Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms

Abstract The global arrangement of the degrees of freedom in a standard Argyris finite element method (FEM) enforces C 2 {C^{2}} at interior vertices, while solely global C 1 {C^{1}} continuity is required for the conformity in H 2 {H^{2}} . Since the Argyris finite element functions are not C 2 {C^{2}} at the midpoints of edges in general, the bisection of an edge for mesh-refinement leads to non-nestedness: the standard Argyris finite element space A ′ ⁢ ( 𝒯 ) {A^{\prime}(\mathcal{T})} associated to a triangulation 𝒯 {\mathcal{T}} with a refinement 𝒯 ^ {\widehat{\mathcal{T}}} is not a subspace of the standard Argyris finite element space A ′ ⁢ ( 𝒯 ^ ) {A^{\prime}(\widehat{\mathcal{T}})} associated to the refined triangulation 𝒯 ^ {\widehat{\mathcal{T}}} . This paper suggests an extension A ⁢ ( 𝒯 ) {A(\mathcal{T})} of A ′ ⁢ ( 𝒯 ) {A^{\prime}(\mathcal{T})} that allows for nestedness A ⁢ ( 𝒯 ) ⊂ A ⁢ ( 𝒯 ^ ) {A(\mathcal{T})\subset A(\widehat{\mathcal{T}})} under mesh-refinement. The extended Argyris finite element space A ⁢ ( 𝒯 ) {A(\mathcal{T})} is called hierarchical, but is still based on the concept of the Argyris finite element as a triple ( T , P 5 ⁢ ( T ) , ( Λ 1 , … , Λ 21 ) ) {(T,P_{5}(T),(\Lambda_{1},\dots,\Lambda_{21}))} in the sense of Ciarlet. The other main results of this paper are the optimal convergence rates of an adaptive mesh-refinement algorithm via the abstract framework of the axioms of adaptivity and uniform convergence of a local multigrid V-cycle algorithm for the effective solution of the discrete system.

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