An abstract analysis of parameter dependent problems and its applications to mixed finite element methods

This paper is concerned with numerical analysis of a class of parameter dependent problems. Such problems arise, for example, from the analysis of nearly incompressible media and from the penalization of some constrained minimization problems. We first consider the solvability and the uniqueness of an abstract problem under certain hypotheses, and then analyze the dependence of the solutions on the parameter. We also propose an iteration scheme, by which we can obtain solutions for fairly wide range of parameter values. We introduce a mixed finite element approximation to the present problem, and discuss its approximation properties. In particular, we consider the locking phenomenon, which often arises from numerical approximation of the present type of problems. We also give some simple numerical results to see the dependence of numerical solutions on the parameter. In Appendices, we present some sufficient conditions for the hypotheses employed to assure the solvability of the present problem and its approximations