Divergence-free finite elements on tetrahedral grids for

Author:
Shangyou Zhang

Journal:
Math. Comp. **80** (2011), 669-695

MSC (2010):
Primary 65N30, 76M10, 76D07

Published electronically:
August 26, 2010

MathSciNet review:
2772092

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Abstract: It was shown two decades ago that the - mixed element on triangular grids, approximating the velocity by the continuous piecewise polynomials and the pressure by the discontinuous piecewise polynomials, is stable for all , provided the grids are free of a nearly-singular vertex. The problem with the method in 3D was posted then and remains open. The problem is solved partially in this work. It is shown that the - element is stable and of optimal order in approximation, on a family of uniform tetrahedral grids, for all . The analysis is to be generalized to non-uniform grids, when we can deal with the complicity of 3D geometry.

For the divergence-free elements, the finite element spaces for the pressure can be avoided in computation, if a classic iterated penalty method is applied. The finite element solutions for the pressure are computed as byproducts from the iterate solutions for the velocity. Numerical tests are provided.

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Additional Information

**Shangyou Zhang**

Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716

Email:
szhang@udel.edu

DOI:
https://doi.org/10.1090/S0025-5718-2010-02412-3

Keywords:
Mixed finite elements,
Stokes equations,
divergence-free element,
tetrahedral grids.

Received by editor(s):
June 18, 2008

Received by editor(s) in revised form:
January 25, 2010

Published electronically:
August 26, 2010

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.