Plane Wave Discontinuous Galerkin Methods Research Articles

A PWDG method for the Maxwell system in anisotropic media with piecewise constant coefficient matrix

In this paper we are concerned with plane wave discontinuous Galerkin (PWDG) methods for time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are piecewise constant symmetric matrices, where each constant symmetric matrix is defined on a medium (subdomain). By using suitable scaling transformations and coordinate (complex) transformations on every subdomain, the original Maxwell equation in anisotropic media is transformed into a Maxwell equation in isotropic media occupying a union domain of specific subdomains of complex Euclidean space. Based on these transformations, we define anisotropic plane wave basis functions and discretize the considered Maxwell equations by PWDG method with the proposed plane wave basis functions. We derive error estimates of the resulting approximate solutions, and further introduce a practically feasible local hp-refinement algorithm, which substantially improves accuracies of the approximate solutions. Numerical results indicate that the approximate solutions generated by the proposed PWDG methods possess high accuracy for the case of strong discontinuity media.

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

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.

Plane wave discontinuous Galerkin methods for the Helmholtz equation and Maxwell equations in anisotropic media

In this paper we are concerned with plane wave discontinuous Galerkin (PWDG) methods for Helmholtz equation and time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are matrices instead of numbers. We first define novel plane wave basis functions based on rigorous choices of scaling transformations and coordinate transformations. Then we derive the error estimates of the resulting approximate solutions with respect to the condition number of the coefficient matrices, under a new 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 PWDG method possess high accuracy.

A combined scheme of the local spectral element method and the generalized plane wave discontinuous Galerkin method for the anisotropic Helmholtz equation

In this paper we first consider a special class of nonhomogeneous and anisotropic Helmholtz equation with variable coefficients and the source term. Combined with local spectral element method, a generalized plane wave discontinuous Galerkin method for the discretization of such Helmholtz equation is designed. Then we define new generalized plane wave basis functions for two-dimensional anisotropic Helmholtz equation with the variable coefficient. Besides, the error estimates of the approximation solutions generated by the proposed discretization method are derived. Especially, the orders of the condition number ρ of the anisotropic matrix in the error estimates are optimal. Finally, numerical results verify the validity of the theoretical results, and indicate that the new method possesses high accuracy and is slightly affected by the pollution effect for the large wavenumbers.

A Trefftz-discontinuous Galerkin method for time-harmonic elastic wave problems

In this paper, we develop a plane wave discontinuous Galerkin method combined with local spectral element method for the elastic wave propagation in two and three space dimensions. We derive the error estimates of the approximation solutions in the mesh-dependent norm and the mesh-independent norm. Some dependence of the error bounds on the orders q of local spectral elements and the number p of plane wave propagation directions is given. Numerical results assess the validity of the theoretical results and indicate that the resulting approximate solutions generated by the PWDG–LSFE possess high accuracy.

The Plane Wave Discontinuous Galerkin Method Combined with Local Spectral Finite Elements for the Wave Propagation in Anisotropic Media

The Plane Wave Discontinuous Galerkin Method Combined with Local Spectral Finite Elements for the Wave Propagation in Anisotropic Media

Generalized plane wave discontinuous Galerkin methods for nonhomogeneous Helmholtz equations with variable wave numbers

ABSTRACT In this paper we are concerned with numerical method for nonhomogeneous Helmholtz equations with variable wave numbers. We derive generalized plane wave basis functions for three-dimensional homogeneous Helmholtz equations with variable wave numbers. Then, by combining the local spectral element method, we design a generalized plane wave discontinuous Galerkin method for the discretization of such nonhomogeneous Helmholtz equations (in both dimensions two and three dimensions). We show that the approximation solutions generated by the proposed discretization method yield error estimates with high accuracies. Numerical results indicate that the resulting approximate solutions generated by the new method possess high accuracy and verify the validity of the theoretical results.

