SPLITTING EXTRAPOLATIONS FOR SOLVING BOUNDARY INTEGRAL EQUATIONS OF MIXED BOUNDARY CONDITIONS ON POLYGONS BY MECHANICAL QUADRATURE METHODS

Abstract

To solve the boundary integral equations (BIE) of mixed boundary conditions, we propose the mechanical quadrature methods (MQM) using specific quadrature rule to deal with weakly singular integrals. Denote $h_{m}$ as the mesh width of a curved edge $\Gamma_{m}$ ($m=1,...,d)$ of polygons. Then the multivariate asymptotic expansions of solution errors are found to be $O(h^{3}),$ where $h=\max_{1\leq m\leq d}h_{m}.$ Hence, by using the splitting extrapolation methods (SEM), the high convergence rates as $O(h^{5})$ can be achieved. Moreover, numerical examples are provided to support our theoretical analysis.