇ h w h · n e ] (cid:107) L ( e ) + C (cid:88) e ∈E Ih β (cid:107) [ w h ] (cid:107) L ( e ) d − (cid:88) j =1 (cid:107) [ ∇ h w h · τ je ] (cid:107) L ( e ) ≤ C (cid:114) βδ (cid:88) e ∈E Ih (cid:16) β (cid:107) [ w h ] (cid:107) L ( e ) + δ (cid:107) [ ∇ h w h · n e ] (cid:107) L ( e ) (cid:17) + Cβ (cid:88) e ∈E Ih (cid:16) (cid:107) [ w h ] (cid:107) L ( e ) + d − (cid:88) j =1 (cid:107) [ ∇ h w h · τ je ] (cid:107) L ( e ) (cid:17) , where { τ je } d − j =1 denotes an orthogonal tangential frame on the edge/face e , and wehave used the decomposition α = ( α · n e ) n e + (cid:80) d − j =1 ( α · τ je ) τ je . Now substituting estimates (3.15)–(3.19) into (3.14) we obtain2 k (cid:107) w h (cid:107) L (Ω) ≤ A h ( w h , χ h ; v h , −∇ h w h )(3.21) + ( d −
2) Re A h ( w h , χ h ; w h , −∇ h w h )+ Ck (cid:107) w h (cid:107) L (Γ) − c Ω (cid:107)∇ h w h (cid:107) L (Γ) + k (cid:107) w h (cid:107) L (Ω) + C (cid:88) e ∈E Ih k βh e β (cid:107) [ w h ] (cid:107) L ( e ) − (cid:88) e ∈E Ih (cid:104) [[ w h ]] , {∇ h w h }(cid:105) e + 14 | w h | ,h + C (cid:88) e ∈E Ih βh e d − (cid:88) j =1 β (cid:107) [ ∇ h · τ je ] (cid:107) L ( e ) + 14 | w h | ,h + C (cid:88) e ∈E Ih βh e β (cid:107) [ w h ] (cid:107) L ( e ) + C (cid:88) e ∈E Ih (cid:114) βδ (cid:16) β (cid:107) [ w h ] (cid:107) L ( e ) + δ (cid:107) [ ∇ h w h · n e ] (cid:107) L ( e ) (cid:17) + C (cid:88) e ∈E Ih (cid:16) β (cid:107) [ w h ] (cid:107) L ( e ) + d − (cid:88) j =1 β (cid:107) [ ∇ h w h · τ je ] (cid:107) L ( e ) (cid:17) . On noting that (3.8) provides upper bounds for terms (cid:107) [ ∇ h w h · n e ] (cid:107) L ( e ) , (cid:107) [ w h ] (cid:107) L ( e ) and k (cid:107) w h (cid:107) L (Γ) in terms of Im A h ( w h , χ h ; w h , −∇ h w h ), using these bounds in(3.21) we get2 k (cid:107) w h (cid:107) L (Ω) + k (cid:107) w h (cid:107) L (Γ) + c Ω (cid:107)∇ h w h (cid:107) L (Γ) (3.22) ≤ A h ( w h , χ h ; v h , −∇ h w h ) + ( d −
2) Re A h ( w h , χ h ; w h , −∇ h w h )+ 12 | w h | ,h + k (cid:107) w h (cid:107) L (Ω) − (cid:88) e ∈E Ih (cid:104){∇ h w h } , [[ w h ]] (cid:105) e + C (cid:88) e ∈E Ih (cid:16) βh e + 1 (cid:17) d − (cid:88) j =1 β (cid:107) [ ∇ h w h · τ je ] (cid:107) L ( e ) + M Im A h ( w h , χ h ; w h , −∇ h w h ) , where M := C (cid:16) k + (cid:114) βδ + max e ∈E Ih k + 1 βh e (cid:17) . (3.23)To bound the jumps of the tangential derivatives (cid:107) [ ∇ h w h · τ je ] (cid:107) L ( e ) in (3.21),we appeal to the inverse inequality(3.24) (cid:107) [ ∇ h w h · τ je ] (cid:107) L ( e ) ≤ Ch − e (cid:107) [ w h ] (cid:107) L ( e ) , DG METHODS FOR THE HELMHOLTZ EQUATION 13 and using (3.8) and (3.7) to get2 k (cid:107) w h (cid:107) L (Ω) + k (cid:107) w h (cid:107) L (Γ) + c Ω (cid:107)∇ h w h (cid:107) L (Γ) ≤ A h ( w h , χ h ; v h , −∇ h w h ) + ( d −
2) Re A h ( w h , χ h ; w h , −∇ h w h )+ k (cid:107) w h (cid:107) L (Ω) + M Im A h ( w h , χ h ; w h , −∇ h w h ) − (cid:88) e ∈E Ih (cid:104){∇ h w h } , [[ w h ]] (cid:105) e + 12 | w h | ,h ≤ A h ( w h , χ h ; v h , −∇ h w h ) + ( d −
1) Re A h ( w h , χ h ; w h , −∇ h w h )+ 3 k (cid:107) w h (cid:107) L (Ω) + M Im A h ( w h , χ h ; w h , −∇ h w h ) − | w h | ,h , where M = M + max e ∈E Ih (cid:16) Ch e + Cβh e (cid:17) (3.25) = C (cid:16) k + (cid:114) βδ + max e ∈E Ih (cid:16) k + 1 βh e + 1 h e + 1 βh e (cid:17)(cid:17) . Using the linearity of the sesquilinear form A h we have k (cid:107) w h (cid:107) L (Ω) + k (cid:107) w h (cid:107) L (Γ) + c Ω (cid:107)∇ h w h (cid:107) L (Γ) + | w h | ,h (3.26) ≤ A h ( w h , χ h ; v h , −∇ h w h ) + 2( d −
1) Re A h ( w h , χ h ; w h , −∇ h w h )+ 2 M Im A h ( w h , χ h ; w h , −∇ h w h )= Re A h ( w h , χ h ; ˜ w h , −∇ h w h ) + 2 M Im A h ( w h , χ h ; w h , −∇ h w h ) . with ˜ w h = 4 v h + 2( d − w h . Step 4: Finishing up.
