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Conjugate gradient method for dual-dual mixed formulations
Author(s):
Gabriel
N.
Gatica;
Norbert
Heuer.
Journal:
Math. Comp.
71
(2002),
1455-1472.
MSC (2000):
Primary 65N30, 65N22, 65F10
Posted:
December 5, 2001
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Abstract:
We deal with the iterative solution of linear systems arising from so-called dual-dual mixed finite element formulations. The linear systems are of a two-fold saddle point structure; they are indefinite and ill-conditioned. We define a special inner product that makes matrices of the two-fold saddle point structure, after a specific transformation, symmetric and positive definite. Therefore, the conjugate gradient method with this special inner product can be used as iterative solver. For a model problem, we propose a preconditioner which leads to a bounded number of CG-iterations. Numerical experiments for our model problem confirming the theoretical results are also reported.
References:
-
- 1.
- T. ARBOGAST, M.F. WHEELER AND I. YOTOV, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal., 34 (1997), pp. 828-852. MR 98g:65105
- 2.
- S. F. ASHBY, T. A. MANTEUFFEL, AND P. E. SAYLOR, A taxonomy for conjugate gradient methods, SIAM J. Numer. Anal., 27 (1990), pp. 1542-1568. MR 91i:65062
- 3.
- G. R. BARRENECHEA, G. N. GATICA, AND J.-M. THOMAS, Primal mixed formulations for the coupling of FEM and BEM. part I: Linear problems, Numer. Funct. Anal. Optim., 19 (1998), pp. 7-32. MR 99d:65310
- 4.
- D. BRAESS, Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 1997. MR 98f:65002
- 5.
- J. H. BRAMBLE AND J. E. PASCIAK, A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comp., 50 (1988), pp. 1-17. MR 89m:65097a
- 6.
- F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer Verlag, 1991. MR 92d:65187
- 7.
- Z. CHEN, Expanded mixed finite element methods for linear second-order elliptic problems, RAIRO Math. Model. Numer. Anal., 32 (1998), pp. 479-499. MR 99j:65201
- 8.
- Z. CHEN, Expanded mixed finite element methods for quasilinear second-order elliptic problems, RAIRO Math. Model. Numer. Anal., 32 (1998), pp. 501-520. MR 99j:65202
- 9.
- L. FRANCA AND A. LOULA, A new mixed finite element method for the Timoshenko beam problem, RAIRO Modélisation Mathématique et Analyse Numérique, 25 (1991), pp. 561-578. MR 92e:65150
- 10.
- G. N. GATICA, Solvability and Galerkin approximations of a class of nonlinear operator equations.
Technical Report 99-03, Departamento de Ingeniería Matemática, Universidad de Concepción, Chile. Submitted for publication. http://www.ing-mat.udec.cl/inf-loc-dim.html - 11.
- G. N. GATICA AND N. HEUER, A dual-dual mixed formulation for the coupling of mixed-FEM and BEM in hyperelasticity, SIAM J. Numer. Anal., 38 (2000), pp. 380-400. MR 2001e:65193
- 12.
- G. N. GATICA AND N. HEUER, Minimum residual iteration for a dual-dual mixed formulation of exterior transmission problems, Numer. Linear Algebra Appl., 8 (2001), pp. 147-164. CMP 2001:09
- 13.
- G. N. GATICA AND N. HEUER, An expanded mixed finite element method via a dual-dual formulation and the minimum residual method, J. Comput. Appl. Math., 132 (2001), pp. 371-385.
- 14.
- G. N. GATICA, N. HEUER AND S. MEDDAHI, Solvability and fully discrete Galerkin schemes of nonlinear two-fold saddle point problems. Technical Report 00-03, Departamento de Ingeniería Matemática, Universidad de Concepción, Chile. http://www.ing-mat.udec.cl/ inf-loc-dim.html
- 15.
- G. N. GATICA AND S. MEDDAHI, A dual-dual mixed formulation for nonlinear exterior transmission problems, Math. Comp., 70 (2001), pp. 1461-1480.
- 16.
- G. N. GATICA AND W. L. WENDLAND, Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems, Appl. Anal., 63 (1996), pp. 39-75. MR 99a:65167
- 17.
- N. HEUER, M. MAISCHAK, AND E. P. STEPHAN, Preconditioned minimum residual iteration for the
 - version of the coupled FEM/BEM with quasi-uniform meshes, Numer. Linear Algebra Appl., 6 (1999), pp. 435-456. MR 2000j:65112 - 18.
- V. I. LEBEDEV, Iterative methods for solving operator equations with a spectrum contained in several intervals, USSR Comput. Math. and Math. Phys., 9 (1969), pp. 17-24. MR 42:7052
- 19.
- C. C. PAIGE AND M. A. SAUNDERS, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617-629. MR 52:4595
- 20.
- J. E. ROBERTS AND J.-M. THOMAS, Mixed and Hybrid Methods. In: Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.L. Lions, vol. II, Finite Element Methods (Part 1), North-Holland, Amsterdam, 1991. CMP 91:14
- 21.
- T. RUSTEN AND R. WINTHER, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 887-904. MR 93a:65043
- 22.
- A. J. WATHEN, B. FISCHER, AND D. J. SILVESTER, The convergence of iterative solution methods for symmetric and indefinite linear systems, in Numerical Analysis 1997, D. F. Griffiths and G. A. Watson, eds., Pitman Research Notes in Mathematics, Harlow, England, 1997, pp. 230-243. MR 99e:65058
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Additional Information:
Gabriel
N.
Gatica
Affiliation:
GI$^2$MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
Email:
ggatica@ing-mat.udec.cl
Norbert
Heuer
Affiliation:
GI$^2$MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
Email:
norbert@ing-mat.udec.cl
DOI:
10.1090/S0025-5718-01-01394-1
PII:
S 0025-5718(01)01394-1
Keywords:
Mixed finite elements,
dual-dual variational formulation,
conjugate gradient method
Received by editor(s):
September 22, 1999
Received by editor(s) in revised form:
October 3, 2000
Posted:
December 5, 2001
Additional Notes:
This research was partially supported by CONICYT-Chile through Program A on Numerical Analysis of the FONDAP in Applied Mathematics and Fondecyt project No. 1980122, and by the Dirección de Investigación of the Universidad de Concepción through the Advanced Research Groups Program
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Copyright
2001,
American Mathematical Society
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