Galerkin method and implementation of functional type a posteriori error estimates with black-box solvers for Linear Elasticity

A posteriori error estimates provide an explicit accuracy control for numerical solutions of boundary-value problems for Partial Differential Equations. It is an important and practically significant area of research in applied mathematics. Regardless of the fact that the classical methods of a posteriori error control are investigated very intensively for several decades, functional-type a posteriori estimates are very promising. This approach is fully reliable and it can be used to solvers with some hidden details of numerical implementations. Such estimates are known for many problems of the elasticity theory. However, as follows from the work of Prof. S. Repin and Dr. A. Muzalevsky, when implementing a posteriori estimates of the functional type, the use of classical approximations of the Finite Element Method leads to a growing overestimation of the absolute value of the error. Later, in the work of M. Frolov, this effect is highlighted more transparently. It is shown that the use of approximations that are more natural for mixed finite element methods avoids a growing overestimation of the absolute error value with mesh refinements. Comprehensive numerical testing and justification of this approach are provided in joint papers with Dr. M. Churilova and Ph.D. student D. Petukhov. In particular, a comparative analysis is performed for zero-order and first-order Raviart-Thomas finite elements implemented in MATLAB by D. Petukhov. For plane problems of linear elasticity, it is shown that the use of the first-order Raviart-Thomas approximation significantly reduces an overestimation of the absolute error value.