Convergence rates of curved boundary element methods for the 3D Laplace and Helmholtz equations

TL;DR

This paper proves improved convergence rates for curved boundary element methods solving 3D Laplace and Helmholtz equations, supported by numerical experiments with high-order curved elements.

Contribution

It provides a rigorous analysis of consistency errors and establishes better convergence rates for curved boundary element methods in 3D.

Findings

Enhanced convergence rates for curved boundary element methods.

Validation through numerical experiments with high-order curved elements.

Analysis of consistency errors due to geometry perturbations.

Abstract

We establish improved convergence rates for curved boundary element methods applied to the three-dimensional (3D) Laplace and Helmholtz equations with smooth geometry and data. Our analysis relies on a precise analysis of the consistency errors introduced by the perturbed bilinear and sesquilinear forms. We illustrate our results with numerical experiments in 3D based on basis functions and curved triangular elements up to order four.