Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T21:51:33.775Z Has data issue: false hasContentIssue false
  • English
  • Français

Local Multilevel Method on Adaptively Refined Meshes for Elliptic Problems with Smooth Complex Coefficients

Part of: Numerical linear algebra Partial differential equations, boundary value problems

Published online by Cambridge University Press:  05 August 2015

Shishun Li ,
Xinping Shao  and
Zhiyong Si
Shishun Li*
Affiliation:
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, P. R.China
Xinping Shao
Affiliation:
School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
Zhiyong Si
Affiliation:
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, P. R.China
*
*Email addresses: lss6@sina.com (S.-S. Li), shaoxinping1983@126.com (X.-P. Shao), sizhiyong@hpu.edu.cn (Z.-Y. Si)
Get access

Abstract

In this paper, a local multilevel algorithm is investigated for solving linear systems arising from adaptive finite element approximations of second order elliptic problems with smooth complex coefficients. It is shown that the abstract theory for local multilevel algorithm can also be applied to elliptic problems whose dominant coefficient is complex valued. Assuming that the coarsest mesh size is sufficiently small, we prove that this algorithm with Gauss-Seidel smoother is convergent and optimal on the adaptively refined meshes generated by the newest vertex bisection algorithm. Numerical experiments are reported to confirm the theoretical analysis.

Keywords

Local multilevel algorithmAdaptive finite element methodComplex coefficientsGauss-Seidel smoother

