Numerical evaluation of singular and finite-part integrals on curved surfaces using symbolic manipulation
This paper considers a class of integrals that arise with the boundary element method in three dimensions. This method reduces a three- dimensional partial differential equation problem to a two-dimensional surface integration problem. The simplification in the numerical problem due to the decrease in dimension is offset by an increase in difficulty because the resulting surface integrals over boundary patches can be weakly, Cauchy or hypersingular. The authors first show how to regularize the problem by reducing the finite part integrals to sums of weakly singular integrals. Some of the integrals in the sums involve derivatives of certain expansion functions. The authors describe how to evaluate these derivatives symbolically. Then they show how Duffy's transformation can be used to reduce the strength of the corner singularities in the final set of integrals so that all integrals can be evaluated numerically with Gauss rules. The authors analyse the convergence of this process and present numerical examples that demonstrate the validity of their analysis and overall efficiency of their methods.
scientific article; zbMATH DE number 150546
scientific article