By the definition of (cid:107) · (cid:107) DG in (3.6) and the fact that ∇ h v h = ∇ h w h we have (cid:107) v h (cid:107) DG = k (cid:107) α · ∇ h w h (cid:107) L (Ω) + k (cid:107) α · ∇ h w h (cid:107) L (Γ) (3.27) + c Ω (cid:107)∇ h w h (cid:107) L (Γ) + | w h | ,h ≤ Ck (cid:107)∇ h w h (cid:107) L (Ω) + Ck (cid:107)∇ h w h (cid:107) L (Γ) + c Ω (cid:107)∇ h w h (cid:107) L (Γ) + | w h | ,h ≤ C (1 + k ) (cid:16) | w h | ,h + c Ω (cid:107)∇ h w h (cid:107) L (Γ) (cid:17) ≤ C (1 + k ) (cid:107) w h (cid:107) DG . Thus, it follows from the triangle inequality that (cid:107) ˜ w h (cid:107) DG ≤ (cid:107) v h (cid:107) DG + 2( d − (cid:107) w h (cid:107) DG ≤ C (1 + k ) (cid:107) w h (cid:107) DG . (3.28) Now from (3.26) and (3.28) we haveRe A h ( w h , χ h ; ˜ w h , −∇ h w h ) (cid:107) ˜ w h (cid:107) DG + Im A h ( w h , χ h ; w h , −∇ h w h ) (cid:107) w h (cid:107) DG (3.29) ≥ Re A h ( w h , χ h ; ˜ w h , −∇ h w h ) C (1 + k ) (cid:107) w h (cid:107) DG + Im A h ( w h , χ h ; w h , −∇ h w h ) (cid:107) w h (cid:107) DG ≥ M · Re A h ( w h , χ h ; ˜ w h , −∇ h w h ) + 2 M Im A h ( w h , χ h ; w h , −∇ h w h ) (cid:107) w h (cid:107) DG ≥ c γ (cid:107) w h (cid:107) DG for some constant c > γ is defined by (3.5). Hence, (3.4) holds. The proofis complete. (cid:3) Remark 3.1. (a) We note that γ depends on both h and k .(b) The generalized inf-sup condition is a weak estimate because it does not pro-vide a control for the variable χ h . As a comparison, we recall that the standard inf-sup condition for the sesquilinear form A h should be sup ( v h ,τ h ) ∈ V h × Σ h (cid:12)(cid:12) A h ( w h , χ h ; v h , τ h ) (cid:12)(cid:12) (cid:107) ( v h , τ h ) (cid:107) ≥ c (cid:107) ( w h , χ h ) (cid:107) ∀ ( w h , χ h ) ∈ V h × Σ h for some positive constant c = c ( k, β, δ, Ω) . Where (cid:107) ( w h , χ h ) (cid:107) := (cid:0) k (cid:107) w h (cid:107) L (Ω) + (cid:107) χ h (cid:107) L (Ω) (cid:1) . However, the above standard inf-sup condition can be proved only under the meshconstraint h = O ( k − ) and we believe that it does not hold without a mesh con-straint. Stability estimates.
The goal of this subsection is to establish the absolutestability for the LDG method inf-sup condition proved inthe previous subsection.
Theorem 3.1.
Let ( u h , σ h ) ∈ V h × Σ h solve (3.1) . Define (3.30) M ( f, g ) := (cid:107) f (cid:107) Ω + (cid:107) g (cid:107) L (Γ) . Then there hold the following stability estimates: (cid:107) u h (cid:107) DG (cid:46) γ k − M ( f, g ) . (3.31) (cid:107) σ h (cid:107) L (Ω) (cid:46) γ k − (cid:16) δ + k − ) (cid:0) max K ∈T h h − K (cid:1)(cid:17) M ( f, g ) . (3.32) Proof.
By Schwarz inequality we have | F ( v h , χ h ) | ≤ (cid:107) f (cid:107) L (Ω) (cid:107) v h (cid:107) L (Ω) + (cid:107) g (cid:107) L (Γ) (cid:107) v h (cid:107) L (Γ) (3.33) ≤ Ck − M ( f, g ) (cid:0) k (cid:107) v h (cid:107) L (Ω) + k (cid:107) v h (cid:107) L (Γ) (cid:1) . ≤ Ck − M ( f, g ) (cid:107) v h (cid:107) DG . DG METHODS FOR THE HELMHOLTZ EQUATION 15
Let ( w h , χ h ) = ( u h , σ h ) in (3.4). By equation (3.1) and (3.33) we get c γ (cid:107) u h (cid:107) DG ≤ sup ( v h , τ h ) ∈ V h × Σ h Re A h ( u h , σ h ; v h , τ h ) (cid:107) v h (cid:107) DG + sup ( v h , τ h ) ∈ V h × Σ h Im A h ( u h , σ h ; v h , τ h ) (cid:107) v h (cid:107) DG = sup ( v h , τ h ) ∈ V h × Σ h Re F ( v h , τ h ) (cid:107) v h (cid:107) DG + sup ( v h , τ h ) ∈ V h × Σ h Im F ( v h , τ h ) (cid:107) v h (cid:107) DG ≤ ( v h , τ h ) ∈ V h × Σ h (cid:12)(cid:12) F ( v h , τ h ) (cid:12)(cid:12) (cid:107) v h (cid:107) DG ≤ Ck − M ( f, g ) . Hence (3.31) holds.To show (3.32), setting ( v h , τ h ) = (0 , σ h ) in (3.1) and using the trace andSchwarz inequalities yields (cid:107) σ h (cid:107) L (Ω) = ( ∇ h u h , σ h ) Ω + (cid:88) e ∈E Ih (cid:16) ı δ (cid:104) [[ ∇ h u h ]] , [[ σ h ]] (cid:105) e − (cid:104) [[ u h ]] , { σ h }(cid:105) e (cid:17) ≤ (cid:107)∇ h u h (cid:107) L (Ω) + 14 (cid:107) σ h (cid:107) L (Ω) + C (cid:88) e ∈E Ih (cid:88) K = K e ,K (cid:48) e h − K (cid:16) δ (cid:107)∇ h u h (cid:107) L ( K ) + (cid:107) u h (cid:107) L ( K ) (cid:17) (cid:107) σ h (cid:107) L ( K ) ≤ (cid:107)∇ h u h (cid:107) L (Ω) + 12 (cid:107) σ h (cid:107) L (Ω) + C (cid:88) e ∈E Ih (cid:88) K = K e ,K (cid:48) e h − K (cid:16) δ (cid:107)∇ h u h (cid:107) L ( K ) + (cid:107) u h (cid:107) L ( K ) (cid:17) . Thus, (cid:107) σ h (cid:107) L (Ω) (cid:46) (cid:16) δ (cid:0) max K ∈T h h − K (cid:1)(cid:17) | u h | ,h + (cid:0) max K ∈T h h − K (cid:1) (cid:107) u h (cid:107) L (Ω) . The desired estimate (3.32) follows from combining the above inequality with (3.31).The proof is complete. (cid:3)
An immediate consequence of the stability estimates is the following uniquesolvability theorem.
Theorem 3.2.
There exists a unique solution to the LDG method (3.1) for all k, h, δ, β > .Proof. Since problem (3.1) is equivalent to a linear system, hence, it suffices toshow the uniqueness. But the uniqueness follows immediately from the stabilityestimates as the zero sources imply that any solution must be a trivial solution. (cid:3)
Absolute stability of LDG method
In this subsection, we considerthe LDG method u h , σ h ) ∈ V h × Σ h such that(3.34) B h ( u h , σ h ; v h , τ h ) = F ( v h , τ h ) ∀ ( v h , τ h ) ∈ V h × Σ h , where F is defined in (3.3) and B h ( w h , χ h ; v h , τ h ) = ( χ h , ∇ h v h ) Ω − k ( w h , v h ) Ω + ı k (cid:104) w h , v h (cid:105) Γ (3.35) − (cid:88) e ∈E Ih (cid:104){ χ h } − ı β [[ w h ]] , [[ v h ]] (cid:105) e − (cid:88) e ∈E Ih (cid:16) ı δ (cid:104) [[ χ h ]] , [[ τ h ]] (cid:105) e − (cid:104) [[ w h ]] , { τ h }(cid:105) e (cid:17) + ( χ h , τ h ) Ω − ( ∇ h w h , τ h ) Ω . A generalized inf-sup condition.
The goal of this subsection is to showthat the sesquilinear form B h defined in (3.35) for the LDG method inf-sup condition. To the end, we introduce the following spacenotation: S h := { ( w h , χ h ) ∈ V h × Σ h ; ( w h , χ h ) satisfies (3.37) } , (3.36)where ( χ h , τ h ) Ω − ( ∇ h w h , τ h ) Ω (3.37) − (cid:88) e ∈E Ih (cid:16) ı δ (cid:104) [[ χ h ]] , [[ τ h ]] (cid:105) e − (cid:104) [[ w h ]] , { τ h }(cid:105) e (cid:17) = 0 ∀ ( v h , τ h ) ∈ V h × Σ h . We note that it is easy to check that ( w h , χ h ) ∈ S h implies that it satisfies (2.7)with ˆ u K being defined by the LDG method Lemma 3.1.
