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Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

Xiaobing Feng, Haijun Wu

arXiv:0810.1475v1math.NA

TL;DR

The paper develops piecewise-linear IPDG methods for the Helmholtz equation with first-order absorbing boundaries in two and three dimensions. It proves stability without mesh constraints and derives error estimates, including an optimal broken H1 estimate and an L2 estimate that becomes optimal in the preasymptotic regime, while numerically examining pollution and implementation trade-offs.

  • Problem

    Large-wave-number Helmholtz discretizations face pollution and stability difficulties, while finite element well-posedness may require increasingly restrictive mesh constraints.

  • Method

    The paper constructs symmetric and non-symmetric piecewise-linear IPDG methods that penalize function-value, normal-derivative, and tangential-derivative jumps using complex penalty parameters.

  • Results

    The methods are stable and well-posed without mesh constraints, with optimal broken H1 error and an L2 estimate that becomes optimal when k^2h ≳1.

  • Takeaways & Limitations

    Numerical tests suggest that penalty-parameter tuning can significantly reduce pollution error, although IPDG uses more degrees of freedom than finite elements on the same mesh.

  • Takeaways & Limitations

    On the same mesh, the IPDG linear system has about three times as many degrees of freedom as the finite element system in two dimensions.

Abstract

from arXiv · show

This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the two and three dimensions. It is proved that the proposed discontinuous Galerkin methods are stable (hence well-posed) without any mesh constraint. For each fixed wave number $k$, optimal order (with respect to $h$) error estimate in the broken $H^1$-norm and sub-optimal order estimate in the $L^2$-norm are derived without any mesh constraint. The latter estimate improves to optimal order when the mesh size $h$ is restricted to the preasymptotic regime (i.e., $k^2 h \gtrsim 1$). Numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect (with respect to $k$) in the error bounds. The novelties of the proposed interior penalty discontinuous Galerkin methods include: first, the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives; second, the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters. Since the Helmholtz problem is a non-Hermitian and indefinite linear problem, as expected, the crucial and the most difficult part of the whole analysis is to establish the stability estimates (i.e., a priori estimates) for the numerical solutions. To the end, the cruxes of our analysis are to establish and to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in \cite{cummings00,Cummings_Feng06,hetmaniuk07}.

1. Introduction.

The paper develops piecewise-linear IPDG methods for high-dimensional Helmholtz problems with first-order absorbing boundaries, targeting stability, error estimates, and pollution effects without restrictive mesh constraints.

  • Problem setting: The study addresses Helmholtz wave propagation and acoustic scattering in two and three dimensions using artificial boundary conditions on a finite domain.The modeled problem uses a first-order absorbing Robin condition on the outer boundary and a sound-soft scatterer boundary.
  • Motivation: Finite element discretizations require kh ≲1 to resolve wavelengths, while large wave numbers can cause pollution and deteriorating solution accuracy.The paper notes that practical meshes commonly use 6–10 grid points per wavelength, yet pollution remains an issue as k increases.
  • Method: The proposed IPDG methods penalize jumps of function values, normal derivatives, and tangential derivatives across element edges or faces.Derivative-jump penalties are presented as critical to stability for these indefinite Helmholtz discretizations, unlike their merely helpful role in coercive elliptic problems.
  • Motivation: The paper emphasizes DG flexibility, including easy trial and test spaces, curved-boundary handling, unstructured meshes, and built-in coarse-grain parallelism.These properties motivate DG as an alternative to conventional finite element discretizations for wave problems.
  • Method: Complex penalty parameters with positive imaginary parts ensure stability for both symmetric and non-symmetric IPDG formulations.Because the Helmholtz problem is complex-valued, non-symmetric terms do not cancel when the same function is used in both arguments.
  • Results: The analysis proves stability and well-posedness without mesh constraints, with optimal broken H1 error and conditionally optimal L2 error estimates.The L2 estimate becomes optimal when k^2h ≳1; numerical tests also suggest reduced pollution through penalty-parameter tuning.

2. Notation and Preliminaries.

The preliminaries establish notation, geometric assumptions, triangulation conditions, and previously known stability estimates used in the subsequent IPDG analysis.

  • Notation: The paper uses C for a positive constant independent of h and k, with ≲, ≳, and ≃ denoting inequalities up to such constants.These symbols provide compact notation for mesh- and wave-number-dependent estimates.
  • Geometric preliminaries: A domain is star-shaped with respect to xQ when (x − xQ) · nQ ≥ cQ on its boundary, and strictly star-shaped when cQ is positive.The definition supplies the geometric structure needed for stability arguments.
  • Geometric assumptions: The analysis assumes that Ω1 is strictly star-shaped and the scatterer D is star-shaped with respect to the same point.The resulting boundary conditions use positive constants cΩ1 on ΓR and cD on ΓD.
  • Stability preliminaries: Earlier work provides stability estimates for the Helmholtz problem under the stated star-shaped-domain assumptions.The paper recalls these estimates as preliminaries for obtaining explicit wave-number dependence later.
  • Mesh notation: The computational domain is triangulated by elements K with diameters hK and edges or faces e with diameters he, under a minimal angle condition.These mesh quantities parameterize the family Th used to formulate the IPDG methods.

3. Formulation of discontinuous Galerkin methods.

The paper formulates symmetric and non-symmetric IPDG methods for the Helmholtz equation using piecewise linear trial spaces, jump penalties, and a first-order absorbing boundary condition. Its formulation penalizes value, normal-derivative, and tangential-derivative jumps, with complex penalty parameters supporting stability.

