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.

中文翻译：

---

自适应和多重网格算法的分层 Argyris 有限元方法

标准 Argyris 有限元方法 (FEM) 中自由度的全局排列在内部顶点强制执行 C 2 {C^{2}}，而仅需要全局 C 1 {C^{1}} 连续性才能符合要求在 H 2 {H^{2}} 中。由于 Argyris 有限元函数一般不是 C 2 {C^{2}} 在边的中点，网格细化的边二分会导致非嵌套：标准的 Argyris 有限元空间 A ′ ⁢ ( 𝒯 ) {A^{\prime}(\mathcal{T})} 与三角剖分 𝒯 {\mathcal{T}} 相关联并进行了细化 𝒯 ^ {\widehat{\mathcal{T}}} 不是标准的 Argyris 有限元空间 A ′ ⁢ ( 𝒯 ^ ) {A^{\prime}(\widehat{\mathcal{T}})} 关联到细化三角剖分 𝒯 ^ {\widehat{\mathcal{T}}} . 本文建议扩展 A ⁢ ( 𝒯 ) {A(\mathcal{T})} 的 A ′ ⁢ ( 𝒯 ) {A^{\prime}(\mathcal{T})} 允许嵌套 A ⁢ ( 𝒯 ) ⊂ A ⁢ ( 𝒯 ^ ) {A(\mathcal{T})\subset A(\widehat{\mathcal{T}})} 在网格细化下。扩展的 Argyris 有限元空间 A ⁢ ( 𝒯 ) {A(\mathcal{T})} 被称为分层的，但仍然基于 Argyris 有限元作为三元组 ( T , P 5 ⁢ ( T ) ， ( Λ 1 , … , Λ 21 ) ) {(T,P_{5}(T),(\Lambda_{1},\dots,\Lambda_{21}))} 在 Ciarlet 的意义上。本文的其他主要结果是通过局部多重网格 V 循环算法的自适应公理和均匀收敛公理的抽象框架，获得了自适应网格细化算法的最佳收敛速度，以有效解决离散系统。