Optimality of adaptive $H(\mathrm{div}\mathrm{div})$ mixed finite element methods for the Kirchhoff-Love plate bending problem

Abstract

This paper presents a reliable and efficient residual-based a posteriori error analysis for the symmetric $H(\mathrm{div}\mathrm{div})$ mixed finite element method for the Kirchhoff-Love plate bending problem with mixed boundary conditions. The key ingredient lies in the construction of boundary-condition-preserving complexes at both continuous and discrete levels. Additionally, the discrete symmetric $H(\mathrm{div}\mathrm{div})$ space is extended to ensure nestedness, which leads to optimality for the adaptive algorithm. Numerical examples confirm the effectiveness of the a posteriori error estimator and demonstrate the optimal convergence rate under adaptive refinements.