Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mortar coupling of $hp$-discontinuous Galerkin and boundary element methods for the Helmholtz equation

Christoph Erath, Lorenzo Mascotto, Jens Markus Melenk, Ilaria Perugia, Alexander Rieder

TL;DR

A coupling of a discontinuous Galerkin finite element method with a boundary element method to solve the Helmholtz equation with variable coefficients in three dimensions is designed and analyzed.

Abstract

We design and analyze a coupling of a discontinuous Galerkin finite element method with a boundary element method to solve the Helmholtz equation with variable coefficients in three dimensions. The coupling is realized with a mortar variable that is related to an impedance trace on a smooth interface. The method obtained has a block structure with nonsingular subblocks. We prove quasi-optimality of the $h$- and $p$-versions of the scheme, under a threshold condition on the approximability properties of the discrete spaces. Amongst others, an essential tool in the analysis is a novel discontinuous-to-continuous reconstruction operator on tetrahedral meshes with curved faces.

Mortar coupling of $hp$-discontinuous Galerkin and boundary element methods for the Helmholtz equation

TL;DR

A coupling of a discontinuous Galerkin finite element method with a boundary element method to solve the Helmholtz equation with variable coefficients in three dimensions is designed and analyzed.

Abstract

We design and analyze a coupling of a discontinuous Galerkin finite element method with a boundary element method to solve the Helmholtz equation with variable coefficients in three dimensions. The coupling is realized with a mortar variable that is related to an impedance trace on a smooth interface. The method obtained has a block structure with nonsingular subblocks. We prove quasi-optimality of the - and -versions of the scheme, under a threshold condition on the approximability properties of the discrete spaces. Amongst others, an essential tool in the analysis is a novel discontinuous-to-continuous reconstruction operator on tetrahedral meshes with curved faces.

Paper Structure

This paper contains 18 sections, 16 theorems, 172 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Helmholtz model problem, boundary integral operators, and mortar coupling
  3. Helmholtz model problem
  4. Boundary integral operators
  5. Mortar coupling
  6. DGFEM-BEM mortar coupling
  7. DGFEM discretization of \ref{['first:system:Helmholtz']}
  8. BEM discretization of \ref{['second:system:Helmholtz']} and discretization of \ref{['third:system:Helmholtz']}
  9. Complete discrete formulation
  10. A Gårding inequality
  11. Continuity of $\mathcal{T}_h( (\cdot,\cdot,\cdot), (\cdot,\cdot,\cdot))$
  12. Adjoint problem
  13. Error analysis
  14. Numerical results
  15. Conclusions
  16. ...and 3 more sections

Key Result

Theorem 2.1

( FEMBEM:mortar) Let $\mathcal{T}(\cdot, \cdot)$ be defined as in form:T:for:Helmholtz, and assume that the interface $\Gamma$ is smooth. Then, there exists $c > 0$ depending only on $k_0$ and $\Omega$ and, for each $k \ge k_0$, there is a positive constant $c_G(k)$ depending on $k$ and $\Omega$, s

Figures (3)

  • Figure 1: $h$-version. Wave number $k=2\sqrt{3}\pi$.
  • Figure 2: $h$-version. Wave number $k=\sqrt{3}\pi$.
  • Figure 3: $p$-version.

Theorems & Definitions (35)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Theorem 4.4
  • proof
  • ...and 25 more