Biharmonic obstacle problem: guaranteed and computable error bounds for approximate solutions

Abstract: The paper is concerned with a free boundary problem generated by the biharmonic operator and an obstacle. The main goal is to deduce a fully guaranteed upper bound of the difference between the exact minimizer u and any function (approximation) from the corresponding energy class (which consists of the functions in $H^2$ satisfying the prescribed boundary conditions and the restrictions stipulated by the obstacle). For this purpose we use the duality method of the calculus of variations and general type error identities earlier derived for a wide class of convex variational problems. By this method, we define a combined primal--dual measure of error. It contains four terms of different nature. Two of them are the norms of the difference between the exact solutions (of the direct and dual variational problems) and corresponding approximations. Two others are nonlinear measures, related to approximation of the coincidence set (they vanish if the coincidence set defined by means of the approximate solution coincides with the exact one). The measure satisfies the error identity, which right hand side depends on approximate solutions only and, therefore, is fully computable. Thus, the identity provides direct estimation of the primal--dual errors. However, it contains a certain restriction on the form of the dual approximation. In the second part of the paper, we present a way to skip the restriction. As a result, we obtain a fully guaranteed and directly computable error majorant valid for a wide class of approximations regardless of the method used for their construction. The estimates are verified in a series of tests with different approximate solutions. Some of them are quite close to the exact solution and others are rather coarse and have coincidence sets that differ much from the exact one. The results show that the estimates are robust and effective in all the cases.