Adaptive multiscale moment method for solving two-dimensional fredholm integral equation of the first kind – abstract

The aim of this paper is to deal with the problem of numerical solution of two-dimensional Fredholm integral equation of the first kind. First, two-dimensional multiscale basis is introduced following the tensor product of the one-dimensional multiscale triangular basis. We also discuss the geometrical meaning of the coefficients of the multiscale basis and on the compression technique for representing the original two dimensional function of the multiscale basis. Second, by use of this kind of basis, the multiscale moment method for solving two-dimensional Fredholm integral equation of the first kind has been proposed. Furthermore, the adaptive algorithm of the multiscale moment method (AMMM) has been presented according to the characteristics of solution coefficients of the integral equation. From (V -1) -th scale to V-th scale, the approximate solution on the finer grid is predicted from the known solution on the coarse grid by use of a tensor B-spline interpolant product. Many of the numerical simulations are carried out to test the feasibility of the multiscale moment method and on how to implement the adaptive algorithm. It will be found that the adaptive algorithm can reduce the size of linear equations constructed from multiscale moment method for two-dimensional Fredholm integral equation. The two-dimensional AMMM is an efficient, stable and adaptive algorithm to solve integral equations.