The completion of locally refined simplicial partitions created by bisection
By Rob P. Stevenson. Mathematics of Computation, 2008
397 citations recorded by the source corpus.
Recently, in [Found. Comput. Math., 7(2) (2007), 245-269], we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations, which is essentially the method from [SINUM, 38 (2000), 466-488] by Morin, Nochetto, and Siebert, converges with the optimal rate. The number of triangles N in the output partition of such a method is generally larger than the number M of triangles that in all intermediate partitions have been marked for bisection, because additional bisections are needed to retain conforming meshes. A key ingredient to our proof was a result from [Numer. Math., 97(2004), 219-268] by Binev, Dahmen and DeVore saying that N - N-0 <= CM for some absolute constant C, where N-0 is the number of triangles from the initial partition that have never been bisected. In this paper, we extend this result to bisection algorithms of n-simplices, with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions.
Subfields: Computational Theory and Mathematics, Computer Vision and Pattern Recognition, Discrete Mathematics and Combinatorics
View Paper: https://www.semanticscholar.org/paper/015ea47f205cd9f5b2e000e98e9c627df3aff0a0
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Citations in the corpus
- Theory of adaptive finite element methods: An introduction (2009, methodology): “We summarize in this chapter the generalization to any space dimension by Stevenson [70].”
- Morley Finite Element Method for the Eigenvalues of the Biharmonic Operator (2014, methodology): “Generate T ℓ +1 from T ℓ and M ℓ with newest-vertex bisection [4, 39]. end for Output: Triangulations ( T ℓ ) ℓ and discrete solutions ( λ ℓ,j , u ℓ,j ) j ∈ J ℓ .”
- Quasioptimal cardinality of AFEM driven by nonresidual estimators (2012, background): “The labeling of refinement edges onT0 satisfies (Stevenson, 2008, Condition (b), Section 4) ford > 2.”
- Primer of Adaptive Finite Element Methods (2011, background): “For d > 2 the situation is more complicated and one needs the concepts of type and vertex order [45, 53].”