Convergence analysis of a multigrid algorithm for the acoustic single layer equation

TL;DR

This paper develops and analyzes a multigrid algorithm for solving the acoustic single layer equation in 2D, providing error analysis and numerical insights into eigenfunction behavior with a focus on the weak inner product.

Contribution

It introduces a multigrid method tailored for the acoustic single layer equation and offers a comprehensive error analysis and numerical study of eigenfunction oscillations.

Findings

The multigrid algorithm converges effectively for the acoustic single layer equation.

The weak inner product influences the oscillatory behavior of eigenfunctions.

Numerical results support the theoretical error estimates.

Abstract

We present and analyze a multigrid algorithm for the acoustic single layer equation in two dimensions. The boundary element formulation of the equation is based on piecewise constant test functions and we make use of a weak inner product in the multigrid scheme as proposed in \cite{BLP94}. A full error analysis of the algorithm is presented. We also conduct a numerical study of the effect of the weak inner product on the oscillatory behavior of the eigenfunctions for the Laplace single layer operator.