An analysis of nonconforming multigrid methods, leading to an improved method for the Morley element
Author:
Rob Stevenson
Journal:
Math. Comp. 72 (2003), 5581
MSC (2000):
Primary 65N55, 65N30, 65F10
Published electronically:
May 1, 2002
MathSciNet review:
1933814
Fulltext PDF Free Access
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Similar Articles 
Additional Information
Abstract: We recall and slightly refine the convergence theory for nonconforming multigrid methods for symmetric positive definite problems developed by Bramble, Pasciak and Xu. We derive new results to verify the regularity and approximation assumption, and the assumption on the smoother. From the analysis it will appear that most efficient multigrid methods can be expected for fully regular problems, and for prolongations for which the energy norm of the iterated prolongations is uniformly bounded. Guided by these observations, we develop a new multigrid method for the biharmonic equation discretized with Morley finite elements, or equivalently, for the Stokes equations discretized with the nonconforming pair. Numerical results show that the new method is superior to standard ones.
 [BDH99]
D. Braess, M. Dryja, and W. Hackbusch.
A multigrid method for nonconforming FEdiscretisations with applications to nonmatching grids. Computing, 63(1):125, 1999. MR 2000h:65048
 [BP92]
J.H. Bramble and J.E. Pasciak.
The analysis of smoothers for multigrid algorithms. Math. Comp., 58:467488, 1992. MR 92f:65146
 [BPX91]
J.H. Bramble, J.E. Pasciak, and J. Xu.
The analysis of multigrid algorithms with nonnested spaces or inherited quadratic forms. Math. Comp., 56(193):134, 1991. MR 91h:65159
 [Bre89]
S.C. Brenner.
An optimalorder nonconforming multigrid method for the biharmonic equation. SIAM J. Numer. Anal, 26(5):11241138, 1989. MR 89f:65119
 [Bre90]
S.C. Brenner.
A nonconforming multigrid method for the stationary Stokes equations. Math. Comp., 55(192):411437, 1990. MR 91d:65167
 [Bre99]
S.C. Brenner.
Convergence of nonconforming multigrid methods without full elliptic regularity. Math. Comp., 68(225):2553, 1999. MR 99c:65229
 [BS94]
S.C. Brenner and L.R. Scott.
The Mathematical Theory of Finite Element Methods. SpringerVerlag, New York, 1994. MR 95f:65001
 [Che97]
Z. Chen.
The analysis of intergrid transfer operators and multigrid methods for nonconforming finite elements. ETNA, 5:7896, 1997. MR 99c:65230
 [Che99]
Z. Chen.
On the convergence of Galerkinmultigrid methods for nonconforming finite elements. EastWest J. Numer. Math., 7(2):79104, 1999. MR 2000c:65116
 [Cia78]
P.G. Ciarlet.
The finite element method for elliptic problems. NorthHolland, Amsterdam, 1978. MR 58:25001
 [CO98]
Z. Chen and P. Oswald.
Multigrid and multilevel methods for nonconforming rotated Q1 elements. Math. Comp., 67:667693, 1998. MR 98g:65118
 [FM90]
R.S. Falk and M.E. Morley.
Equivalence of finite element methods for problems in elasticity. SIAM J. Numer. Anal., 27(6):14861505, 1990. MR 91i:65117
 [GR86]
V. Girault and P.A. Raviart.
Finite element methods for NavierStokes equations, Theory and Algorithms. SpringerVerlag, Berlin, 1986. MR 88b:65129
 [Hac85]
W. Hackbusch.
MultiGrid Methods and Applications. SpringerVerlag, Berlin, 1985.
 [MMB87]
J. Mandel, S. McCormick, and R. Bank.
Variational multigrid theory. In S.F. McCormick, editor, Multigrid Methods, chapter 5. SIAM, Philadelphia, Pennsylvania, 1987, pp. 131177. MR 89m:65004
 [Osw92]
P. Oswald.
On a hierarchical basis multilevel method with nonconforming P1 elements. Numer. Math., 62:189212, 1992. MR 93b:65059
 [Osw97]
P. Oswald.
Intergrid transfer operators and multilevel preconditioners for nonconforming discretizations. Appl. Numer. Math, 23, 1997, 139158. MR 98g:65110
 [Ste99]
R.P. Stevenson.
Nonconforming finite elements and the cascadic multigrid method. Technical Report 1120, University of Utrecht, November 1999. Revised version, January 2001, to appear in Numer. Math.
 [Ste00]
R.P. Stevenson.