For any ( w h , χ h ) ∈ S h , there holds the following estimates: | w h | ,h ≤ (cid:114) (cid:107) χ h (cid:107) L (Ω) (3.38) + C (cid:16) (cid:88) e ∈E Ih (cid:16) h e (cid:107) [[ w h ]] (cid:107) L ( e ) + δ h e (cid:107) [[ χ h ]] (cid:107) L ( e ) (cid:17)(cid:17) , (cid:107) χ h − ∇ h w h (cid:107) L (Ω) ≤ C (cid:16) (cid:88) e ∈E Ih (cid:16) h e (cid:107) [[ w h ]] (cid:107) L ( e ) + δ h e (cid:107) [[ χ h ]] (cid:107) L ( e ) (cid:17)(cid:17) . (3.39) DG METHODS FOR THE HELMHOLTZ EQUATION 17
Proof.
On noting that ( w h , χ h ) satisfies (3.37), setting τ h = ∇ h w h in (3.37), weget | w h | ,h = Re( χ h , ∇ h w h ) Ω − Re (cid:88) e ∈E Ih (cid:16) ı δ (cid:104) [[ χ h ]] , [[ ∇ h w h ]] (cid:105) e − (cid:104) [[ w h ]] , {∇ h w h }(cid:105) e (cid:17) ≤ | w h | ,h + 12 (cid:107) χ h (cid:107) L (Ω) + (cid:88) e ∈E Ih δh − e (cid:88) K = K e ,K (cid:48) e (cid:107) [[ χ h ]] (cid:107) L ( e ) (cid:107)∇ h w h (cid:107) L ( K ) + (cid:88) e ∈E Ih h − e (cid:88) K = K e ,K (cid:48) e (cid:107) [[ w h ]] (cid:107) L ( e ) (cid:107)∇ h w h (cid:107) L ( K ) ≤ | w h | ,h + 12 (cid:107) χ h (cid:107) L (Ω) + 134 | w h | ,h + C (cid:88) e ∈E Ih (cid:16) h e (cid:107) [[ w h ]] (cid:107) L ( e ) + δ h e (cid:107) [[ χ h ]] (cid:107) L ( e ) (cid:17) . Therefore, | w h | ,h ≤ (cid:107) χ h (cid:107) L (Ω) + C (cid:88) e ∈E Ih (cid:16) h e (cid:107) [[ w h ]] (cid:107) L ( e ) + δ h e (cid:107) [[ χ h ]] (cid:107) L ( e ) (cid:17) , which gives (3.38).The estimate (3.39) follows from the same derivation by setting τ h = χ h −∇ h w h in (3.37). The proof is complete. (cid:3) We now are ready to state a generalized inf-sup condition for the sesquilinearform B h . Proposition 3.2.
There exits constant c > such that for any ( w h , χ h ) ∈ S h there holds sup ( v h , τ h ) ∈ V h × Σ h Re B h ( w h , χ h ; v h , τ h ) (cid:107) ( v h , τ h ) (cid:107) DG (3.40) + sup ( v h , τ h ) ∈ V h × Σ h Im B h ( w h , χ h ; v h , τ h ) (cid:107) ( v h , τ h ) (cid:107) DG ≥ c γ (cid:107) ( w h , χ h ) (cid:107) DG , where γ := k + max e ∈E Ih (cid:16) k + 1 βh e + β + δh e + δh e + 1 βh e (cid:17) , (3.41) (cid:107) ( w h , χ h ) (cid:107) DG := (cid:16) k (cid:107) w h (cid:107) L (Ω) + k (cid:107) w h (cid:107) L (Γ) (3.42) + (cid:107) χ h (cid:107) L (Ω) + c Ω (cid:107)∇ h w h (cid:107) L (Γ) (cid:17) . Proof.
Since the proof follows the same lines as that of Proposition 3.1, we shallonly highlight the main steps and point out the differences.
Step 1: Taking the first test function.
We first choose the test function ( v h , τ h ) = ( w h , χ h ) to get B h ( w h , χ h ; v h , τ h ) = B h ( w h , χ h ; w h , χ h )= ( χ h , χ h ) Ω − k ( w h , w h ) Ω + ı k (cid:104) w h , w h (cid:105) Γ − (cid:88) e ∈E Ih (cid:104){ χ h } − ı β [[ w h ]] , [[ w h ]] (cid:105) e − (cid:88) e ∈E Ih (cid:16) ı δ (cid:104) [[ χ h ]] , [[ χ h ]] (cid:105) e − (cid:104) [[ w h ]] , { χ h }(cid:105) e (cid:17) + ( χ h , ∇ h w h ) Ω − ( ∇ h w h , χ h ) Ω . Taking the real and imaginary parts yieldsRe B h ( w h , χ h ; w h , χ h ) = (cid:107) χ h (cid:107) L (Ω) − k (cid:107) w h (cid:107) L (Ω) , (3.43) Im B h ( w h , χ h ; w h , χ h ) = k (cid:107) w h (cid:107) L (Γ) (3.44) + (cid:88) e ∈E Ih (cid:16) − δ (cid:107) [[ χ h ]] (cid:107) L ( e ) + β (cid:107) [[ w h ]] (cid:107) L ( e ) (cid:17) + 2 Im( χ h , ∇ h w h ) Ω + (cid:88) e ∈E Ih (cid:104) [[ w h ]] , { χ h }(cid:105) e . On noting that ( w h , χ h ) satisfies (3.37), setting τ h = χ h in (3.37), we get (cid:107) χ h (cid:107) L (Ω) − ( ∇ h w h , χ h ) Ω = (cid:88) e ∈E Ih (cid:16) ı δ (cid:107) [[ χ h ]] (cid:107) L ( e ) − (cid:104) [[ w h ]] , { χ h }(cid:105) e (cid:17) , Taking the imaginary part yieldsIm( χ h , ∇ h w h ) Ω + (cid:88) e ∈E Ih Im (cid:104) [[ w h ]] , { χ h }(cid:105) e = (cid:88) e ∈E Ih δ (cid:107) [[ χ h ]] (cid:107) L ( e ) . Hence, (3.44) becomesIm B h ( w h , χ h ; w h , χ h ) = k (cid:107) w h (cid:107) L (Γ) (3.45) + (cid:88) e ∈E Ih (cid:16) δ (cid:107) [[ χ h ]] (cid:107) L ( e ) + β (cid:107) [[ w h ]] (cid:107) L ( e ) (cid:17) . Step 2: Taking the second test function.
