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The global arrangement of the degrees of freedom in a standard Argyris finite element method (FEM) enforcesC2{C^{2}}at interior vertices, while solely globalC1{C^{1}}continuity is required for the conformity inH2{H^{2}}. Since the Argyris finite element functions are  notC2{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 spaceA′⁢(){A^{\prime}(\mathcal{T})}associated to a triangulation{\mathcal{T}}with a refinement^{\widehat{\mathcal{T}}}is  not  a subspace of the standard Argyris finite element spaceA′⁢(^){A^{\prime}(\widehat{\mathcal{T}})}associated to the refined triangulation^{\widehat{\mathcal{T}}}. This paper suggests an extensionA⁢(){A(\mathcal{T})}ofA′⁢(){A^{\prime}(\mathcal{T})}that allows for nestednessA⁢()⊂A⁢(^){A(\mathcal{T})\subset A(\widehat{\mathcal{T}})}under mesh-refinement. The extended Argyris finite element spaceA⁢(){A(\mathcal{T})}is called hierarchical, but is still based on the concept of the Argyris finite element as a triple(T,P5⁢(T),(Λ1,…,Λ21)){(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.

Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms