A posteriori error analysis and adaptive finite element solution of variational inequalities of the second kind

Abstract

In the employment of the finite element method (and other numerical methods) for solving application problems, an important issue is to assess the reliability of the numerical solution. A posteriori error estimates provide quantitative information on the accuracy of the numerical solution and are the basis for the development of automatic, adaptive procedures for engineering applications of the finite element method. We perform an a posteriori error analysis for adaptive finite element solution of elliptic variational inequalities of the second kind. Using duality theory in convex analysis, we establish a general framework for a posteriori error estimation. We then derive a posteriori error estimates of residual type and of gradient recovery type, with particular choices of the dual variable present in the general framework. The reliability of the error estimates is rigorously shown. The efficiency of the error estimators is theoretically investigated and numerically validated. Detailed derivation and analysis of the error estimates are given for a model variational inequality of the second kind. We present extension of the results in solving other elliptic variational inequalities such as those arising in the study of frictional contact problems in elasticity. First, we derive a posteriori error estimates for a static frictional contact problem, and then consider a quasistatic case. We report numerous numerical examples, illustrating the effectiveness of the a posteriori error estimates.