Adaptive Time Discretization for Retarded Potentials

TL;DR

This paper introduces advanced adaptive time discretization techniques for retarded potential integral equations, combining smooth partition of unity methods with efficient quadrature and error estimation for improved numerical solutions.

Contribution

It develops a novel adaptive time discretization approach using $C^{ abla}$-partition of unity, quadrature for $C^{ abla}$ functions, and an a posteriori error estimator for enhanced accuracy.

Findings

Quadrature method converges rapidly.

Adaptive refinement improves solution accuracy.

Numerical experiments confirm efficiency and convergence.

Abstract

In this paper, we will present advanced discretization methods for solving retarded potential integral equations. We employ a $C^{\infty}$-partition of unity method in time and a conventional boundary element method for the spatial discretization. One essential point for the algorithmic realization is the development of an efficient method for approximation the elements of the arising system matrix. We present here an approach which is based on quadrature for (non-analytic) $C^{\infty}$ functions in combination with certain Chebyshev expansions. Furthermore we introduce an a posteriori error estimator for the time discretization which is employed also as an error indicator for adaptive refinement. Numerical experiments show the fast convergence of the proposed quadrature method and the efficiency of the adaptive solution process.