Some error estimates for the numerical approximation of surface integrals

Abstract:

Recently, the first author introduced a new approach to the numerical quadrature of surface integrals in the context of boundary element methods. It is assumed that a global parametrization m of the surface is only indirectly given (e.g., via an iterative method) and that m is not accessible analytically. Of particular interest are parametrizations which are based on automatic triangulations of surfaces. In order to avoid an explicit reference to the partial derivatives of m, modified trapezoidal and midpoint rules were introduced. The present paper discusses some error estimates for these methods. The estimates are surprisingly difficult since $\mathcal {O}({h^3})$-terms have to be shown to cancel; this does not occur in the expansion of the standard rules.