A generalized plane wave discontinuous Galerkin method for three-dimensional anisotropic Helmholtz equations with variable wave numbers

doi:10.1016/j.aml.2021.107595

Applied Mathematics Letters

Volume 123, January 2022, 107595

https://doi.org/10.1016/j.aml.2021.107595 Get rights and content

Abstract

In this paper we are concerned with the numerical method for three-dimensional anisotropic Helmholtz equations with variable wave numbers, where positive definite matrices define anisotropic media. We define novel generalized plane wave basis functions based on rigorous choice of the coordinate transformation. Then we derive the desired error estimates of the resulting approximate solutions with respect to the condition number of the coefficient matrices, under an assumption on the shape regularity of polyhedral meshes. Numerical results verify the validity of the theoretical results, and indicate that the approximate solutions generated by the proposed method possess high accuracies.

Introduction

The plane wave method, which falls into the Trefftz method [1], is based on the Trefftz approximation space made of plane wave basis functions. This type of numerical method was first introduced to solve Helmholtz equations and time-harmonic Maxwell’s equations, and has seen rapid algorithmic development and extensions since then; see [2], [3], [4], [5], [6]. Based on the fact that the plane wave method is restricted to the case of piecewise constant wavenumbers, in order to extend the plane wave method to the case of variable coefficients, a generalized plane wave method was developed in [7] to solve the homogeneous Helmholtz equation with smooth variable wave numbers. The key ingredient of generalized plane wave method is to construct by the Taylor expansion generalized plane wave basis functions, which are approximate solutions to the original governing equation without boundary conditions.

The three-dimensional Helmholtz and Maxwell equations in anisotropic media with positive definite matrices play an important role in practical physical applications, for example, determining the response of the inclusion to an impinging acoustic or electromagnetic wave (see [8], [9], [10], [11]). Besides, example electromagnetic problems within this class also include the design of waveguides and antennas, scattering of electromagnetic waves from automobiles and aircraft, and the penetration and absorption of electromagnetic waves by dielectric objects (see [12]). Recently, the PWDG methods [13] have been developed to solve the Helmholtz equation and time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are positive definite matrices, and the error estimates of the resulting approximate solutions with respect to the condition number of the coefficient matrices are proved.

In this paper we first consider three-dimensional Helmholtz equations with variable wave numbers in anisotropic media with positive definite matrices. Based on the coordinate transformation and the Taylor expansion, we first derive generalized plane wave basis functions for the considered model, and prove that the transformation has the desired stability estimates on the condition number. In order to build better convergence results, an assumption on the triangulation that the transformed triangulation ${\hat{T}}_{\hat{h}}$ is shape regular rather than the physical triangulation $T_{h}$ is employed. Further, we prove that the resulting approximate solutions generated by the generalized plane wave discontinuous Galerkin method (GPWDG) have the desired convergence order with respect to the condition number and the mesh-width of the triangulation, respectively. Numerical experiments show that the approximate solutions are slightly affected by the condition number, and possess high convergence orders with respect to the mesh-width of the triangulation.

The paper is organized as follows. In Section 2, we describe the proposed GPWDG method for anisotropic problems with variable coefficients. In Section 3, we explain how to discretize the variational problem. In Section 4, we give error estimates for the corresponding approximate solutions. Finally, we report some numerical results to confirm the effectiveness of the proposed method.

Section snippets

The model and its variational formula

We want to compute a numerical approximation of the smooth solution $u$ of $\{\begin{aligned} - \nabla \cdot A \nabla u - ω^{2} ϑ u = 0 in Ω, \\ n \cdot A \nabla u + i ω u = g on Γ = \partial Ω . \end{aligned}$ Here, $Ω$ is a bounded domain in three dimensions, $n$ denotes the unit outer normal vector to the boundary $Γ$ ; $A$ is positive definite matrix independent of space variable $x$ ; $ω > 0$ is the temporal frequency of the field, $ϑ \in L^{\infty} (Ω)$ is assumed to be a strictly positive, smooth and bounded real function, and $g \in L^{2} (Γ)$ .

