A fully discrete Calderon Calculus for two dimensional time harmonic waves

TL;DR

This paper introduces a fully discretized Calderón Calculus for 2D Helmholtz equations, using non-conforming Petrov-Galerkin methods on staggered grids, achieving second-order accuracy and demonstrated through numerical experiments.

Contribution

It develops a novel fully discrete Calderón Calculus for 2D Helmholtz problems using non-conforming Petrov-Galerkin schemes with staggered grids.

Findings

Numerical schemes achieve order h^2 accuracy.

The method performs well in numerical experiments.

The approach is based on Dirac delta and piecewise constant functions.

Abstract

In this paper, we present a fully discretized Calder\'{o}n Calculus for the two dimensional Helmholtz equation. This full discretization can be understood as highly non-conforming Petrov-Galerkin methods, based on two staggered grids of mesh size $h$, Dirac delta distributions substituting acoustic charge densities and piecewise constant functions for approximating acoustic dipole densities. The resulting numerical schemes from this calculus are all of order $h^2$ provided that the continuous equations are well posed. We finish by presenting some numerical experiments illustrating the performance of this discrete calculus.