Next, we choose another test function ( v h , τ h ) = ( α · ∇ h w h , χ h ), which is dif-ferent from the one used in the proof of Proposition 3.1, and use the fact that ∇ h v h = ∇ h w h to get B h ( w h , χ h ; v h , τ h ) = B h ( w h , χ h , α · ∇ h w h , χ h )= ( χ h , χ h ) Ω − k ( w h , α · ∇ h w h ) Ω + ı k (cid:104) w h , α · ∇ h w h (cid:105) Γ + (cid:88) e ∈E Ih ı β (cid:104) [[ w h ]] , [[ α · ∇ h w h ]] (cid:105) e − (cid:88) e ∈E Ih (cid:104){ χ h } , [[ α · ∇ h w h ]] (cid:105) e − (cid:88) e ∈E Ih ı δ (cid:104) [[ χ h ]] , [[ χ h ]] (cid:105) e + (cid:88) e ∈E Ih (cid:104) [[ w h ]] , { χ h }(cid:105) e + 2ı Im( χ h , ∇ h w h ) . DG METHODS FOR THE HELMHOLTZ EQUATION 19
Taking the real part immediately gives ( v h = α · ∇ h w h )Re B h ( w h , χ h ; v h , χ h )(3.46) = (cid:107) χ h (cid:107) L (Ω) − k Re( w h , v h ) Ω − k Im (cid:104) w h , v h (cid:105) Γ + (cid:88) e ∈E Ih (cid:16) Re (cid:104){ χ h } , [[ w h ]] − [[ v h ]] (cid:105) e − β Im (cid:104) [[ w h ]] , [[ v h ]] (cid:105) e (cid:17) . Step 3: Deriving an upper bound for k (cid:107) w h (cid:107) L (Ω) . To bound (3.43) from below, we again need to get an upper bound for the term k (cid:107) w h (cid:107) L (Ω) on the right hand side of (3.43).Using the integral identity (3.10), (3.12), (3.43) and (3.46), we have2 k (cid:107) w h (cid:107) L (Ω) = 2 Re B h ( w h , χ h ; v h , χ h )(3.47) + ( d −
2) Re B h ( w h , χ h ; w h , χ h ) − d (cid:107) χ h (cid:107) L (Ω) + 2 k Im (cid:104) w h , v h (cid:105) Γ + k (cid:10) α · n Ω , | w h | (cid:11) Γ + 2 k Re (cid:88) e ∈E Ih (cid:104) α · n e { w h } , [ w h ] (cid:105) e + 2 Re (cid:88) e ∈E Ih (cid:104){ χ h } , [[ v h ]] (cid:105) e + 2( d −
1) Re (cid:88) e ∈E Ih (cid:104){ χ h } , [[ w h ]] (cid:105) e + 2 Im (cid:88) e ∈E Ih β (cid:104) [[ w h ]] , [[ v h ]] (cid:105) e + d | w h | ,h − (cid:88) e ∈E Bh (cid:10) α · n e , |∇ h w h | (cid:11) e − (cid:88) e ∈E Ih (cid:104) α · n e {∇ h w h } , [ ∇ h w h ] (cid:105) e ± (cid:88) e ∈E Ih (cid:104) α · n e { χ h } , [ ∇ h w h ] (cid:105) e . We note that by (3.13) the sum of the second and third lines to the last is zero,and the contribution of the last line is obviously zero. These terms are purposelyadded in order to get sharper upper bounds when they are combined with the termspreceding them.We now need to bound the terms on the right-hand side of (3.47). Some of thesehave been obtained in the proof of the Proposition 3.1, and the others are derivedas follows. 2( d −
1) Re (cid:88) e ∈E Ih (cid:104){ χ h } , [[ w h ]] (cid:105) e (3.48) (cid:46) d − (cid:88) e ∈E Ih h − e (cid:88) K = K e ,K (cid:48) e (cid:107) χ h (cid:107) L ( K ) (cid:107) [ w h ] (cid:107) L ( e ) ≤ (cid:107) χ h (cid:107) L (Ω) + C (cid:88) e ∈E Ih βh e β (cid:107) [ w h ] (cid:107) L ( e ) . (cid:88) e ∈E Ih (cid:104){ χ h } , [[ v h ]] (cid:105) e − (cid:88) e ∈E Ih (cid:104) α · n e { χ h } , [ ∇ h w h ] (cid:105) e (3.49) = 2 Re (cid:88) e ∈E Ih d − (cid:88) j =1 (cid:90) e (cid:16) ( α · τ je ) { χ h · n e }− ( α · n e ) { χ h · τ je } (cid:17) ∇ h [ w h ] · τ je (cid:105) (cid:46) (cid:88) e ∈E Ih d − (cid:88) j =1 h − e (cid:88) K = K e ,K (cid:48) e (cid:107) χ h (cid:107) L ( K ) (cid:107) [ ∇ h w h · τ je ] (cid:107) L ( e ) ≤ (cid:107) χ h (cid:107) L (Ω) + C (cid:88) e ∈E Ih βh e d − (cid:88) j =1 β (cid:107) [ ∇ h w h · τ je ] (cid:107) L ( e ) . It follows from (3.38) and (3.39) that2 Re (cid:88) e ∈E Ih (cid:104) α · n e { χ h − ∇ h w h } , [ ∇ h w h ] (cid:105) e (3.50) ≤ (cid:88) e ∈E Ih h − e (cid:88) K = K e ,K (cid:48) e (cid:107) χ h − ∇ h w h (cid:107) L ( K ) (cid:107)∇ h w h (cid:107) L ( K ) ≤ C max e ∈E Ih h − e (cid:107) χ h − ∇ h w h (cid:107) L (Ω) + 117 (cid:107)∇ h w h (cid:107) L (Ω) ≤ C (cid:88) e ∈E Ih (cid:16) h e (cid:107) [[ w h ]] (cid:107) L ( e ) + δ h e (cid:107) [[ χ h ]] (cid:107) L ( e ) (cid:17) + 116 (cid:107) χ h (cid:107) L (Ω) + C (cid:88) e ∈E Ih (cid:16) h e (cid:107) [[ w h ]] (cid:107) L ( e ) + δ h e (cid:107) [[ χ h ]] (cid:107) L ( e ) (cid:17) ≤ (cid:107) χ h (cid:107) L (Ω) + C (cid:88) e ∈E Ih (cid:16) h e (cid:107) [[ w h ]] (cid:107) L ( e ) + δ h e (cid:107) [[ χ h ]] (cid:107) L ( e ) (cid:17) . Similar to the derivation of (3.20), we have2 Im (cid:88) e ∈E Ih β (cid:104) [[ w h ]] , [[ v h ]] (cid:105) e = 2 Im (cid:88) e ∈E Ih β (cid:104) [ w h ] , [ α · ∇ h w h ] (cid:105) e (3.51) ≤ C (cid:88) e ∈E Ih βh − e (cid:88) K = K e ,K (cid:48) e (cid:107) [ w h ] (cid:107) L ( e ) (cid:107)∇ h w h (cid:107) L ( K ) ≤ C (cid:88) e ∈E Ih β h − e (cid:107) [ w h ] (cid:107) L ( e ) + 117 (cid:107)∇ h w h (cid:107) L (Ω) ≤ (cid:107) χ h (cid:107) L (Ω) + C (cid:88) e ∈E Ih (cid:16) β + 1 h e (cid:107) [[ w h ]] (cid:107) L ( e ) + δ h e (cid:107) [[ χ h ]] (cid:107) L ( e ) (cid:17) . DG METHODS FOR THE HELMHOLTZ EQUATION 21
Now substituting estimates (3.48)–(3.51), together with (3.15), (3.16) and (3.17),into (3.47) we obtain2 k (cid:107) w h (cid:107) L (Ω) ≤ B h ( w h , χ h ; v h , χ h )(3.52) + ( d −
