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2016 | OriginalPaper | Chapter

5. Iterative Solution Methods

Author : Sören Bartels

Published in: Numerical Approximation of Partial Differential Equations

Publisher: Springer International Publishing

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Abstract

Linear systems of equations resulting from finite element discretizations of partial differential equations are typically large, sparse, and ill-conditioned. Their efficient numerical solution exploits properties of the underlying continuous problem or a sequence of discretizations. The chapter discusses multigrid, domain decomposition, and preconditioning methods.

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previous chapter Local Resolution Techniques

next chapter Saddle-Point Problems

Fundamental contributions to the development of multigrid methods, preconditioning of finite element matrices, and domain decomposition methods are the articles [2‐4, 6, 9, 12, 18]. Specialized textbooks on the subjects are the references [7, 10, 11, 13, 15, 16]. The historical development of domain decomposition methods is recapitulated in [8], and the survey article [17] discusses various aspects of preconditioning techniques. Chapters on iterative solution methods are contained in the textbooks [1, 5, 14].

Braess, D.: Finite Elements, 3rd edn. Cambridge University Press, Cambridge (2007). URL http://dx.doi.org/10.1017/CBO9780511618635

Braess, D., Hackbusch, W.: A new convergence proof for the multigrid method including the V -cycle. SIAM J. Numer. Anal. 20 (5), 967–975 (1983). URL http://dx.doi.org/10.1137/0720066

Bramble, J.H., Pasciak, J.E., Schatz, A.H.: The construction of preconditioners for elliptic problems by substructuring. I. Math. Comp. 47 (175), 103–134 (1986). URL http://dx.doi.org/10.2307/2008084

Bramble, J.H., Pasciak, J.E., Xu, J.: Parallel multilevel preconditioners. Math. Comp. 55 (191), 1–22 (1990). URL http://dx.doi.org/10.2307/2008789

Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008). URL http://dx.doi.org/10.1007/978-0-387-75934-0

Dahmen, W., Kunoth, A.: Multilevel preconditioning. Numer. Math. 63 (3), 315–344 (1992). URL http://dx.doi.org/10.1007/BF01385864

Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Numerical Mathematics and Scientific Computation, 2nd edn. Oxford University Press, Oxford (2014). URL http://dx.doi.org/10.1093/acprof:oso/9780199678792.001.0001

Gander, M.J.: Schwarz methods over the course of time. Electron. Trans. Numer. Anal. 31, 228–255 (2008)MathSciNetMATH

Griebel, M., Oswald, P.: On the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math. 70 (2), 163–180 (1995). URL http://dx.doi.org/10.1007/s002110050115

10.

Hackbusch, W.: Multigrid methods and applications. Springer Series in Computational Mathematics, vol. 4. Springer, Berlin (1985). URL http://dx.doi.org/10.1007/978-3-662-02427-0

11.

Hackbusch, W.: Iterative solution of large sparse systems of equations. Applied Mathematical Sciences, vol. 95. Springer, New York (1994). URL http://dx.doi.org/10.1007/978-1-4612-4288-8

12.

Lions, P.L.: On the Schwarz alternating method. I. In: First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987), pp. 1–42. SIAM, Philadelphia, PA (1988)

13.

Quarteroni, A., Valli, A.: Domain decomposition methods for partial differential equations. Numerical Mathematics and Scientific Computation. The Clarendon Press, Oxford University Press, New York (1999)MATH

14.

Rannacher, R.: Numerische Mathematik 2 (Numerik partieller Differentialgleichungen) (2008). URL http://numerik.iwr.uni-heidelberg.de/~lehre/notes/. Lecture Notes, University of Heidelberg, Germany

15.

Saad, Y.: Iterative methods for sparse linear systems, second edn. Society for Industrial and Applied Mathematics, Philadelphia, PA (2003). URL http://dx.doi.org/10.1137/1.9780898718003

16.

Toselli, A., Widlund, O.: Domain decomposition methods—algorithms and theory. Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2005)

17.

Xu, J., Chen, L., Nochetto, R.H.: Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids. In: Multiscale, Nonlinear and Adaptive Approximation, pp. 599–659. Springer, Berlin (2009). URL http://dx.doi.org/10.1007/978-3-642-03413-8_14

18.

Yserentant, H.: On the multilevel splitting of finite element spaces. Numer. Math. 49 (4), 379–412 (1986). URL http://dx.doi.org/10.1007/BF01389538

Title: Iterative Solution Methods
Author: Sören Bartels
Publisher: Springer International Publishing
Book: Numerical Approximation of Partial Differential Equations
Print ISBN: 978-3-319-32353-4

Electronic ISBN: 978-3-319-32354-1

Copyright Year: 2016
DOI: https://doi.org/10.1007/978-3-319-32354-1_5

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