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A Posteriori Estimates for Variational Inequalities

In: Numerical Mathematics and Advanced Applications 2009

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  • S. Repin

    (Russian Academy of Sciences, St. Petersburg Department of Steklov Institute of Mathematics)

Abstract

This paper is concerned with guaranteed and computable error bounds for approximate solutions of variational inequalities. The estimates are derived by purely functional methods. The first method is based upon methods of convex analysis and calculus of variations and the second one derives estimates with the help of certain transformations of the corresponding variational inequality. Both methods (variational and nonvariational) has been earlier developed and applied for linear problems where they lead to the same estimates [Two-sided estimates of deviation from exact solutions of uniformly elliptic equations, 2001]. In the paper, we shortly discuss variational inequalities associated with obstacle type problems and show that both methods also result in the same error majorants. The majorants are valid for any approximation from the admissible functional class and does not exploit Galerkin orthogonality, higher regularity of solutions, or a priori information on the structure of coincidence set. Also, the paper contains a concise overview of results related to similar a posteriori error estimates derived for other classes of nonlinear problems.

Suggested Citation

  • S. Repin, 2010. "A Posteriori Estimates for Variational Inequalities," Springer Books, in: Gunilla Kreiss & Per Lötstedt & Axel Målqvist & Maya Neytcheva (ed.), Numerical Mathematics and Advanced Applications 2009, pages 755-762, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-11795-4_81
    DOI: 10.1007/978-3-642-11795-4_81
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