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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 - type when 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.

**[1]**D. N. Arnold, F. Brezzi and M. Fortin,*A stable finite element for the Stokes equations*, Calcolo**21**(1984), 337 - 344. MR**86m:65136****[2]**F. Brezzi and M. Fortin,*Mixed and hybrid finite element methods*, Springer-Verlag, Berlin, 1991. MR**92d:65187****[3]**P. G. Ciarlet,*The Finite Element Method for Elliptic Problems*, North-Holland, Amsterdam, New York, Oxford, 1978. MR**58:25001****[4]**V. Girault and P. A. Raviart,*Finite Element Methods for Navier-Stokes Equations*, Springer-Verlag, New York-Heidelberg-Berlin, 1986. MR**88b:65129****[5]**J. Qin,*On the Convergence of Some Low Order Mixed Finite Elements for Incompressible Fluids*, Ph. Dissertation, Department of Mathematics, Pennsylvania State University, 1994.**[6]**J. Qin and S. Zhang,*Stability and approximability of the**-**element for Stokes equations*, preprint.**[7]**J. Qin and S. Zhang,*A general convergence theory on preconditioned linear iterations for saddle point problems*, unpublished manuscript.**[8]**L. R. Scott and M. Vogelius,*Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials*, Math. Modelling Numer. Anal.**19**(1985), 111 - 143. MR**87i:65190****[9]**L. R. Scott and S. Zhang,*Finite-element interpolation of non-smooth functions satisfying boundary conditions*, Math. Comp.**54**(1990), 483 - 493. MR**90j:65021****[10]**L. R. Scott and S. Zhang,*Multilevel iterated penalty method for mixed elements*, The Proceedings for the Ninth International Conference on Domain Decomposition Methods, Bergen, 1998, pp. 133-139.**[11]**R. Stenberg,*Analysis of mixed finite element methods for the Stokes problem: A unified approach*, Math. Comp.**42**(1984), 9 - 23. MR**84k:76014****[12]**R. Stenberg,*On some three-dimensional finite elements for incompressible media*, Comput. Methods Appl. Mech. Engrg.**63**(1987), 261 - 269. MR**88i:73050****[13]**S. Zhang,*Multi-level Iterative Techniques*, Ph. Dissertation, Department of Mathematics, Pennsylvania State University, 1988.**[14]**S. Zhang,*Successive subdivisions of tetrahedra and multigrid methods on tetrahedral meshes*, Houston J. of Math. (1995), 541 - 556.MR**96f:65183****[15]**S. Zhang,*An optimal order multigrid method for biharmonic,**finite-element equations*, Numer. Math.**56**(1989), 613 - 624.MR**90j:65135****[16]**S. Zhang,*Optimal order non-nested multigrid methods for solving finite element equations I: on quasiuniform meshes*, Math. Comp.**55**(1990), 23 - 36. MR**91g:65268**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65N30,
65F10

Retrieve articles in all journals with MSC (2000): 65N30, 65F10

Additional Information

**Shangyou Zhang**

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

Email:
szhang@math.udel.edu

DOI:
https://doi.org/10.1090/S0025-5718-04-01711-9

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