A plane wave discontinuous Galerkin method with a Dirichlet-to-Neumann boundary condition for the scattering problem in acoustics

We consider the numerical solution of an acoustic scattering problem by the Plane Wave Discontinuous Galerkin Method (PWDG) in the exterior of a bounded domain in R2. In order to apply the PWDG method, we introduce an artificial boundary to truncate the domain, and we impose a non-local Dirichlet-to-Neumann (DtN) boundary condition on the artificial curve. To define the method, we introduce new consistent numerical fluxes that incorporate the truncated series of the DtN map. Error estimates with respect to the truncation order of the DtN map, and with respect to mesh width are derived. Numerical results suggest that the accuracy of the PWDG method for the acoustic scattering problem can be significantly improved by using DtN boundary conditions.

Optimization of plane wave directions in plane wave discontinuous Galerkin methods for the Helmholtz equation

Recently, the use of special local test functions other than polynomials in Discontinuous Galerkin (DG) approaches has attracted a lot of attention and became known as DG-Trefftz methods. In particular, for the 2D Helmholtz equation plane waves have been used in \[11] to derive an Interior Penalty (IP) type Plane Wave DG (PWDG) method and to provide an a priori error analysis of its $p$-version with respect to equidistributed plane wave directions. The dependence on the distribution of the plane wave directions has been studied in \[1] based on a least squares method. However, the emphasis in \[1] has been on the $h$-version of the PWDG method, i.e., decreasing the mesh width $h$ for a fixed number $p$ of plane wave directions. In this contribution, we are interested in the $p$-version, i.e., increasing $p$ for a fixed mesh-width $h$. We formulate the choice of the plane wave directions as a control constrained optimal control problem with a continuously differentiable objective functional and the variational formulation of the PWDG method as a further constraint. The necessary optimality conditions are derived and numerically solved by a projected gradient method. Numerical results are given which illustrate the benefits of the approach.

Novel Multilevel Preconditioners for the Systems Arising from Plane Wave Discretization of Helmholtz Equations with Large Wave Numbers

In this paper we are concerned with fast algorithms for the systems arising from the plane wave discretizations for two-dimensional Helmholtz equations with large wave numbers. We consider the plane wave weighted least squares (PWLS) method and the plane wave discontinuous Galerkin (PWDG) method. The main goal of this paper is to construct multilevel parallel preconditioners for solving the resulting Helmholtz systems. To this end, we first build a multilevel space decomposition for the plane wave discretization space based on a multilevel overlapping domain decomposition method. Then, corresponding to the space decomposition, we construct an additive multilevel preconditioner for the underlying Helmholtz systems. Further, we design both additive and multiplicative multilevel preconditioners with smoothers, which are different from the standard multigrid preconditioners. We apply the proposed multilevel preconditioners with a constant coarsest mesh size to solve two-dimensional Helmholtz systems generated by the PWLS method or PWDG method, and we find that the new preconditioners possess nearly stable convergence; i.e., the iteration counts of the preconditioned iterative methods (PCG or PGMRES) with the preconditioners increase very slowly when the wave number increases (and the fine mesh size decreases).

Substructuring Preconditioners for the Systems Arising from Plane Wave Discretization of Helmholtz Equations

In this paper we are concerned with the plane wave methods for Helmholtz equations with large wave numbers. We extend the plane wave weighted least-squares (PWLS) method and the plane wave discontinuous Galerkin (PWDG) method to Helmholtz equations in inhomogeneous mediums and with mixed boundary conditions. The main goal of this paper is to construct parallel preconditioners for solving the resulting discrete systems. To this end, we design a kind of substructuring domain decomposition preconditioner for both the two-dimensional case and the three-dimensional case, which is different from the existing substructuring preconditioners. We apply the proposed preconditioner to solve the Helmholtz systems generated by the PWLS method or the PWDG method, and we find that the new preconditioners with practical coarse mesh sizes possess relatively stable convergence, i.e., the iteration counts of the corresponding iterative methods (PCG or preconditioned GMRES) increase slowly when the wave number increases (and the mesh size decreases).

