A Self-adaptive Solver For Fredholm Integral Equations Of The Second Kind With Weakly Singular Kernels

In this thesis, we present a fast self-adaptive algorithm for Fredholm integral equations of the second kind with weakly singular kernels. Briefly speaking, we design an algorithm which can determine the correlated parameters such that the corresponding numerical solution has given accuracy.The thesis is divided into two parts. In the first part of the thesis, we discuss the solution of the discrete linear systems of the equations by the Generalized Minimal Residual (GMRES) method. Since the main cost per iteration of the GMRES method is the cost of the matrix-vector multiplication, the fast matrix-vector multiplication is the key of the algorithm. To obtain the fast matrix-vector multiplication algorithm, we approximate the kernel function with piece-wise polynomial and get the approximate coefficient matrix A using the approximate kernel function. Let n be the number of quadrature points and k2 be the number of interpolating knots at each subdomain. It is proved that the construction of A and the matrix-vector multiplication Ax require O(nk) operations respectively. The storage of A is also O(nk}. At the end of this part, the stability of the algorithm is discussed and numerical results are given to illustrate the stability and efficiency of our algorithms.In the second part of the thesis, we deduce the self-adaptive algorithm. We discuss the connection between the accuracy of approximate matrix and the degree of interpolating polynomials and study the method of estimating the accuracy of the numerical solution. Our algorithm is recursive: doubling the number of quadrature points until the numerical solution reach the preset accuracy. Finally, numerical results are given to illustrate the efficiency and feasibility of our algorithms.