Optimal multilevel adaptive FEM for the Argyris element

TL;DR

This paper demonstrates that an adaptive multilevel solver significantly improves the efficiency and convergence rates of the Argyris finite element method, making it more practical for complex problems with corner singularities.

Contribution

It introduces an efficient adaptive multilevel solver for the hierarchical Argyris FEM, achieving optimal convergence rates and linear time complexity on unstructured grids.

Findings

The adaptive solver is highly efficient with linear time complexity.

Optimal convergence rates are achieved in benchmarks with corner singularities.

Rehabilitation of the Argyris FEM from a computational perspective.

Abstract

The main drawback for the application of the conforming Argyris FEM is the labourious implementation on the one hand and the low convergence rates on the other. If no appropriate adaptive meshes are utilised, only the convergence rate caused by corner singularities [Blum and Rannacher, 1980], far below the approximation order for smooth functions, can be achieved. The fine approximation with the Argyris FEM produces high-dimensional linear systems and for a long time an optimal preconditioned scheme was not available for unstructured grids. This paper presents numerical benchmarks to confirm that the adaptive multilevel solver for the hierarchical Argyris FEM from [Carstensen and Hu, 2021] is in fact highly efficient and of linear time complexity. Moreover, the very first display of optimal convergence rates in practically relevant benchmarks with corner singularities and general boundary conditions leads to the rehabilitation of the Argyris finite element from the computational perspective.