A comparison of adaptive algorithms for solving plane problems in the linear elasticity theory using the zero- and first-order Raviart – Thomas elements

Functional-type a posteriori error estimates are known for many problems of the elasticity theory. However, as followed from the work of S. I. Repin and A. V. Muzalevsky, the use of classical Finite Element Method (FEM) approximations for their implementation may lead to a growing overestimation of the absolute value of an error. Later, in the work of M. E. Frolov, it was shown that the use of approximations for mixed FEMs avoids a growing overestimation of the absolute error with mesh refinements. Further research in this direction was carried out by M. E. Frolov and M. A. Churilova using the simplest Raviart – Thomas and Arnold – Boffi – Falk approximations. In this paper, a comparative analysis is performed for zero-order and first-order Raviart – Thomas finite elements. It is shown for plane problems of linear elasticity that the use of the first-order Raviart – Thomas approximation significantly reduces an overestimation of the absolute error value.