Residual-based a posteriori error estimators for mixed finite element methods for fourth order elliptic singularly perturbed problems

Abstract

We consider mixed finite element approximation of a singularly perturbed fourth-order elliptic problem with two different boundary conditions, and present a new measure of the error, whose components are balanced with respect to the perturbation parameter. With different boundary conditions, the simply supported plate model and the clamped plate model are considered. In particular, a balanced energy norm has been defined. Based on the new norm, residual-based a posteriori estimators are developed for both problems, which are uniform with respect to both the perturbation parameter and the mesh function. A novel analysis approach is introduced for the clamped plate model to address certain difficulty of the problem. Numerical examples are provided to confirm theoretical findings.