Implementation of error control for solving plane problems in linear elasticity by mixed finite elements

The paper is devoted to investigation of one of the modern approaches to a posteriori error control for approximate solutions of boundary-value problems. Such type of approaches provides, for the given approximate solution and problem data, a direct computation of a quantitative error estimate for deviation from the unknown exact solution of a problem. In this work, we use the functional approach proposed by S. Repin and colleagues - see, for example, books by P. Neittaanmäki and S. Repin (Elsevier, 2004) and by S. Repin (de Gruyter, 2008) and references cited therein. It allows us to determine the reliable and sufficiently sharp upper bounds to errors for a wide class of approximate solutions. To demonstrate the potentials of the approach, the well-known software for engineering computations ANSYS is employed. Most of the standard approaches, including the technique implemented in the package, are strictly applicable only to a restricted set of approximate solutions with additional mathematical properties. The analysis of a series of plane strain problems shows that the functional approach with implementation based on the Arnold-Boffi-Falk mixed finite element (SIAM J. Numer. Anal., 2005, vol. 42, no. 6, pp. 2429-2451) significantly extends the capabilities of the standard methodology of the package. Numerical calculations are based on examples proposed in the paper by V. Manet (Composites Science and Technology, 1998. V. 58, N. 12. P. 1899-1905). The data thus obtained point to highly efficient robust estimates of the true error. At the same time, the standard procedure of the package may lead to a significant growth of overestimation of the error during the process of finite element mesh refinement.