## Estimates for the rate of convergence of finite element approximations of the solution of a time-dependent variational inequality

Master Thesis

1993

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University of Cape Town

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The main aim of this thesis is to analyse two types of general finite element approximations to the solution of a time-dependent variational inequality. The two types of approximations considered are the following: 1. Semi-discrete approximations, in which only the spatial domain is discretised by finite elements; 2. fully discrete approximations, in which the spatial domain is again discretised by finite elements and, in addition, the time domain is discretised and the time-derivatives appearing in the variational inequality are approximated by backward differences. Estimates of the error inherent in the above two types of approximations, in suitable Sobolev norms, are obtained; in particular, these estimates express the rate of convergence of successive finite element approximations to the solution of the variational inequality in terms of element size h and, where appropriate, in terms of the time step size k. In addition, the above analysis is preceded by related results concerning the existence and uniqueness of the solution to the variational inequality and is followed by an application in elastoplasticity theory.

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Bibliography: pages 93-101.

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Schroeder, G. 1993. Estimates for the rate of convergence of finite element approximations of the solution of a time-dependent variational inequality. University of Cape Town.