2) Re B h ( w h , χ h ; w h , χ h ) − d (cid:107) χ h (cid:107) L (Ω) + Ck (cid:107) w h (cid:107) L (Γ) − c Ω (cid:88) e ∈E Bh (cid:107)∇ h w h (cid:107) L ( e ) + k (cid:107) w h (cid:107) L (Ω) + C (cid:88) e ∈E Ih k βh e β (cid:107) [ w h ] (cid:107) L ( e ) + 116 (cid:107) χ h (cid:107) L (Ω) + C (cid:88) e ∈E Ih βh e β (cid:107) [ w h ] (cid:107) L ( e ) + 1716 d (cid:107) χ h (cid:107) L (Ω) + Cd (cid:88) e ∈E Ih (cid:16) β + 1 h e (cid:107) [[ w h ]] (cid:107) L ( e ) + δ h e (cid:107) [[ χ h ]] (cid:107) L ( e ) (cid:17) + 116 (cid:107) χ h (cid:107) L (Ω) + C (cid:88) e ∈E Ih βh e d − (cid:88) j =1 β (cid:107) [ ∇ h w h · τ je ] (cid:107) L ( e ) + 116 (cid:107) χ h (cid:107) L (Ω) + C (cid:88) e ∈E Ih (cid:16) h e (cid:107) [[ w h ]] (cid:107) L ( e ) + δ h e (cid:107) [[ χ h ]] (cid:107) L ( e ) (cid:17) + 116 (cid:107) χ h (cid:107) L (Ω) + C (cid:88) e ∈E Ih (cid:16) β + 1 h e (cid:107) [[ w h ]] (cid:107) L ( e ) + δ h e (cid:107) [[ χ h ]] (cid:107) L ( e ) (cid:17) . On noting that (3.45) provides upper bounds for terms (cid:107) [[ χ h ]] (cid:107) L ( e ) , (cid:107) [ w h ] (cid:107) L ( e ) and k (cid:107) w h (cid:107) L (Γ) in terms of Im B h ( w h , χ h ; w h , χ h ), and the jumps of the tangentialderivatives (cid:107) [ ∇ h w h · τ je ] (cid:107) L ( e ) in (3.52) can be bounded by the inverse inequality(3.24), we get2 k (cid:107) w h (cid:107) L (Ω) + k (cid:107) w h (cid:107) L (Γ) + c Ω (cid:88) e ∈E Bh (cid:107)∇ h w h (cid:107) L ( e ) (3.53) ≤ B h ( w h , χ h ; v h , χ h ) + ( d −
2) Re B h ( w h , χ h ; w h , χ h )+ M Im B h ( w h , χ h ; w h , χ h ) + k (cid:107) w h (cid:107) L (Ω) + (cid:16)
14 + d (cid:17) (cid:107) χ h (cid:107) L (Ω) ≤ B h ( w h , χ h ; v h , χ h ) + ( d −
1) Re B h ( w h , χ h ; w h , χ h )+ M Im B h ( w h , χ h ; w h , χ h ) + 3 k (cid:107) w h (cid:107) L (Ω) − (cid:107) χ h (cid:107) L (Ω) , where M = C (cid:16) k + max e ∈E Ih (cid:16) k + 1 βh e + β + δh e + δh e + 1 βh e (cid:17)(cid:17) . (3.54)Using the linearity of the sesquilinear form B h , we have k (cid:107) w h (cid:107) L (Ω) + k (cid:107) w h (cid:107) L (Γ) + (cid:107) χ h (cid:107) L (Ω) + c Ω (cid:107)∇ h w h (cid:107) L (Γ) ≤ B h ( w h , χ h ; v h , χ h ) + 2( d −
1) Re B h ( w h , χ h ; w h , χ h )+ 2 M Im B h ( w h , χ h ; w h , χ h )= Re B h ( w h , χ h ; ˜ w h , χ h ) + 2 M Im B h ( w h , χ h ; w h , χ h ) , where ˜ w h = 4 v h + 2( d − w h and v h = α · ∇ h w h . Step 4: Finishing up.
It follows from the inverse inequality (3.24) that k (cid:107) v h (cid:107) L (Ω) + k (cid:107) v h (cid:107) L (Γ) ≤ Ck (cid:16) max K ∈T h h − K (cid:107) w h (cid:107) L (Ω) + c Ω (cid:107)∇ h w h (cid:107) L (Γ) (cid:17) . Then by the definition of the norm (cid:107) ( w h , χ h ) (cid:107) DG in (3.42) and the fact that ∇ h v h = ∇ h w h , we get (cid:107) ( ˜ w h , χ h ) (cid:107) DG ≤ (cid:107) ( v h , χ h ) (cid:107) DG + 2( d − (cid:107) ( w h , χ h ) (cid:107) DG ≤ C (cid:0) max K ∈T h h − K + k + 1 (cid:1) (cid:107) ( w h , χ h ) (cid:107) DG . Therefore,Re B h ( w h , χ h ; ˜ w h , χ h ) (cid:107) ( ˜ w h , χ h ) (cid:107) DG + Im B h ( w h , χ h ; w h , χ h ) (cid:107) ( w h , χ h ) (cid:107) DG (3.55) ≥ Re B h ( w h , χ h ; ˜ w h , χ h ) C (cid:0) max K ∈T h h − K + k + 1 (cid:1) (cid:107) ( w h , χ h ) (cid:107) DG + Im B h ( w h , χ h ; w h , χ h ) (cid:107) ( w h , χ h ) (cid:107) DG ≥ M Re B h ( w h , χ h ; ˜ w h , χ h ) + M Im B h ( w h , χ h ; w h , χ h ) (cid:107) ( w h , χ h ) (cid:107) DG ≥ c γ (cid:107) ( w h , χ h ) (cid:107) DG , for some constant c > γ defined by (3.41). Hence, (3.40) holds and theproof is complete. (cid:3) Stability estimates.
The generalized inf-sup condition proved in the lastsubsection immediately infers the following (absolute) stability and well-posednesstheorems for the LDG method
Theorem 3.3.
Let ( u h , σ h ) ∈ V h × Σ h solve (3.34) . Then there holds (cid:107) ( u h , σ h ) (cid:107) DG (cid:46) γ k − M ( f, g ) . (3.56) Proof.
On noting that any solution ( u h , σ h ) of (3.34) belongs to the set S h , thedesired estimate (3.3) follows readily from (3.40) with ( w h , χ h ) = ( u h , σ h ), (3.34)and (3.33). The proof is complete. (cid:3) Theorem 3.4.
The LDG method (3.34) has a unique solution for all k, h, δ, β > . Since the proof of the above theorem is a verbatim copy of that of Theorem 3.2,we omit it. 4.
Error estimates
The goal of this section is to derive error estimates for the LDG method u, σ ) using a corresponding coercive sesquilinearform of A h (resp. B h ) and derive error bounds for the projection. We note thatthe error analysis for the elliptic projections has an independent interest in itself(cf. [3]). Second, we bound the error between the projection and the LDG solutionusing the stability estimates obtained in Section 3. Since the error analysis for thetwo LDG methods are similar, we shall give more details of the error analysis for DG METHODS FOR THE HELMHOLTZ EQUATION 23 the LDG method j = 1 , H j ( T h ) = (cid:89) K ∈T h H j ( K ) , h = max K ∈T h h K ≈ max e ∈E h h e , β = β h − , δ = δ h for some positive constants β and δ .4.1. Error estimates for the LDG method
Elliptic projection and its error estimates.
