Affiliation:
1. Humboldt-Universität zu Berlin , 10099 Berlin , Germany
2. LMAM and School of Mathematical Sciences , Peking University , Beijing 100871 , P. R. China
Abstract
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.
Subject
Applied Mathematics,Computational Mathematics,Numerical Analysis
Cited by
3 articles.
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