Boundary Element Methods for Semilinear Elliptic Partial Differential Equations (II): Algorithms and Computations for Unstable Solutions from Various Models

Abstract

We continue the study of semilinear elliptic BVPs of the form

$$\left\{\begin{array}{lllll}\Delta u(x) + f(x,u(x)) = 0,& x\in\Omega, \\ u(x) = 0,& x\in\partial\Omega \end{array}\right.$$

(12.1)

where Ω is a bounded open domain in R^N, N = 2, and f is a nonlinear function of x and u. We will deal with f ≡ u^p, –u + u^p, or variants thereof. We wish to compute numerical solutions of (12.1) by BEMand plot their graphics for visualization. on domains with various geometries and topologies. We also hope to survey existing algorithms and to introduce new ones, set certain numerical benchmarks, and explore singular perturbation cases. This chapter is based mainly on our prior work [37].