Two-level Additive Schwarz Preconditioners

for Nonconforming Finite Element Methods

Author:
Susanne C. Brenner

Journal:
Math. Comp. **65** (1996), 897-921

MSC (1991):
Primary 65F10, 65N30, 65N55

DOI:
https://doi.org/10.1090/S0025-5718-96-00746-6

MathSciNet review:
1348039

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Abstract | References | Similar Articles | Additional Information

Abstract: Two-level additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar second-order symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergence-free nonconforming P1 finite element approximation of the stationary Stokes equations. The condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap.

**1.**J.H. Argyris, I. Fried, and D.W. Scharpf,*The TUBA family of plate elements for the matrix displacement method*, Aero. J. Roy. Aero. Soc.**72**(1968), 701--709.**2.**J.H. Bramble and J. Xu,*Some estimates for a weighted projection*, Math. Comp.**56**(1991), 463--476. MR**91k:65140****3.**S.C. Brenner,*Two-level additive Schwarz preconditioners for nonconforming finite elements*, Domain Decomposition Methods in Scientific and Engineering Computing, Contemporary Mathematics 180 (D.E. Keyes et al., eds.), American Mathematical Society, Providence, 1994, pp. 9-14. MR**95j:65134****4.**------,*A two-level additive Schwarz preconditioner for nonconforming plate elements*, Numer. Math.**72**(1996), 419--447.**5.**------,*A two-level additive Schwarz preconditioner for the stationary Stokes equations*, Adv. Comp. Math.**4**(1995), 111--126.**6.**S.C. Brenner and L.R. Scott,*The Mathematical Theory of Finite Element Methods*, Springer-Verlag, New York, 1994. MR**95f:65001****7.**P.G. Ciarlet,*The Finite Element Method for Elliptic Problems*, North-Holland, Amsterdam-New York-Oxford, 1978. MR**58:25001****8.**L.C. Cowsar,*Domain decomposition methods for nonconforming finite elements spaces of Lagrange-type*, Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, NASA Conference Publication 3224 (1993), 93--109.**9.**M. Crouzeix and P.-A. Raviart,*Conforming and nonconforming finite element methods for solving the stationary Stokes equations I*, R.A.I.R.O.**R-3**(1973), 33--75. MR**49:8401****10.**M. Dryja and O.B. Widlund,*An additive variant of the Schwarz alternating method in the case of many subregions*, Technical Report 339, Department of Computer Science, Courant Institute (1987).**11.**------,*Some domain decomposition algorithms for elliptic problems*, Technical Report 438, Department of Computer Science, Courant Institute (1989).**12.**R.S. Falk and M.E. Morley,*Equivalence of finite element methods for problems in elasticity*, SIAM J. Numer. Anal.**27**(1990), 1486--1505. MR**91i:65177****13.**L.S.D. Morley,*The triangular equilibrium problem in the solution of plate bending problems*, Aero. Quart.**19**(1968), 149--169.**14.**S.V. Nepomnyaschikh,*On the application of the bordering method to the mixed boundary value problem for elliptic equations and on mesh norms in*, Sov. J. Numer. Anal. Math. Modelling**4**(1989), 493--506.**15.**M. Sarkis,*Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients*, Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, NASA Conference Publication 3224 (1993), 543--565.**16.**F. Thomasset,*Implementation of Finite Element Methods for Navier-Stokes Equations*, Springer-Verlag, New York, 1981. MR**84k:76015****17.**O.B. Widlund,*Some Schwarz methods for symmetric and nonsymmetric elliptic problems*, Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations (D.E. Keyes et al., eds.), SIAM, Philadelphia, 1991, pp. 19--36.**18.**J. Xu,*Iterative methods by space decomposition and subspace correction*, SIAM Review**34**(1992), 581--613. MR**93k:65029****19.**X. Zhang,*Studies in Domain Decomposition: Multi-level Methods and the Biharmonic Dirichlet Problem*, Dissertation, (Technical Report 584, Department of Computer Science) Courant Institute (1991).

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Additional Information

**Susanne C. Brenner**

Affiliation:
Department of Mathematics and Computer Science, Clarkson University, Potsdam, New York 13699-5815

Address at time of publication:
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Email:
brenner@math.sc.edu

DOI:
https://doi.org/10.1090/S0025-5718-96-00746-6

Keywords:
Domain decomposition,
additive Schwarz preconditioner,
nonconforming finite elements,
Laplace equation,
biharmonic equation,
stationary Stokes equations

Received by editor(s):
July 6, 1993

Received by editor(s) in revised form:
November 18, 1993, and August 1, 1994

Additional Notes:
This work was supported in part by the National Science Foundation under Grant No. DMS-92-09332.

Article copyright:
© Copyright 1996
American Mathematical Society