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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 Numerical optimization and variational techniques
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
49J40 Variational inequalities
49M15 Newton-type methods
35K90 Abstract parabolic equations
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