$hp$-Discontinuous Galerkin methods for the Helmholtz equation with large wave number

hp-Discontinuous Galerkin methods for the Helmholtz equation with large wave number

This paper develops some interior penalty hp-discontinuous Galerkin (hp-DG) methods for the Helmholtz equation in two and three dimensions. The proposed hp-DG methods are defined using a sesquilinear form which is not only mesh-dependent but also degree-dependent. In addition, the sesquilinear form contains penalty terms which not only penalize the jumps of the function values across the elemen...

Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

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 (...

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 −∆− ω, ω > 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...

Plane Wave Discontinuous Galerkin Methods

Standard low order Lagrangian finite element discretization of boundary value problems for the Helmholtz equation −∆u−ωu = f are afflicted with the so-called pollution phenomenon: though for sufficiently small hω an accurate approximation of u is possible, the Galerkin procedure fails to provide it. Attempts to remedy this have focused on incorporating extra information in the form of plane wav...

hp 3D Flow Simulations with Discontinuous Galerkin Methods