Abstract
This chapter focuses on building elliptic problem solvers with ELLPACK. Embedding a partial differential equation (PDE) solving system into a general purpose language allows to build PDE solvers for a much broader range of applications and, often, to achieve high efficiency. ELLPACK is designed to solve linear, second order elliptic problems in two or three dimensions. The geometry in two dimensions is general, but is rectangular in three dimensions. It can be used effectively to solve broad classes of nonlinear elliptic problems, initial value problems, and simultaneous nonlinear elliptic problems. The chapter focuses on the use of Fréchet derivatives and MACSYMA to provide a simple and effective implementation of Newton's method for nonlinear elliptic problems. ELLPACK is used to solve elliptic problems with interface conditions and on three dimensional cylindrical domains.
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