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A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems. (English) Zbl 0954.65052

The purpose of this paper is to study a question concerning the discretization theory for a class of nonlinear evolution inequalities that encompasses time dependent monotone operator equations and parabolic variational inequalities. This discretization theory combines a backward Euler scheme for time discretization and the Galerkin method for space discretization. It is included the set of convex subsets in the sense of Glowinski-Mosco-Stummel to allow a nonconforming approximation of unilateral constraints. As an application the parabolic Signorini problems involving the $p$-Laplacian (with standard piecewise polynomial finite elements for space discretization) are proposed.

Main result: A new approach in view to the convergence analysis of Glowinski-Lions-Trémolières (the variational inequalities given by bilinear forms) to nonlinear evolution problems is presented. Without imposing any regularity assumption for the solution various norm convergence results for piecewise linear (or piecewise quadratic) trial functions are established. The authors present a new theory of full space time discretization for the evolution problem. Finally, new discretization theory to $p$-harmonic Signorini initial boundary value problems is applied.

##### MSC:
 65K10 Optimization techniques (numerical methods) 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 35K85 Linear parabolic unilateral problems; linear parabolic variational inequalities 49J40 Variational methods including variational inequalities 49M15 Newton-type methods in calculus of variations 35K90 Abstract parabolic equations
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