On the coupling of boundary integral and finite element methods for nonlinear boundary value problems.

摘要

This dissertation is concerned with the application of the coupling of boundary integral and finite element methods to a class of nonlinear exterior boundary value problems. Specifically, the problem consists of nonlinear partial differential equations in a bounded inner region, and homogeneous linear equations in the corresponding unbounded exterior region, in addition to appropriate boundary and transmission conditions. As a prototype, a two-dimensional exterior Dirichlet problem for a class of nonlinear second order elliptic equations in divergence form is studied. Furthermore, as an application of our approach, a three-dimensional elasto-plastic interface problem is also included.;Emphases are given to the variational formulations, mathematical foundations, and Galerkin approximations of the coupling procedure. The method used boundary integral formulations to convert the problem under consideration into a nonlocal boundary value problem on a finite region where the nonlinearity occurs. Then, by means of a convenient weak formulation, this nonlocal problem is reduced to an equivalent operator equation. Existence, uniqueness, and approximation results for the solution of this operator equation are established from fundamental results in nonlinear functional analysis, including the theory of monotone operators. In the case of a strongly monotone and Lipschitz-continuous operator, an asymptotic error analysis for a boundary-finite element solution of the operator equation is provided.