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Two-level preconditioning for \(h\)-version boundary element approximation of hypersingular operator with GenEO

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Abstract

In the present contribution, we consider symmetric positive definite operators stemming from boundary integral equation, and we study a two-level preconditioner where the coarse space is built using local generalized eigenproblems in the overlap. We will refer to this coarse space as the GenEO coarse space.

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Notes

  1. https://github.com/htool-ddm/htool.

  2. https://github.com/xclaeys/BemTool.

References

  1. Ainsworth, M., Guo, B.: An additive Schwarz preconditioner for p-version boundary element approximation of the hypersingular operator in three dimensions. Numer. Math. 85(3), 343–366 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alléon, G., Benzi, M., Giraud, L.: Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics. Numer. Algorithms 16(1), 1–15 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alouges, F., Aussal, M.: The sparse cardinal sine decomposition and its application for fast numerical convolution. Numer. Algorithms 70(2), 427–448 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammerling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, vol. 9. SIAM, New Delhi (1999)

    Book  MATH  Google Scholar 

  5. Aurada, M., Feischl, M., Führer, T., Karkulik, M., Praetorius, D.: Energy norm based error estimators for adaptive BEM for hypersingular integral equations. Appl. Numer. Math. 95, 15–35 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bebendorf, M.: Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems. Lecture Notes in Computational Science and Engineering, vol. 63. Springer-Verlag, Berlin (2008)

    MATH  Google Scholar 

  7. Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V. (eds.) NIST Digital Library of Mathematical Functions. Release 1.0.21 of 15 Dec 2018. http://dlmf.nist.gov/

  8. Dolean, V., Jolivet, P., Nataf, F.: An Introduction to Domain Decomposition Methods. Algorithms, Theory, and Parallel Implementation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2015)

    Book  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  10. Geuzaine, C., Remacle, J.-F.: GMSH: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Greengard, L., Gueyffier, D., Martinsson, P.-G., Rokhlin, V.: Fast direct solvers for integral equations in complex three-dimensional domains. Acta Numer. 18, 243 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. Griebel, M., Oswald, P.: On the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math. 70(2), 163–180 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hackbusch, W.: Hierarchical Matrices: Algorithms and Analysis. Springer Series in Computational Mathematics, vol. 49. Springer-Verlag, Berlin (2016)

    Google Scholar 

  15. Hestenes, M., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bureau Stand. 49(6), 409–436 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  16. Heuer, N.: Additive Schwarz Methods for Weakly Singular Integral Equations In \(\mathbb{R}^3\)—The p-Version, pp. 126–135. Vieweg+Teubner Verlag, Wiesbaden (1996)

    MATH  Google Scholar 

  17. Hiptmair, R.: Operator preconditioning. Comput. Math. Appl. 52(5), 699–706 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hiptmair, R., Jerez-Hanckes, C., Urzúa-Torres, C.: Optimal operator preconditioning for hypersingular operator over 3d screens. Technical report 2016-09, Seminar for Applied Mathematics, ETH Zürich, Switzerland (2016)

  19. Jolivet, P., Hecht, F., Nataf, F., Prud’homme, C.: Scalable domain decomposition preconditioners for heterogeneous elliptic problems. In: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis on—SC ’13. ACM Press (2013)

  20. Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Martinsson, P., Rokhlin, V.: A fast direct solver for boundary integral equations in two dimensions. J. Comput. Phys. 205(1), 1–23 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  23. McLean, W., Steinbach, O.: Boundary element preconditioners for a hypersingular integral equation on an interval. Adv. Comput. Math. 11(4), 271–286 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nepomnyaschikh, S.V.: Decomposition and fictitious domains methods for elliptic boundary value problems. In: Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations: Philadelphia, pp. 62–72. Society for Industrial and Applied Mathematics, PA (1992)

  25. Rjasanow, S., Steinbach, O.: The Fast Solution of Boundary Integral Equations. Mathematical and Analytical Techniques with Applications to Engineering. Springer, Berlin (2007)

    MATH  Google Scholar 

  26. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sauter, S.A., Schwab, C.: Boundary Element Methods. Springer Series in Computational Mathematics. Translated and Expanded from the 2004 German Original, vol. 39. Springer-Verlag, Berlin (2011)

    Book  MATH  Google Scholar 

  28. Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C., Scheichl, R.: Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Numer. Math. 126(4), 741–770 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Steinbach, O., Wendland, W.L.: The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math. 9(1–2), 191–216 (1998). (Numerical treatment of boundary integral equations)

    Article  MathSciNet  MATH  Google Scholar 

  30. Stephan, E.P.: Boundary integral equations for screen problems in \(\mathbb{R}^3\). Integr. Eqn. Oper. Theory 10(2), 236–257 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  31. Toselli, A., Widlund, O.B.: Domain Decomposition Methods: Algorithms and Theory, vol. 34. Springer, Berlin (2005)

    Book  MATH  Google Scholar 

  32. Tran, T., Stephan, E.P.: Additive Schwarz methods for the h-version boundary element method. Appl. Anal. 60(1–2), 63–84 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. West, D.B.: Introduction to Graph Theory, 2nd edn. Pearson, London (2000)

    Google Scholar 

  34. Widlund, O., Dryja, M.: An additive variant of the Schwarz alternating method for the case of many subregions. Technical report 339, Department of Computer Science, Courant Institute (1987)

  35. Xu, J.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56(3), 215–235 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work received support from the ANR research grant ANR-15-CE23-0017-01 and was granted access to the HPC resources of CINES under the allocation 2018-A0060607330 made by GENCI.

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Correspondence to Pierre Marchand.

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Marchand, P., Claeys, X., Jolivet, P. et al. Two-level preconditioning for \(h\)-version boundary element approximation of hypersingular operator with GenEO. Numer. Math. 146, 597–628 (2020). https://doi.org/10.1007/s00211-020-01149-5

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