Abstract
Anisotropically refined mixed finite elements are beneficial for the resolution of local features such as boundary layers. Unfortunately, the stability of the resulting scheme is highly sensitive to the aspect ratio of the elements. Previous analysis revealed that the degeneration arises from a relatively small number of spurious (piecewise constant) pressure modes. The present article is concerned with resolving the problem of how to suppress the spurious pressure modes in order to restore stability yet at the same time not incur any deterioration in the approximation properties of the reduced pressure space. Two results are presented. The first gives the minimal constraints on the pressure space needed to restore stability with respect to aspect ratio and shows that the approximation properties of the constrained pressure space and the unconstrained pressure space are essentially identical. Alternatively, one can impose the constraint weakly through the use of a stabilized finite element scheme. A second result shows that the stabilized finite element scheme is robust with respect to the aspect ratio of the elements and produces an approximation that satisfies an error bound of the same type to the mixed finite element scheme using the constrained space.
Original language  English 

Pages (fromto)  1107–1120 
Number of pages  14 
Journal  SIAM Journal on Numerical Analysis 
Volume  53 
Issue number  2 
Early online date  16 Apr 2015 
DOIs  
Publication status  Published  2015 
Keywords
 anisotropic meshes
 stabilization
 spurious pressure mode
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Gabriel Barrenechea
Person: Academic