NDLI: On the Existence of Explicit $hp$-Finite Element Methods Using GaussLobatto Integration on the Triangle

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Helenbrook, B. T.
Copyright Year	2009
Abstract	Spectral-element simulations on quadrilaterals and hexahedra rely on the GaussLobatto (GL) integration rule to enable explicit simulations with optimal spatial convergence rates. In this work, it is proved that a similar integration rule does not exist on triangles. The following properties of the rule are sought: a $(p+1)(p+2)/2$ point integration rule capable of exactly integrating the space given by ${\cal T}(2p -1)\equiv\{x^my^n\|0\leq m,n;m+n\leq2p-1\}$, where p is an integer; integration points located at each of the triangle vertices; $p-1$ integration points located on each side; and $(p-1)(p-2)/2$ integration points located in the interior of the element. The proof hinges on the fact that the existence of such a rule implies the existence of a nodal basis with an approximate diagonal mass matrix that can be inverted to obtain exact Galerkin projections of functions in ${\cal T}(p-1)$. The proof shows that vertex functions of a basis having this property exist and are unique, but on a triangle these functions are not nodal, and therefore the GL rule does not exist. In spite of this, the existence of the vertex functions indicates that there may be a nonnodal basis that has the above property. This basis would enable explicit $hp$-finite element simulations on the triangle with optimal spatial accuracy. The methodology developed in the paper gives insight into a possible way to find such a basis.
Starting Page	1304
Ending Page	1318
Page Count	15
File Format	PDF
ISSN	00361429
DOI	10.1137/070685439
e-ISSN	10957170
Issue Number	2
Volume Number	47
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2009-02-25
Access Restriction	Subscribed
Subject Keyword	triangles quadrature mass-lumping integration Lobatto Finite element methods Gauss Quadrature and cubature formulas
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Numerical Analysis Computational Mathematics

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