Mechanical Quadrature Methods and their Extrapolations for Solving the First Kind Boundary Integral Equations of Stokes Equation

In this article the mechanical quadrature methods (MQMs) and their extrapolations are proposed and analyzed for solving the first kind boundary integral equations of Stokes equation with closed smooth boundary or closed piecewise curved boundary. It is straightforward and cost efficient to obtain the entries in the linear system arising from the MQMs. The condition numbers of the discrete matrices are of only O(h-1) and the MQMs achieve higher accuracy than the collocation and Galerkin methods. The analysis of MQMs is more challenging than that of the collocation and Galerkin methods since its theory is no longer within the framework of the projection theory. In this article the convergence of the MQM solutions and the asymptotic expansions of the MQM solution errors are proved for both of the two types of boundary. In order to further improve the accuracy, a Richardson extrapolation is constructed for the mechanical quadrature solution on the smooth boundary and a splitting extrapolation is constructed for the mechanical quadrature solution on the piecewise curved boundary based on the asymptotic expansions of the errors. Numerical examples are provided to illustrate the features of the proposed numerical methods