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

TL;DR

This paper develops an adaptive mixed finite element method for the Kirchhoff-Love plate bending problem, providing reliable error estimation and proving optimal convergence rates through boundary-condition-preserving complexes and nested discrete spaces.

Contribution

It introduces a novel residual-based a posteriori error analysis for the symmetric $H( ext{div} ext{div})$ mixed finite element method, ensuring optimal adaptive algorithm performance.

Findings

Numerical examples confirm the effectiveness of the error estimator.

Adaptive refinements achieve optimal convergence rates.

The method handles mixed boundary conditions effectively.

Abstract

This paper presents a reliable and efficient residual-based a posteriori error analysis for the symmetric $H(\operatorname{div}\operatorname{div})$ 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 $H(\operatorname{div}\operatorname{div})$ 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.