For any ( w, χ ) ∈ H ( T h ) × H ( T h ) d , we define the elliptic projection ( ˜ w h , ˜ χ h ) ∈ V h × Σ h of ( w, χ ) by a h ( ˜ w h , ˜ χ h ; v h , τ h ) = a h ( w, χ ; v h , τ h ) ∀ ( v h , τ h ) ∈ V h × Σ h , (4.1)where a h ( w h , χ h ; v h , τ h ) : = A h ( w h , χ h ; v h , τ h ) + k ( w h , v h ) Ω (4.2) = ( χ h , ∇ h v h ) Ω + ı k (cid:104) w h , v h (cid:105) Γ − (cid:88) e ∈E Ih (cid:10) {∇ h w h } − ı β h − [[ w h ]] , [[ v h ]] (cid:11) e − (cid:88) e ∈E Ih (cid:16) ı δ (cid:104) [[ ∇ h w h ]] , [[ τ h ]] (cid:105) e − (cid:104) [[ w h ]] , { τ h }(cid:105) e (cid:17) + ( χ h , τ h ) Ω − ( ∇ h w h , τ h ) Ω . To derive error bounds for the above elliptic projection, we first notice that a h ( w h , χ h ; v h , −∇ h v h ) = ( ∇ h w h , ∇ h v h ) Ω + ı k (cid:104) w h , v h (cid:105) Γ − (cid:88) e ∈E Ih (cid:10) {∇ h w h } − ı β h − [[ w h ]] , [[ v h ]] (cid:11) e + (cid:88) e ∈E Ih (cid:16) ı δ (cid:104) [[ ∇ h w h ]] , [[ ∇ h v h ]] (cid:105) e − (cid:104) [[ w h ]] , {∇ h v h }(cid:105) e (cid:17) =: A h ( w h , v h ) . As a result, ˜ w h ∈ V h satisfies A h ( ˜ w h , v h ) = A h ( w, v h ) ∀ v h ∈ V h . (4.3)Moreover, since a h ( w h , χ h ; 0 , τ h ) = ( χ h , τ h ) Ω − ( ∇ h w h , τ h ) Ω − (cid:88) e ∈E Ih (cid:16) ı δ (cid:104) [[ ∇ h w h ]] , [[ τ h ]] (cid:105) e − (cid:104) [[ w h ]] , { τ h }(cid:105) e (cid:17) , we have that ˜ χ h ∈ Σ h satisfies( ˜ χ h , τ h ) Ω = ( ∇ h ˜ w h , τ h ) Ω (4.4) + (cid:88) e ∈E Ih (cid:16) ı δ (cid:104) [[ ∇ h ˜ w h − ∇ h w ]] , [[ τ h ]] (cid:105) e − (cid:104) [[ ˜ w h − w ]] , { τ h }(cid:105) e (cid:17) + ( χ , τ h ) Ω − ( ∇ h w, τ h ) Ω ∀ τ h ∈ Σ h . Lemma 4.1.
For any w, v ∈ H ( T h ) , there exists a k - and h -independent constant C such that |A h ( w, v ) | ≤ C ||| w |||| ,h ||| v ||| ,h . (4.5) Moreover, for any (cid:15) ∈ (0 , , there exists a constant c (cid:15) > such that Re A h ( v h , v h ) + (1 − (cid:15) + c (cid:15) ) Im A h ( v h , v h ) ≥ (1 − (cid:15) ) (cid:107) v h (cid:107) ,h , (4.6) where (cid:107) w (cid:107) ,h := (cid:16) (cid:107)∇ h w (cid:107) L (Ω) + k (cid:107) w (cid:107) L (Γ) (4.7) + (cid:88) e ∈E Ih (cid:16) β (cid:107) [ w ] (cid:107) L ( e ) + δ (cid:107) [[ ∇ h w ]] (cid:107) L ( e ) (cid:17)(cid:17) , ||| w ||| ,h := (cid:16) (cid:107) w (cid:107) ,h + (cid:88) e ∈E Ih β − (cid:107){∇ h w · n e }(cid:107) L ( e ) (cid:17) . (4.8)Since the proof of the above lemma is elementary, we omit it. We now recall thefollowing stability estimate for u (cf. [7, 10]): (cid:107) u (cid:107) H (Ω) (cid:46) ( k − + k ) M ( f, g ) , which is needed to prove the next lemma and will be used several times in the restof this section. Proposition 4.1.
Let u ∈ H (Ω) be the solution to problem (1.1) - (1.2) and σ = ∇ u . Let (˜ u h , ˜ σ h ) ∈ V h × Σ h denote the elliptic projection of ( u, σ ) defined by (4.1) . Then there hold the following error estimates: (cid:107) u − ˜ u h (cid:107) ,h + k (cid:107) u − ˜ u h (cid:107) L (Γ) (cid:46) (1 + kh ) kh, (4.9) (cid:107) u − ˜ u h (cid:107) L (Ω) (cid:46) (1 + kh ) kh , (4.10) (cid:107) σ − ˜ σ h (cid:107) L (Ω) (cid:46) (1 + kh ) kh. (4.11) Proof.
Since the proof of (4.9) and (4.10) is essentially same as that of [10, Lemma5.2], we omit it to save the space and refer the reader to [10] for the details.To show (4.11), on noting that (4.4) and the identity ( σ , τ h ) = ( ∇ h u, τ h ) imply( σ − ˜ σ h , τ h ) Ω = ( ∇ h u − ∇ h ˜ u h , τ h ) Ω − (cid:88) e ∈E Ih (cid:16) ı δ h (cid:104) [[ ∇ h ( u − ˜ u h )]][[ τ h ]] (cid:105) e (4.12) − (cid:104) [[ u − ˜ u h ]] , { τ h }(cid:105) e (cid:17) ∀ τ h ∈ Σ h . DG METHODS FOR THE HELMHOLTZ EQUATION 25
For any χ h ∈ Σ h , we set τ h = χ h − ˜ σ h . Then by (4.12), the trace inequality,Schwarz inequality we get (cid:107) σ − ˜ σ h (cid:107) L (Ω) = ( σ − ˜ σ h , σ − χ h ) Ω + ( σ − ˜ σ h , τ h ) Ω = ( σ − ˜ σ h , σ − χ h ) Ω + ( ∇ h ( u − ˜ u h ) , τ h ) Ω − (cid:88) e ∈E Ih (cid:16) ı δ h (cid:104) [[ ∇ h ( u − ˜ u h )]] , [[ τ h ]] (cid:105) e − (cid:104) [[ u − ˜ u h ]] , { τ h }(cid:105) e (cid:17) ≤ (cid:107) σ − ˜ σ h (cid:107) L (Ω) (cid:107) σ − χ h (cid:107) L (Ω) + (cid:107)∇ h ( u − ˜ u h ) (cid:107) L (Ω) (cid:107) τ h (cid:107) L (Ω) + C (cid:16) (cid:88) e ∈E Ih δ h (cid:107) [[ ∇ h ( u − ˜ u h )]] (cid:107) L ( e ) + β h − (cid:107) [ u − ˜ u h ] (cid:107) L ( e ) + ε (cid:88) e ∈E Ih h (cid:107) τ h (cid:107) L ( e ) (cid:17) ≤ (cid:107) σ − ˜ σ h (cid:107) L (Ω) + 14 (cid:107) τ h (cid:107) L (Ω) + (cid:107) σ − χ h (cid:107) L (Ω) + C (cid:107) u − ˜ u h (cid:107) ,h ≤ (cid:107) σ − ˜ σ h (cid:107) L (Ω) + 54 (cid:107) σ − χ h (cid:107) L (Ω) + C (cid:107) u − ˜ u h (cid:107) ,h . Hence, it follows from the above inequality, (4.9), and the polynomial approximationtheory (cf. [2]) that (cid:107) σ − ˜ σ h (cid:107) L (Ω) ≤ C (cid:107) u − ˜ u h (cid:107) ,h + 2 inf χ h ∈ Σ h (cid:107) σ − χ h (cid:107) L (Ω) (cid:46) (1 + kh ) hk + ( k + k − ) h (cid:46) (1 + kh ) hk, which gives (4.11). The proof is complete. (cid:3) Global error estimates for the LDG method
In the precedingsubsection we have derived the error bounds for ( u − ˜ u h , σ − ˜ σ h ). By the decom-position u − u h = ( u − ˜ u h ) + (˜ u h − u h ) and σ − σ h = ( σ − ˜ σ h ) + ( ˜ σ h − σ h ) andthe triangle inequality, it suffices to get error bounds for (˜ u h − u h , ˜ σ h − σ h ). Weshall accomplish this task by exploiting the linearity of the Helmholtz equation andusing the stability estimate for the LDG method u, σ ) satisfies A h ( u, σ ; v h , τ h ) = F ( v h , τ h ) ∀ ( v h , τ h ) ∈ V h × Σ h . (4.13)Subtracting (3.1) from (4.13) yields the following error equation (or Galerkin or-thogonality): A h ( u − u h , σ − σ h ; v h , τ h ) = 0 ∀ ( v h , τ h ) ∈ V h × Σ h . (4.14)Next, to proceed we introduce the notation u − u h = e h + q h , e h := u − ˜ u h , q h := ˜ u h − u h , σ − σ h = ψ h + φ h , ψ h := σ − ˜ σ h , φ h := ˜ σ h − σ h . Then by (4.14) and the definitions of the sesquilinear form a h and the ellipticprojection we have A h ( q h , φ h ; v h , τ h ) = − A h ( e h , ψ h ; v h , τ h )(4.15) = − a h ( e h , ψ h ; v h , τ h ) + k ( e h , v h )= k ( e h , v h ) , ∀ ( v h , τ h ) ∈ V h × Σ h . The above equation implies that ( q h , φ h ) ∈ V h × Σ h is the LDG solution to theHelmholtz problem with source terms f = k e h and g = 0. Then an application ofthe stability estimates of Theorem 3.1 immediately yields the following lemma. Proposition 4.2.
