Saturation and Reliable Hierarchical a Posteriori Morley Finite Element Error Control

Abstract

This paper proves the saturation assumption for the nonconforming Morley finite element discretization of the biharmonic equation. This asserts that the error of the Morley approximation under uniform refinement is strictly reduced by a contraction factor smaller than one up to explicit higher-order data approximation terms. The refinement has at least to bisect any edge such as red refinement or 3-bisections on any triangle. This justifies a hierarchical error estimator for the Morley finite element method, which simply compares the discrete solutions of one mesh and its red-refinement. The related adaptive mesh-refining strategy performs optimally in numerical experiments. A remark for Crouzeix-Raviart nonconforming finite element error control is included.

Generation of Error Indicators for Partial Differential Equations by Machine Learning Methods

“…These methods can be viewed as further development of the idea suggested by C. Runge. Among recent publications related to this method we mention Buffa and Garau [8], Carstensen et al [10], Yu et al [31]. In these publications, the reader will find many other references related to the method (see also Agouzal [1], Bank and Weiser [7], Duran and Rodriguez [11] and the references therein).…”