Residual-Based Adaptivity and PWDG Methods for the Helmholtz Equation

We present a study of two residual a posteriori error indicators for the plane wave discontinuous Galerkin (PWDG) method for the Helmholtz equation. In particular, we study the $h$-version of PWDG in which the number of plane wave directions per element is kept fixed. First, we use a slight modification of the appropriate a priori analysis to determine a residual indicator. Numerical tests show that this is reliable but pessimistic in that the ratio between the true error and the indicator increases as the mesh is refined. We therefore introduce a new analysis based on the observation that sufficiently many plane waves can approximate piecewise linear functions as the mesh is refined. Numerical results demonstrate an improvement in the efficiency of the indicators.

Plane Wave Discontinuous Galerkin Method with Lagrange Multipliers for Solving Time-Harmonic Maxwell's Equations in Three Dimensions

A discontinuous Galerkin method is formulated with plane wave basis functions and Lagrange multipliers, and it is investigated for solving vector curl-curl equations derived from Maxwell's equations. This method was previously developed for solving the scalar Helmholtz equation in both two and three dimensions. By defining vector plane waves and vector Lagrange multipliers within tetrahedral elements and on element boundaries, this algorithm is extended to solve three-dimensional vector problems for electromagnetic analysis. Numerical results for wave propagation in free space and scattering by a perfectly electrically conducting sphere are presented to validate the proposed method and evaluate its potential capability.

Dispersion analysis of plane wave discontinuous Galerkin methods

SUMMARYThe plane wave DG (PWDG) method for the Helmholtz equation was introduced and analyzed in [GITTELSON, C., HIPTMAIR, R., AND PERUGIA, I. Plane wave discontinuous Galerkin methods: analysis of the h‐version. Math. Model. Numer. Anal. 43 (2009), 297–331] as a generalization of the so‐called ultra‐weak variational formulation, see [O. CESSENAT AND B. DESPRéS, Application of an ultra weak variational formulation of elliptic PDEs to the two‐dimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), pp. 255–299]. The method relies on Trefftz‐type local trial spaces spanned by plane waves of different directions, and links cells of the mesh through numerical fluxes in the spirit of DG methods.We conduct a partly empirical dispersion analysis of the method in a discrete translation‐invariant setting by studying the mismatch of wave numbers of discrete and continuous plane waves traveling in the same direction. We find agreement of the wave numbers for directions represented in the local trial spaces. For other directions, the PWDG methods turn out to incur both phase and amplitude errors. This manifests itself as a pollution effect haunting the h‐version of the method. Our dispersion analysis allows a quantitative prediction of the strength of this effect and its dependence on the wave number and number of plane waves. Copyright © 2014 John Wiley & Sons, Ltd.

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.

Plane Wave Discontinuous Galerkin Methods for the 2D Helmholtz Equation: Analysis of the p-Version

Plane wave discontinuous Galerkin (PWDG) methods are a class of Trefftz-type methods for the spatial discretization of boundary value problems for the Helmholtz operator $-\Delta-\omega^2$, $\omega>0$. They include the so-called ultra weak variational formulation from [O. Cessenat and B. Després, SIAM J. Numer. Anal., 35 (1998), pp. 255–299]. This paper is concerned with the a priori convergence analysis of PWDG in the case of p-refinement, that is, the study of the asymptotic behavior of relevant error norms as the number of plane wave directions in the local trial spaces is increased. For convex domains in two space dimensions, we derive convergence rates, employing mesh skeleton-based norms, duality techniques from [P. Monk and D. Wang, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 121–136], and plane wave approximation theory.

Plane wave discontinuous Galerkin methods: Analysis of theh-version

We are concerned with a finite element approximation for time-harmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in medium-frequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. We focus on a class of Trefftz-type discontinuous Galerkin methods that employs trial and test spaces spanned by local plane waves. In this paper we give ap riori convergence estimates for the h-version of these plane wave discontinuous Galerkin methods in two dimensions. To that end, we develop new inverse and approximation estimates for plane waves and use these in the context of duality techniques. Asymptotic optimality of the method in a mesh dependent norm can be established. However, the estimates require a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We give numerical evidence that this requirement cannot be dispensed with. It reflects the presence of numerical dispersion.