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Generalizations of the Finite Element Method

100%
Schweitzer M.
Open Mathematics
|
2012
|
tom 10
|
nr 1
3-24
EN
This paper is concerned with the generalization of the finite element method via the use of non-polynomial enrichment functions. Several methods employ this general approach, e.g. the extended finite element method and the generalized finite element method. We review these approaches and interpret them in the more general framework of the partition of unity method. Here we focus on fundamental construction principles, approximation properties and stability of the respective numerical method. To this end, we consider meshbased and meshfree generalizations of the finite element method and the use of smooth, discontinuous, singular and numerical enrichment functions.
2
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Properties of triangulations obtained by the longest-edge bisection

100%
Perdomo F., Plaza Á.
Open Mathematics
|
2014
|
tom 12
|
nr 12
1796-1810
EN
The Longest-Edge (LE) bisection of a triangle is obtained by joining the midpoint of its longest edge with the opposite vertex. Here two properties of the longest-edge bisection scheme for triangles are proved. For any triangle, the number of distinct triangles (up to similarity) generated by longest-edge bisection is finite. In addition, if LE-bisection is iteratively applied to an initial triangle, then minimum angle of the resulting triangles is greater or equal than a half of the minimum angle of the initial angle. The novelty of the proofs is the use of an hyperbolic metric in a shape space for triangles.
3
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Anisotropic interpolation error estimates via orthogonal expansions

84%
Li M., Mao S.
Open Mathematics
|
2013
|
tom 11
|
nr 4
621-629
EN
We prove anisotropic interpolation error estimates for quadrilateral and hexahedral elements with all possible shape function spaces, which cover the intermediate families, tensor product families and serendipity families. Moreover, we show that the anisotropic interpolation error estimates hold for derivatives of any order. This goal is accomplished by investigating an interpolation defined via orthogonal expansions.
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