A Hybrid Boundary Element Method for Elliptic Problems with Singularities

TL;DR

This paper introduces a hybrid boundary element method that effectively handles singularities in elliptic boundary value problems, significantly improving accuracy over traditional BEM and GFEM methods.

Contribution

A novel hybrid boundary element method that combines local singularity treatment with standard BEM to enhance solution accuracy near singularities.

Findings

Reduces error in capacitance calculation by a factor of 10

Outperforms standalone BEM and GFEM in singularity regions

Successfully applied to the Motz problem

Abstract

The singularities that arise in elliptic boundary value problems are treated locally by a singular function boundary integral method. This method extracts the leading singular coefficients from a series expansion that describes the local behavior of the singularity. The method is fitted into the framework of the widely used boundary element method (BEM), forming a hybrid technique, with the BEM computing the solution away from the singularity. Results of the hybrid technique are reported for the Motz problem and compared with the results of the standalone BEM and Galerkin/finite element method (GFEM). The comparison is made in terms of the total flux (i.e. the capacitance in the case of electrostatic problems) on the Dirichlet boundary adjacent to the singularity, which is essentially the integral of the normal derivative of the solution. The hybrid method manages to reduce the error in the computed capacitance by a factor of 10, with respect to the BEM and GFEM.