Multi-Level Iteration Methods for Solving Integral Equations of the Second Kind

In this paper we develop multi-level iteration methods for solving Fredhom integral equations of the second kind based on the Galerkin method for which the Galerkin subspace has a multi-resolution decomposition. After expressing the equations using matrices of operators in accordance to the multi-resolution structure, we propose two iteration schemes for solving the equations that are analogues to the Jacobi and Gauss-Seidel iteration schemes for solving algebraic systems. We then discuss the two-grid nature of the schemes, compare them with the well-known two-grid schemes and a two-level scheme and prove their convergence. We also present our numerical implementation of these methods using piecewise linear polynomial wavelets for an integral equation with the logarithmic kernel.