Mixed Finite Element Approximations via Interior and Exterior Penalties for Contact Problems in Elasticity

The use of exterior pcnalty formulations of boundary value problems as a basis for the development of finite element methods has gained much popularity in recent times. Such methods frequently lead to fewer unknowns than Illultiplier methods: they sometimes provide regularity that admits the use of numerical schcmes that might not be otherwisc applicable. alld they can produce one-parameter families of approximate solutions. depending upon the penalty parameter e, all members of which have a useful physical or mathematical significance. For a discussion of some aspects of finite element methods based on exterior penalty ideas. sec, for instance. [I] or [2]. There are two important aspects of penalty mcthods that motivate the present investigation. First of all. penalty methods are widely used in optimization problems in IR", and in the optimization literature one finds not only exterior penalty methods in frequent use. but also so-called interior penalty methods. hybrid (exterior/interior) methods. exponential penalties, etc. Are any of these alternative penalty approaches useful in variational formulation and approximation of boundary value problems? Second. the penalty finite element methods are closely related to mixed finite element methods and a brief examination of some of their properties seems appropriate for a symposium devoted to this general class of methods. We shall be concerned here with penalty-type formulations and finite clement approximations of a class of contact problems in elasticity of the form