Numerical solutions for boundary integral equations.

摘要

Boundary integral equations of the first kind with logarithmic kernels on smooth closed or open contours in {dollar}Rsp2{dollar} were considered. Instead of solving the first kind equations directly, a fully discrete quadrature method was proposed for the equivalent second kind equations with kernels defined by Cauchy singular integrals by simply using the trapezoidal integration rules and a modified quadrature formula for Cauchy singular integrals. Convergence of the method was completely analyzed. It is proved that the order of convergence is {dollar}O(1/nsp{lcub}2k{rcub}),{dollar} where n is the number of nodes in the quadrature formula and 2k + 2 is the degree of smoothness of the right-hand side function of the equation. Numerical examples were presented to confirm the theoretical estimate.; Vector-valued multiwavelets on a compact subset of R{dollar}sp{lcub}d{rcub}{dollar} were constructed and used for a Galerkin method for systems of integral equations of the second kind. A compression strategy was proposed for the coefficient matrix of the linear system obtained from this Galerkin method. It was shown that the compressed Galerkin method preserves, up to a log(N(M)) factor, optimal convergence rate of the original scheme and yields a sparse matrix with O(N(M)log(N(M))) or {dollar}O(N(M)lbrack {lcub}rm log{rcub}(N(M))rbracksp2){dollar} nonzero entries, depending on the parameters chosen in the compression strategy, and bounded condition number if the coefficient matrix is a {dollar}N(M) times N(M){dollar} matrix.