  • Discrete space and form: The IPDG energy space uses elementwise functions, oriented edge normals, and jumps across interior edges or faces.The formulation defines jump conventions and mesh-dependent norms before introducing the sesquilinear form.
  • Penalty structure: The sesquilinear form is consistent with −∆ and includes value, normal-derivative, and tangential-derivative jump terms.Its imaginary part contains the corresponding penalty contributions J0, J1, and L1.
  • Penalty parameters: Penalty parameters are pure imaginary with positive imaginary parts, while nonzero real parts are also admissible and may reduce pollution error numerically.The analysis permits complex parameters with positive imaginary parts; experiments indicate a potential benefit from nonzero real parts.
  • Weak formulation: The methods impose the first-order absorbing boundary condition through the boundary term ik⟨u,v⟩ΓR.The weak formulation combines the DG form, the −k^2 mass term, and source data on the absorbing boundary.
  • Discrete method: The approximation space Vh consists of elementwise linear polynomials, and uh is defined by the resulting discrete variational problem.The discrete equation uses the same sesquilinear form and absorbing-boundary term as the continuous weak formulation.

4. Stability estimates.

The stability analysis establishes well-posedness of the IPDG scheme without a mesh constraint by adapting PDE stability arguments to the discrete setting. The key tools are a local Rellich identity and an elementwise special test function.

  • Stability challenge: The discrete Helmholtz problem is strongly indefinite, making well-posedness under practical mesh constraints nontrivial.The rigidity of piecewise polynomial functions makes the discrete stability problem more delicate than the PDE case.
  • Stability mechanism: The analysis uses the elementwise test function vh=α·∇uh with α(x)=x−xΩ1 and a local Rellich identity.The first lemma establishes integral identities, including the local Laplacian Rellich identity used in the discrete estimates.
  • A priori estimate: For positive penalty parameters, Theorem 4.3 provides a priori stability estimates for the IPDG solution.The estimate is expressed in terms of the source norm M(f,g) and the stability constant Csta.
  • Penalty dependence: If γ1,e and β1,e vanish, the method remains stable but only with a weaker stability estimate derived using inverse inequalities.The stronger analysis therefore benefits from retaining these derivative-jump penalties.
  • Well-posedness: The IPDG method has a unique solution for every k>0 and he>0 when γ0,e>0, with γ1,e and β1,e allowed to be nonnegative.Thus the method is well-posed without imposing a mesh-size restriction.

5. Error analysis.

The error analysis separates elliptic-projection error from the difference between the projection and the IPDG solution, then applies discrete stability to combine them. It derives h-based estimates valid without a mesh constraint and identifies preasymptotic pollution behavior.

  • Assumptions and tools: The error estimates are developed for quasi-uniform meshes with γ1,e≃γ1>0 and an H2-regular problem.The projection analysis uses an auxiliary complex-valued Poisson problem and Nitsche duality for the L2 estimate.
  • Error decomposition: The error analysis first estimates an elliptic projection and then bounds the projection-to-IPDG error using the stability theorem.This decomposition exploits linearity, Galerkin orthogonality, and the projection estimates.
  • General estimates: Theorem 5.4 gives broken H1- and L2-error estimates for u−uh under the assumption u∈H2(Ω).The constants depend on the stability bound and penalty-related quantities.
  • Preasymptotic regime: The preasymptotic estimates apply for any h, while the regime k^2h≳1 yields the stated preasymptotic form.This contrasts with the cited finite-element result, whose preasymptotic estimate was proved only in one dimension under kh≤1.
  • Pollution effect: The second term in the broken H1 estimate is identified as a pollution term.For kh≲1, numerical tests suggest a possible bound of the form eC1kh+eC2k^3h^2 and indicate that tuning penalties can reduce pollution.

6. Numerical experiments.

The experiments examine stability and interpolation errors for increasing wave numbers, then compare IPDG and finite element solutions and test penalty-parameter sensitivity. They show that IPDG remains stable on coarse meshes, converges at the interpolation rate for small h, and can substantially reduce pollution with suitable parameters.

  • Setup: The experiments use a unit regular hexagon and regular triangulations of 6m^2 congruent equilateral triangles with h = 1/m.The sample mesh T1/7 has h = 1/7.
  • 6.1. Stability: The IPDG solution satisfies a stability estimate ∥uh∥1,h ≲ 1, which implies ∥uh∥L2(Ω) ≲ 1/k.These estimates are reported as better than those given by Theorem 4.5.
  • 6.2. Error of the finite element interpolation: Finite element interpolation errors decay with slope −1 in the H1-seminorm for k = 5, 10, 50, and 100, after remaining near 100% on coarse meshes.The critical mesh size is the point where the error begins to decrease and is expected to be at least π/k.
  • 6.2. Error of the finite element interpolation: For finite element interpolation, the error is controlled by kh: it stays around 0.247 for kh = 1 and around 0.124 for kh = 0.25.The ratio 0.124/0.247 ≈ 0.5 verifies the reported error scaling.
  • 6.3. Error of the DG solution: The IPDG error stays near 100% until a critical mesh size, then converges as fast as finite element interpolation for small h, while growing with k along kh = 0.25.Unlike interpolation error, IPDG error is not controlled by the magnitude of kh.
  • 6.5. Reduction of the pollution effect: Using parameters from (6.9) reduces pollution significantly, although the experiments do not show that the pollution error is eliminated.The method can be more effective than finite elements in the preasymptotic range, despite using about three times more degrees of freedom in two dimensions.
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