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A new family of stable mixed finite elements for the 3D Stokes equations

Author: Shangyou Zhang
Journal: Math. Comp. 74 (2005), 543-554
MSC (2000): Primary 65N30, 65F10
Published electronically: August 31, 2004
MathSciNet review: 2114637
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Abstract: A natural mixed-element approach for the Stokes equations in the velocity-pressure formulation would approximate the velocity by continuous piecewise-polynomials and would approximate the pressure by discontinuous piecewise-polynomials of one degree lower. However, many such elements are unstable in 2D and 3D. This paper is devoted to proving that the mixed finite elements of this $\mathbf{P}_k$-$P_{k-1}$ type when $k \ge 3$ satisfy the stability condition--the Babuska-Brezzi inequality on macro-tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedra. This type of mesh simplifies the implementation since it has no restrictions on the initial mesh. The new element also suits the multigrid method.

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Additional Information

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

Keywords: Stokes problem, finite element, mixed element, inf-sup condition, multigrid method
Received by editor(s): October 3, 2002
Received by editor(s) in revised form: January 9, 2003
Published electronically: August 31, 2004
Article copyright: © Copyright 2004 American Mathematical Society