There hold the following estimates for ( q h , φ h ) : (cid:107) q h (cid:107) DG (cid:46) γ (1 + kh ) k h , (4.16) (cid:107) φ h (cid:107) L (Ω) (cid:46) γ (1 + kh + δ kh )(1 + kh ) kh. (4.17)Combining Proposition 4.1 and 4.2, using the triangle inequality and the stan-dard duality argument give the following error estimates for ( u h , σ h ). Theorem 4.1.
Let u ∈ H (Ω) be the solution to problem (1.1) – (1.2) and σ := ∇ u ,and ( u h , σ h ) be the solution to problem (3.1) . Then there hold the following errorestimates for ( u h , σ h ) : (cid:107) u − u h (cid:107) ,h + k (cid:107) u − u h (cid:107) L (Γ) (cid:46) (cid:0) (1 + kh ) + γ (1 + kh ) kh (cid:1) kh, (4.18) (cid:107) u − u h (cid:107) L (Ω) (cid:46) (1 + γ )(1 + kh ) kh , (4.19) (cid:107) σ − σ h (cid:107) L (Ω) (cid:46) (cid:0) (1 + kh ) + γ (1 + kh )(1 + kh + δ kh ) (cid:1) kh. (4.20)4.2. Error estimates for the LDG method
The error analysis for theLDG method a h needs to be replaced by anothersesquilinear form b h in the definition of the elliptic projection (4.1), where b h isdefined by b h ( w h , χ h ; v h , τ h ) : = B h ( w h , χ h ; v h , τ h ) + k ( w h , v h ) Ω (4.21) = ( χ h , ∇ h v h ) Ω + ı k (cid:104) w h , v h (cid:105) Γ − (cid:88) e ∈E Ih (cid:104){ χ h } − ı β [[ w h ]] , [[ v h ]] (cid:105) e − (cid:88) e ∈E Ih (cid:16) ı δ (cid:104) [[ χ h ]] , [[ τ h ]] (cid:105) e − (cid:104) [[ w h ]] , { τ h }(cid:105) e (cid:17) + ( χ h , τ h ) Ω − ( ∇ h w h , τ h ) Ω . Second, due to strong coupling between ˜ u h and ˜ σ h , the error estimates for thenew elliptic projection (˜ u h , ˜ σ h ) must be derived differently. To the end, we needthe following lemma, which replaces Lemma 4.1. Lemma 4.2.
Let β = β h − and δ = δ h for some positive constants β and δ . (i) There exists an h - and k -independent constant c > such that the sesquilin-ear form b h satisfies the following generalized inf-sup condition: for any fixed DG METHODS FOR THE HELMHOLTZ EQUATION 27 ( w h , χ h ) ∈ V h × Σ h sup ( v h , τ h ) ∈ V h × Σ h Re b h ( w h , χ h ; v h , τ h ) ||| ( v h , τ h ) ||| DG (4.22) + sup ( v h , τ h ) ∈ V h × Σ h Im b h ( w h , χ h ; v h , τ h ) ||| ( v h , τ h ) ||| DG ≥ c ||| ( w h , χ h ) ||| DG . (ii) There exists an h - and k -independent constant C > such that for any ( w, χ ) , ( v, τ ) ∈ H ( T h ) × H ( T h ) d , there holds | b h ( w, χ ; v, τ ) | ≤ C ||| ( w, χ ) ||| ,h ||| ( v, τ ) ||| ,h , (4.23) where ||| ( w, χ ) ||| DG := (cid:16) (cid:107) w (cid:107) ,h + (cid:107) χ (cid:107) L (Ω) (cid:17) , (4.24) ||| ( w, χ ) ||| ,h := (cid:16) ||| ( w, χ ) ||| DG + (cid:88) e ∈E Ih β − (cid:107){ χ }(cid:107) L ( e ) (cid:17) . (4.25)The proof of (i) is based on evaluating the first quotient on the left-hand side of(4.22) at ( v h , τ h ) = (cid:0) (1+ C ) w h , C χ h −∇ h w h (cid:1) and evaluating the second quotientat ( v h , τ h ) = ( C w h , C χ h ) for some sufficiently large positive constants C and C .The proof of (ii) is a straightforward application of Schwarz and trace inequalities.We skip the rest of the derivation to save space.The above generalized inf-sup condition, the boundedness of the sesquilinearform b h , and the duality argument (cf. [2]) readily infer the following error estimatesfor the new elliptic projection (˜ u h , ˜ σ h ). We omit the proof since it is standard. Proposition 4.3.
Under the assumptions of Proposition 4.1, there hold the fol-lowing estimates: (cid:107) u − ˜ u h (cid:107) ,h + (cid:107) σ − ˜ σ h (cid:107) L (Ω) (cid:46) kh, (4.26) (cid:107) u − ˜ u h (cid:107) L (Ω) (cid:46) k h . (4.27)The third difference is that the new error function ( q h , φ h ) now satisfies B h ( q h , φ h ; v h , τ h ) = k ( e h , v h ) ∀ ( v h , τ h ) ∈ V h × Σ h . (4.28)As a result, by Theorem 3.3 and (3.38) we get | q h | ,h + (cid:107) ( q h , φ h ) (cid:107) DG (cid:46) γ (1 + kh ) k h , (4.29)which replaces estimates (4.16) and (4.17).After having established Proposition 4.3 and (4.29), once again, by the triangleinequality we arrive at the following error estimates for the solution ( u h , σ h ) to theLDG method Theorem 4.2.