A direct solver for the gradient equation. Technical Report 1163, University of Utrecht, October 2000. to appear in Math. Comp.
 [BDH99]
 D. Braess, M. Dryja, and W. Hackbusch.
A multigrid method for nonconforming FEdiscretisations with applications to nonmatching grids. Computing, 63(1):125, 1999. MR 2000h:65048
 [BP92]
 J.H. Bramble and J.E. Pasciak.
The analysis of smoothers for multigrid algorithms. Math. Comp., 58:467488, 1992. MR 92f:65146
 [BPX91]
 J.H. Bramble, J.E. Pasciak, and J. Xu.
The analysis of multigrid algorithms with nonnested spaces or inherited quadratic forms. Math. Comp., 56(193):134, 1991. MR 91h:65159
 [Bre89]
 S.C. Brenner.
An optimalorder nonconforming multigrid method for the biharmonic equation. SIAM J. Numer. Anal, 26(5):11241138, 1989. MR 89f:65119
 [Bre90]
 S.C. Brenner.
A nonconforming multigrid method for the stationary Stokes equations. Math. Comp., 55(192):411437, 1990. MR 91d:65167
 [Bre99]
 S.C. Brenner.
Convergence of nonconforming multigrid methods without full elliptic regularity. Math. Comp., 68(225):2553, 1999. MR 99c:65229
 [BS94]
 S.C. Brenner and L.R. Scott.
The Mathematical Theory of Finite Element Methods. SpringerVerlag, New York, 1994. MR 95f:65001
 [Che97]
 Z. Chen.
The analysis of intergrid transfer operators and multigrid methods for nonconforming finite elements. ETNA, 5:7896, 1997. MR 99c:65230
 [Che99]
 Z. Chen.
On the convergence of Galerkinmultigrid methods for nonconforming finite elements. EastWest J. Numer. Math., 7(2):79104, 1999. MR 2000c:65116
 [Cia78]
 P.G. Ciarlet.
The finite element method for elliptic problems. NorthHolland, Amsterdam, 1978. MR 58:25001
 [CO98]
 Z. Chen and P. Oswald.
Multigrid and multilevel methods for nonconforming rotated Q1 elements. Math. Comp., 67:667693, 1998. MR 98g:65118
 [FM90]
 R.S. Falk and M.E. Morley.
Equivalence of finite element methods for problems in elasticity. SIAM J. Numer. Anal., 27(6):14861505, 1990. MR 91i:65117
 [GR86]
 V. Girault and P.A. Raviart.
Finite element methods for NavierStokes equations, Theory and Algorithms. SpringerVerlag, Berlin, 1986. MR 88b:65129
 [Hac85]
 W. Hackbusch.
MultiGrid Methods and Applications. SpringerVerlag, Berlin, 1985.
 [MMB87]
 J. Mandel, S. McCormick, and R. Bank.
Variational multigrid theory. In S.F. McCormick, editor, Multigrid Methods, chapter 5. SIAM, Philadelphia, Pennsylvania, 1987, pp. 131177. MR 89m:65004
 [Osw92]
 P. Oswald.
On a hierarchical basis multilevel method with nonconforming P1 elements. Numer. Math., 62:189212, 1992. MR 93b:65059
 [Osw97]
 P. Oswald.
Intergrid transfer operators and multilevel preconditioners for nonconforming discretizations. Appl. Numer. Math, 23, 1997, 139158. MR 98g:65110
 [Ste99]
 R.P. Stevenson.
Nonconforming finite elements and the cascadic multigrid method. Technical Report 1120, University of Utrecht, November 1999. Revised version, January 2001, to appear in Numer. Math.
 [Ste00]
 R.P. Stevenson.
A direct solver for the gradient equation. Technical Report 1163, University of Utrecht, October 2000. to appear in Math. Comp.
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Additional Information
Rob Stevenson
Affiliation:
Department of Mathematics, Utrecht University, P.O. Box 80.010, NL3508 TA Utrecht, The Netherlands
Email:
stevenso@math.uu.nl
DOI:
http://dx.doi.org/10.1090/S0025571802014102
PII:
S 00255718(02)014102
Keywords:
Multigrid method,
nonconforming finite elements,
biharmonic equation,
Morley finite element space,
Stokes equations
Received by editor(s):
November 23, 1998
Received by editor(s) in revised form:
January 23, 2001
Published electronically:
May 1, 2002
Article copyright:
© Copyright 2002
American Mathematical Society