For convenience, assume that $Ω$ is a polyhedron. We partition $Ω$ into computational

Discretization of variational problems

Since $A$ is positive definite matrix, there exist an orthogonal matrix $P$ and the diagonal positive definite matrix $Λ = diag (a_{x}, a_{y}, a_{z})$ such that $A = P^{T} Λ P$ , where $a_{x} \leq a_{y} \leq a_{z}$ are constant and the superscript $T$ denotes matrix transposition. Of course, we can assume that $det (P) = 1$ . It is clear that $A^{\frac{1}{2}} = P^{T} Λ^{\frac{1}{2}} P$ . Define a coordinate transformation: Set

Let $\hat{Ω}$ and ${\hat{Ω}}_{n}$ denote the images of $Ω$ and $Ω_{n}$ under the coordinate transformation (3.1), respectively. Since the map $S$ is linear, the transformed domain $\hat{Ω}$ and

Error estimates of the approximate solutions

We denote the condition number $cond (A)$ by $ρ$ for the anisotropic matrix $A$ . Then $ρ = cond (Λ) = {cond}^{2} (S)$ .

For a positive integer $j$ and a bounded and connected domain $D$ , let ${‖ v ‖}_{j, D}$ and ${| v |}_{j, D}$ denote the norm and the semi-norm of $v$ on the Sobolev space $H^{j} (D)$ , respectively. Define the $ω$ -weighted Sobolev norm ${‖ v ‖}_{s, ω, D}^{2} = \sum_{j = 0}^{s} ω^{2 (s - j)} {| v |}_{j, D}^{2}$ . We shall make the classical parameter choice (see [14]): $α = β = δ = 1 / 2$ and $γ = γ_{0} h^{3}$ , where $γ_{0}$ is a positive constant.

Define the broken Sobolev space $H^{2} (T_{h}) = {v \in L^{2} (Ω) : v |_{Ω_{k}} \in H^{2} (Ω_{k})$

Numerical experiments

Set $Ω = [0, 1] \times [0, 1] \times [0, 1]$ , $ϑ = x^{2} + y^{2} + z^{2}$ and $g = (x + 1) sin (y + z)$ . Since such analytic solution cannot be directly given, as usual we replace the analytic solution by its “good” approximation generated by the standard finite element method with very fine grids in order to compute accuracies of the generalized plane wave approximations generated by the proposed method. To measure the accuracy of the numerical solution $u_{h}$ , we introduce the $L^{2}$ relative error $e r r . ≔ \frac{{‖ u^{fi} - u_{h} ‖}_{L^{2} (Ω)}}{{‖ u^{fi} ‖}_{L^{2} (Ω)}}$ for the finite

Conclusion

In this paper we have introduced a generalized plane wave discontinuous Galerkin method for discretization of three-dimensional anisotropic Helmholtz equations with variable wave numbers, and derived the error estimates of the resulting approximate solutions with respect to $h$ and $ρ$ , respectively. We report some numerical results to verify the validity of theoretical results.

The presently proposed GPWDG discretizations will be extended to time-dependent anisotropic wave equations, in particular

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^☆: The author was supported by Shandong Provincial Natural Science Foundation, China under the grant ZR2020MA046 and Cultivation Project of Young and Innovative Talents in Universities of Shandong Province, China .

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A generalized plane wave discontinuous Galerkin method for three-dimensional anisotropic Helmholtz equations with variable wave numbers☆

Abstract

Introduction

Section snippets

The model and its variational formula

Discretization of variational problems

Error estimates of the approximate solutions

Numerical experiments

Conclusion

Comput. Math. Appl.

Ein gegenstuck zum ritzschen verfahren

Sec. Inte. Cong. Appl. Mech.

Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version

SIAM J. Numer. Anal.

Coupling method of plane wave DG and boundary element for electrogmagnetic scattering

Int. J. Numer. Anal. Model.

A plane wave least-squares method for time-harmonic Maxwell’s equations in absorbing media

SIAM J. Sci. Comput.

Adaptive BDDC algorithms for the system arising from plane wave discretization of Helmholtz equations

Internat. J. Numer. Methods Engrg.

Adaptive-multilevel BDDC algorithm for three-dimensional plane wave Helmholtz systems

J. Comput. Appl. Math.

Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the $p$ -version