Advertisement

Numerical Algorithms

, Volume 78, Issue 3, pp 929–956

# A prewavelet-based algorithm for the solution of second-order elliptic differential equations with variable coefficients on sparse grids

Original Paper

## Abstract

We present a Ritz-Galerkin discretization on sparse grids using prewavelets, which allows us to solve elliptic differential equations with variable coefficients for dimensions d ≥ 2. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality. This algorithm is based on standard 1-dimensional restrictions and prolongations, a simple prewavelet stencil, and the classical operator-dependent stencil for multilinear finite elements. Numerical simulation results are presented for a three-dimensional problem on a curvilinear bounded domain and for a six-dimensional problem with variable coefficients. Simulation results show a convergence of the discretization according to the approximation properties of the finite element space. The condition number of the stiffness matrix can be bounded below 10 using a standard diagonal preconditioner.

## Keywords

Sparse grid Prewavelets Semi-orthogonality Variable coefficients Conjugate gradient method Finite element method

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Achatz, S.: Higher order sparse grid methods for elliptic partial differential equations with variable coefficients. Computing 71(1), 1–15 (2003)
2. 2.
Balder, R., Zenger, C.: The solution of multidimensional real Helmholtz equations of sparse grids. SIAM J. Sci. Comp. 17(3), 631–646 (1996)
3. 3.
Barinka, A., Barsch, T., Charton, P., Cohen, A., Dahlke, S., Dahmen, W., Urban, K.: Adaptive wavelet schemes for elliptic problems—implementation and numerical experiments. SIAM J. Sci. Comput. 23(3), 910–939 (2002)
4. 4.
Bungartz, H.-J.: Du̇nne Gitter und deren Anwendung bei der adaptiven Lȯsung der dreidimensionalen Poisson-Gleichung. Dissertation, Fakultȧt fu̇r Informatik, Technische Universitȧt Mu̇nchen (1992)Google Scholar
5. 5.
Dornseifer, T., Pflaum, C.: Discretization of elliptic differential equations on curvilinear bounded domains with sparse grids. Computing 56(3), 197–213 (1996)
6. 6.
Feuersȧnger, C.: Sparse grid methods for higher dimensional approximation. PhD thesis, University of Bonn (2010)Google Scholar
7. 7.
Gordon, W.J., Hall, C.A.: Construction of curvilinear co-ordinate systems and applications to mesh generation. Int. J. Numer. Methods Eng. 7(4), 461–477 (1973)
8. 8.
Griebel, M.: Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences. Computing 61, 151–179 (1998)
9. 9.
Griebel, M., Hamaekers, J.: A wavelet based sparse grid method for electronic Schrödinger equation. In: Sanz-Solë, M, Soria, J., Varona, J.L., Verdera, J. (eds.) Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006, vol. 1, pp. 1473–1506. European Mathematical Society, Publishing House (2007)Google Scholar
10. 10.
Griebel, M., Hullmann, A.: On a multilevel preconditioner and its condition numbers for the discretized laplacian on full and sparse grids in higher dimensions. In: Singular Phenomena and Scaling in Mathematical Models, pp. 263–296. Springer International Publishing (2014)Google Scholar
11. 11.
Kestler, S., Stevenson, R.: Fast evaluation of system matrices w.r.t. multi-tree collections of tensor product refinable basis functions. J. Comput. Appl Math. 260, 103–116 (2014)
12. 12.
Niedermeier, A., Zimmer, S.: Implementational aspects of prewavelet sparse grid methods. In: Lai, C.-H., Bjoerstad, P., Cross, M., Widlund, O. (eds.) Eleventh International Conference of Domain Decomposition Methods (1999)Google Scholar
13. 13.
Oswald, P.: Multilevel finite element approximation. Teubner Skripten zur Numerik. Teubner, Stuttgart (1994)
14. 14.
Peherstorfer, B., Zimmer, S., Zenger, C., Bungartz, H.-J.: A multigrid method for adaptive sparse grids. SIAM J. Sci. Comput. 37(5), 51–70 (2015)
15. 15.
Pflaum, C.: Diskretisierung elliptischer Differentialgleichungen mit dünnen Gittern. Dissertation, Technische Universität München (1995)Google Scholar
16. 16.
Pflaum, C.: A multilevel algorithm for the solution of second order elliptic differential equations on sparse grids. Numer. Math. 79, 141–155 (1998)
17. 17.
Pflaum, C.: Robust convergence of multilevel algorithms for convection-diffusion equations. SIAM J. Numer. Anal. 37(2), 443–469 (2000)
18. 18.
Pflaum, C., Hartmann, R.: A sparse grid discretization of the helmholtz equation with variable coefficients in high dimensions. SIAM J. Numer. Anal. 54(4), 2707–2727 (2016)
19. 19.
Vassilevski, P.S., Wang, J.: Stabilizing the hierarchical basis by approximate wavelets II implementation and numerical results. SIAM J. Sci. Comput. 20(2), 490–514 (1998)
20. 20.
Zeiser, A.: Fast matrix-vector multiplication in the sparse-grid Galerkin method. J. Sci. Comput. 47(3), 328–346 (2011)
21. 21.
Zenger, C.: Sparse grids. In: Hackbusch, W. (ed.) Proceedings of the Sixth GAMM-Seminar Parallel Algorithms for Partial Differential Equations, Kiel, 1990, vol. 31 of Notes on Numerical Fluid Mechanics, pp. 240–251. Kiel, Vieweg (1991)Google Scholar

## Copyright information

© Springer Science+Business Media, LLC 2017

## Authors and Affiliations

• Rainer Hartmann
• 1
• Christoph Pflaum
• 1
1. 1.Department of Computer Science, System SimulationFriedrich-Alexander-Universität Erlangen-Nürnberg (FAU)ErlangenGermany

## Personalised recommendations

### Citearticle
{"ft_lang_label":"__label__en","ft_lang_prob":0.52363324,"math_prob":0.78872645,"size":5878,"snap":"2019-35-2019-39","text_gpt3_token_len":1552,"char_repetition_ratio":0.15134491,"word_repetition_ratio":0.011049724,"special_character_ratio":0.23936713,"punctuation_ratio":0.2578269,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.977349,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-23T19:55:47Z\",\"WARC-Record-ID\":\"<urn:uuid:4098c56d-2cd8-4185-8c41-f646d373e673>\",\"Content-Length\":\"90751\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:06b2a4c1-0a0a-4db7-b77d-e1ce6a5f90df>\",\"WARC-Concurrent-To\":\"<urn:uuid:aad0ea75-b5a5-4aa5-956d-5ab31ca30614>\",\"WARC-IP-Address\":\"151.101.248.95\",\"WARC-Target-URI\":\"https://link.springer.com/article/10.1007%2Fs11075-017-0407-9\",\"WARC-Payload-Digest\":\"sha1:FEGCZXQUCAPPL7U7TO45JVQTFNYFNEOL\",\"WARC-Block-Digest\":\"sha1:CF74JPIABAGVFQYHS76JFDX7D5CUOTQX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027318986.84_warc_CC-MAIN-20190823192831-20190823214831-00341.warc.gz\"}"}