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 3 , August 2015 , pp. 336 - 355
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bai, D. and Brandt, A., Local mesh refinement multilevel techniques, Siam Journal on Scientific and Statistical Computing, 8 (1987), pp. 109–134.CrossRefGoogle Scholar
[2]Bank, R. E. and Dupont, T., An optimal order process for solving finite element equations, Mathematics of Computation, 36 (1981), pp. 35–51.CrossRefGoogle Scholar
[3]Bänsch, E., Local mesh refinement in 2 and 3 dimensions, Impact of Computing in Science and Engineering, 3 (1991), pp. 181–191.CrossRefGoogle Scholar
[4]Bao, G., Li, P. and Wu, H., An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures, Mathematics of Computation, 79 (2010), pp. 1–34.CrossRefGoogle Scholar
[5]Braess, D. and Hackbusch, W., A new convergence proof for the multigrid method including the V-cycle, Siam Journal on Numerical Analysis, 20 (1983), pp. 967–975.CrossRefGoogle Scholar
[6]Bramble, J. H., Multigrid methods, Pitman, Boston, 1993.Google Scholar
[7]Bramble, J. H., Kwak, D. Y. and Pasciak, J. E., Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems, Siam Journal on Numerical Analysis, 31 (1994), pp. 1746–1763.CrossRefGoogle Scholar
[8]Bramble, J. H. and Pasciak, J. E., Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem, Mathematics of Computation, 77 (2008), pp. 1–10.CrossRefGoogle Scholar
[9]Bramble, J. H., Pasciak, J. E. and Xu, J., The Analysis of multigrid algorithms for non-symmetric and indefinite elliptic problems, mathematics of computation, 51 (1988), pp. 389–414.CrossRefGoogle Scholar
[10]Brandt, A., Multi-level adaptive solutions to boundary-value problems, Mathematics of Computation, 31 (1977), pp. 333–390.CrossRefGoogle Scholar
[11]Brenner, S. C., Convergence of the multigrid V-cycle algorithms for second order boundary value problems without full elliptic regularity, Mathematics of Computation, 71 (2002), pp. 507–525.CrossRefGoogle Scholar
[12]Byfut, A., Gedicke, J., Günther, D., Reininghaus, J. and Wiedemann, S., FFW documentation, Humboldt University of Berlin, German, 2007.Google Scholar
[13]Chen, H. X. and Xu, X. J., Local multilevel methods for adaptive finite element methods for nonsymmetric and indefinite elliptic boundary value problems, Siam Journal on Numerical Analysis, 47 (2010), pp. 4492–4516.CrossRefGoogle Scholar
[14]Chen, Z. and Chen, J., An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems, Mathematics of Computation, 77 (2008), pp. 673–698.CrossRefGoogle Scholar
[15]Ciarlet, P. G., The finite element method for elliptic problems, North-Holland, Amsterdam, 1978.Google Scholar
[16]Douglas, C. C., Multi-grid algorithms with applications to elliptic boundary-value problems, Siam Journal on Numerical Analysis, 21 (1984), pp. 236–254.CrossRefGoogle Scholar
[17]Gopalakrishnan, J. and Pasciak, J. E., Multigrid convergence for second order elliptic problems with smooth complex coefficients, Computer Methods in Applied Mechanics and Engineering, 197 (2008), pp. 4411–4418.CrossRefGoogle Scholar
[18]Hackbusch, W., Multi-grid methods and applications, Springer-Verlag, Berlin, Heidelberg, 1985.CrossRefGoogle Scholar
[19]Kim, S. and Pasciak, J. E., The computation of resonances in open systems using a perfectly matched layer, Mathematics of Computation, 78 (2009), pp. 1375–1398.CrossRefGoogle Scholar
[20]McCormick, S. F., A variational theory for multilevel adaptive techniques (MLAT), Proc. Multigrid Conference, Bristol, Sept., 1983, Ima. J.Google Scholar
[21]McCormick, S. F., Fast adaptive composite grid (FAC) methods: theory for the variational case. Defect correction methods, Computing Supplementum 5, Springer-Verlag, Vienna, 1984, pp. 115–121.Google Scholar
[22]McCormick, S. F., Multilevel adaptive methods for partial differential equations, Frontiers in Applied Mathematics, SIAM Philadelphia, 1989.Google Scholar
[23]Mccormick, S. F., Mckay, S., and Thomas, J., Computational complexity of the fast adaptive composite grid (FAC) method, Applied Numerical Mathematics, 6 (1989), pp. 315–327.CrossRefGoogle Scholar
[24]McCormick, S. F. and Thomas, J., The fast adaptive composite-grid method for elliptic equations, Mathematics of Computation, 46 (1986), pp. 439–456.CrossRefGoogle Scholar
[25]Mekchay, K. and Nochetto, R. H., Convergence of adaptive finite element methods for general second order linear elliptic PDEs, Siam Journal on Numerical Analysis, 43 (2005), pp. 1803–1827.CrossRefGoogle Scholar
[26]Mitchell, W. F., Optimal multilevel iterative methods for adaptive grids, Siam Journal on Scientific and Statistical Computing, 13 (1992), pp. 146–167.CrossRefGoogle Scholar
[27]Oswald, P., Multilevel finite element approximation, theory and applications, Stuttgart: Teubner, Verlag, 1994.CrossRefGoogle Scholar
[28]Scott, R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Mathematics of Computation, 54 (1990), pp. 483–493.CrossRefGoogle Scholar
[29]Toselli, A. and Widlund, O., Domain decomposition and theory, Springer, Berlin-Heidelberg-New York, 2005.CrossRefGoogle Scholar
[30]Wang, J., Convergence analysis of multigrid algorithms for nonselfadjoint and indefinite elliptic problems, Siam Journal on Numerical Analysis, 30 (1993), pp. 275–285.CrossRefGoogle Scholar
[31]Wesseling, P., An introduction to multigrid methods, Wiley, Chichester, 1992.Google Scholar
[32]Xu, X. J., Chen, H. X. and Hoppe, R. H. W., Optimality of local multilevel methods on adaptively refined meshes for elliptic boundary value problems, Journal of Numerical Mathematics, 18 (2010), pp. 59–90.CrossRefGoogle Scholar
[33]Wu, H. J. and Chen, Z. M., Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems, Science in China A, 39 (2006), pp. 1405–1429.CrossRefGoogle Scholar