Let u ∈ H (Ω) be the solution to problem (1.1) – (1.2) and σ := ∇ u ,and ( u h , σ h ) be the solution to problem (3.34) . Then there hold the following errorestimates for ( u h , σ h ) : (cid:107) u − u h (cid:107) ,h + k (cid:107) u − u h (cid:107) L (Γ) (4.30) + (cid:107) σ − σ h (cid:107) L (Ω) (cid:46) (cid:0) γ (1 + kh ) kh ) (cid:1) kh, (cid:107) u − u h (cid:107) L (Ω) (cid:46) (1 + γ (1 + kh )) k h . (4.31) Remark 4.1. (4.29) shows that φ h := ˜ σ h − σ h has an optimal order (in h ) errorbound for the LDG method (4.17) shows that φ h only has a sub-optimalorder error bound for the LDG method σ than the LDG method h . Numerical experiments
In this section we shall provide some numerical results of the two proposed LDGmethods. Our tests are done for the following 2-d Helmholtz problem: − ∆ u − k u = f := sin( kr ) r in Ω , (5.1) ∂u∂n Ω + ı ku = g on Γ R := ∂ Ω . (5.2)Here Ω is the unit square [ − . , . × [ − . , . g is chosen so that the exactsolution is given by(5.3) u = cos( kr ) k − cos k + ı sin kk (cid:0) J ( k ) + ı J ( k ) (cid:1) J ( kr )in polar coordinates, where J ν ( z ) are Bessel functions of the first kind.Assume T /m be the regular triangulation that consists of 2 m right-angledequicrural triangles of size h = 1 /m , for any positive integer m . See Figure 1for the sample triangulation T / and T / . Figure 1.
The computational domain and sample meshes. Left: T / that consists of right-angled equicrural triangles of size h = ;Right: T / with h = .5.1. Sensitivity with respect to the parameters δ and β . In this subsection,we examine the sensitivity of the error of the LDG solutions in H -seminorm withrespect to the parameters δ and β .The LDG method δ = 0 . h e andtesting the sensitivity in the parameter β . With two wave numbers k = 5 and50, we compute the solutions of the LDG method β : DG METHODS FOR THE HELMHOLTZ EQUATION 29 . h − e , 0 . h − e , h − e and 1. The relative errors, defined by the errors in the H -seminorm divided by the exact solution in the H -seminorm, are shown in theleft graph of Figure 2. We observe that the relative errors have similar behaviorsand decay as mesh size h becomes smaller. This shows that the errors are notsensitive to the parameter β . Next, we fix β = 0 . h − e , and repeat the test withdifferent δ . The right graph of Figure 2 shows the relative errors with parameters δ = 0 . h e , 0 . h e , 10 h e and 0 .
1, and wave numbers k = 5 and 50. We observe thatthe errors have similar behaviors for small values δ = 0 . h e and 0 . h e . Larger δ results in larger error.The sensitivity tests of the LDG method Figure 2.
Relative error in the H -seminorm of the LDG method k = 5 and 50.Left: δ = 0 . h e is fixed, β = 0 . h − e , 0 . h − e , h − e and 1; Right: β = 0 . h − e is fixed, δ = 0 . h e , 0 . h e , 10 h e and 0 . Errors of the LDG solutions.
In this subsection, we fix the parameters andinvestigate the changes of the numerical errors as functions of the mesh size.We start from the LDG method δ = 0 . h e , β = 0 . h − e . The relative error of the LDG method, and the finite element interpolation areshown in the left graph of Figure 4, with four different wave numbers k = 5, 10,50 and 100. The relative error of the LDG solution stays around 100% before acritical mesh size is reached, then decays at a rate greater than − −
1) forsmall h . The critical mesh size decreases as k increases.The right graph of Figure 4 contains the relative error when we fix kh = 1 and hk = 0 .
5. It indicates that unlike the error of the finite element interpolation theerror of the LDG is not controlled by the magnitude of kh , which suggests that thereis a pollution contribution in the total error. The left graph of Figure 5 contains Figure 3.
Relative error in the H -seminorm of the LDG method k = 5 and 50.Left: δ = 0 . h e is fixed, β = 0 . h − e , 0 . h − e , h − e and 1; Right: β = 0 . h − e is fixed, δ = 0 . h e , 0 . h e , 10 h e and 0 . k h = 1 fordifferent values of h . The error does not increase with respect to k .The LDG method Figure 4.
Left: relative error of the LDG method H -seminorm for k = 5, 10, 50 and 100; Right: relative error of theLDG method H -seminorm for k = 1 , · · · , kh = 1and kh = 0 . kh >
1. The LDG method δ = 0 . h e , β = 0 . h − e , k = 100 and h = 1 /
45 has a large relative error of size 0 . DG METHODS FOR THE HELMHOLTZ EQUATION 31
Figure 5.
Left: relative error of the LDG method H -seminorm with k h = 1; Right: relative error of the LDG method H -seminorm for k = 5, 10, 50 and 100. Figure 6.
Relative error of the LDG method H -seminorm. Left: kh = 1 and kh = 0 . k h = 1.shows that the LDG solution has the correct shape/phase although its amplitudeis smaller.5.3. Comparison between the two LDG methods.
Two different LDG meth-ods are proposed in this paper. The first one is derived following the IPDG methodproposed in [10], and the second one has a more standard numerical flux formula-tion and is supposed to have a better approximation for the vector/flux variable.In this subsection, we provide a comparison between these two methods, in termsof the error and computational cost.We start by revisiting the test examples of Subsection 5.1. Instead of computingthe relative error of u h in the H -seminorm, we compute the relative error of σ h inthe L -norm for the LDG method Figure 7.
Left: surface plots of the finite element interpolation(left) and the LDG method δ = 0 . h e , β = 0 . h − e , k = 100 and h = 1 / β ,better approximation to σ is achieved for larger δ . It confirms our prediction thatthe LDG method Figure 8.
Relative error of σ h in L -norm of the LDG method k = 5 and 50.Left: δ = 0 . h e is fixed, β = 0 . h − e , 0 . h − e , h − e and 1; Right: β = 0 . h − e is fixed, δ = 0 . h e , 0 . h e , 10 h e and 0 . h , with the parameters δ = 0 . h e , β = 0 . h − e and k = 10. It shows thatthe computational cost of the LDG method Comparison between LDG and finite element solutions.
We have shownthe performance and comparison of the two LDG methods in previous subsections.
DG METHODS FOR THE HELMHOLTZ EQUATION 33
Table 1.
Comparison of the two LDG methods with parameters δ = 0 . h e and β = 0 . h − e .1 /h | u − u h | H order (cid:107) σ − σ h (cid:107) L order CPU time (s)5 4.1059E-01 5.4715E-01 0.064110 1.6915E-01 1.2794 2.4712E-01 1.1467 0.2381LDG Figure 9.
The traces of the LDG xz -plane, for k = 100 and h =1 /
50 (top), 1 /
120 (middle) and 1 /
200 (bottom), respectively. Thedotted lines are the traces of the exact solution.In this subsection, we provide a brief comparison between the LDG solution andthe P conforming finite element solution. We consider the Helmholtz problem (5.1)-(5.2) with wave number k = 100. Withmesh size h = 1 /
50, 1 /
120 and 1 / xz -plane in the left column of Figure 9. The exactsolution is also provided as a reference. The traces of the finite element solutionare shown in the right column of Figure 9. It is clear that the LDG method References [1] D. Arnold, F. Brezzi, B. Cockburn, and D. Marini. Unified analysis of discontinuous Galerkinmethods for elliptic problems.
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DG METHODS FOR THE HELMHOLTZ EQUATION 35
Department of Mathematics, The University of Tennessee, Knoxville, TN 37996,U.S.A.
E-mail address : [email protected] Department of Mathematics, The University of Tennessee, Knoxville, TN 37996,Computer Science and Mathematics Division, Oak Ridge National Laboratory, OakRidge, TN 37830, U.S.A.
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