Geometrically Graded h-p Quadrature Applied to the Complex Boundary Integral Equation Method for the Dirichlet Problem with Corner Singularities

Abstract

Boundary integral methods for the solution of boundary value PDEs are an alternative to `interior' methods, such as finite difference and finite element methods. They are attractive on domains with corners, particularly when the solution has singularities at these corners. In these cases, interior methods can become excessively expensive, as they require a finely discretised 2D mesh in the vicinity of corners, whilst boundary integral methods typically require a mesh discretised in only one dimension, that of arc length. Consider the Dirichlet problem. Traditional boundary integral methods applied to problems with corner singularities involve a (real) boundary integral equation with a kernel containing a logarithmic singularity. This is both tedious to code and computationally inefficient. The CBIEM is different in that it involves a complex boundary integral equation with a smooth kernel. The boundary integral equation is approximated using a collocation technique, and the interior solution is then approximated using a discretisation of Cauchy's integral formula, combined with singularity subtraction. A high order quadrature rule is required for the solution of the integral equation. Typical corner singularities are of square root type, and a `geometrically graded h-p' composite quadrature rule is used. This yields efficient, high order solution of the integral equation, and thence the Dirichlet problem. Implementation and experimental results in \textsc{matlab} code are presented.