In this chapter we develop multiscale methods for solving the Hammerstein equation, and the nonlinear boundary integral equation resulting from a reformulation of a boundary value problem of the Laplace equation with nonlinear boundary conditions. Fast algorithms are proposed using the MAM, in conjunction with matrix truncation strategies and techniques of numerical integration for integrals appearing in the process of solving equations. We prove that the proposed methods require only linear (up to a logarithmic factor) computational complexity and have the optimal convergence order.

In the section that follows we discuss the critical issues in solving nonlinear integral equations. This will shine a light on the ideas developed later in this chapter. In Section 10.2, we introduce the MAM for solving Hammerstein equations and provide a complete convergence analysis for the proposed method. In Section 10.3, we develop the MAM for solving the nonlinear boundary integral equation as a result of a reformulation of a boundary value problem of the Laplace equation with nonlinear boundary conditions. We present numerical experiments in Section 10.4.

Critical issues in solving nonlinear equations

Nonlinear integral equations portray many mathematical physics problems. The Hammerstein equation is a typical kind of nonlinear integral equation. Moreover, boundary value problems of the Laplace equation serve as mathematical models for many important applications. Making use of the fundamental solutions of the equation, we can formulate the boundary value problems as integral equations defined on the boundary (see, Section 2.2.3). For linear boundary conditions, the resulting boundary integral equations are linear, the numerical methods of which have been studied extensively. Nonlinear boundary conditions are also involved in various applications. In these cases, the reformulation of the corresponding boundary value problems leads to nonlinear integral equations.

The nonlinearity introduces difficulties in the numerical solution of the equation, which normally requires an iteration scheme to solve it locally as a linearized integral equation.