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Computation of eigenvalues by numerical upscaling

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Abstract

We present numerical upscaling techniques for a class of linear second-order self-adjoint elliptic partial differential operators (or their high-resolution finite element discretization). As prototypes for the application of our theory we consider benchmark multi-scale eigenvalue problems in reservoir modeling and material science. We compute a low-dimensional generalized (possibly mesh free) finite element space that preserves the lowermost eigenvalues in a superconvergent way. The approximate eigenpairs are then obtained by solving the corresponding low-dimensional algebraic eigenvalue problem. The rigorous error bounds are based on two-scale decompositions of \(\text {H}^1_0(\Omega )\) by means of a certain Clément-type quasi-interpolation operator.

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References

  1. Banjai, L., Börm, S., Sauter, S.: FEM for elliptic eigenvalue problems: how coarse can the coarsest mesh be chosen? An experimental study. Comput. Vis. Sci. 11(4–6), 363–372 (2008)

    Article  MathSciNet  Google Scholar 

  2. Birkhoff, G., de Boor, C., Swartz, B., Wendroff, B.: Rayleigh–Ritz approximation by piecewise cubic polynomials. SIAM J. Numer. Anal. 3, 188–203 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bank, R.E., Grubišić, L., Ovall, J.S.: A framework for robust eigenvalue and eigenvector error estimation and Ritz value convergence enhancement. Appl. Numer. Math. 66, 1–29 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Boffi, D.: Finite element approximation of eigenvalue problems. Acta Numer. 19, 1–120 (2010)

  5. Carstensen, C., Gedicke, J.: An oscillation-free adaptive FEM for symmetric eigenvalue problems. Numer. Math. 118(3), 401–427 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Carstensen, C., Gedicke, J.: An adaptive finite element eigenvalue solver of asymptotic quasi-optimal computational complexity. SIAM J. Numer. Anal. 50(3), 1029–1057 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chu, C.-C., Graham, I.G., Hou, T.-Y.: A new multiscale finite element method for high-contrast elliptic interface problems. Math. Comput. 79(272), 1915–1955 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1987)

    Google Scholar 

  9. Carstensen, C., Verfürth, R.: Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36(5), 1571–1587 (1999). (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  10. Durán, R.G., Padra, C., Rodríguez, R.: A posteriori error estimates for the finite element approximation of eigenvalue problems. Math. Models Methods Appl. Sci. 13(8), 1219–1229 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Giani, S., Graham, I.G.: A convergent adaptive method for elliptic eigenvalue problems. SIAM J. Numer. Anal. 47(2), 1067–1091 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Garau, E.M., Morin, P., Zuppa, C.: Convergence of adaptive finite element methods for eigenvalue problems. Math. Models Methods Appl. Sci. 19(5), 721–747 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hackbusch, W.: On the computation of approximate eigenvalues and eigenfunctions of elliptic operators by means of a multi-grid method. SIAM J. Numer. Anal. 16(2), 201–215 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  14. Henning, P., Målqvist, A.: Localized orthogonal decomposition techniques for boundary value problems. SIAM J. Sci. Comput. 36(4), A1609–A1634 (2014)

    Article  MATH  Google Scholar 

  15. Henning, P., Morgenstern, P., Peterseim, D.: Multiscale partition of unity. In: Griebel, M., Schweitzer, M.A. (eds.) Meshfree Methods for Partial Differential Equations VII, Lecture Notes in Computational Science and Engineering, vol. 100. Springer, New York (2014)

  16. Henning, P., Målqvist, A., Peterseim, D.: Two-level discretization techniques for ground state computations of Bose–Einstein condensates. SIAM J. Numer. Anal. 52(4), 1525–1550 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  17. Henning, P., Målqvist, A., Peterseim, D.: A localized orthogonal decomposition method for semi-linear elliptic problems. ESAIM. Math. Model. Numer. Anal. 48, 1331–1349 (2014). 9

    Article  MATH  MathSciNet  Google Scholar 

  18. Henning, P., Peterseim, D.: Oversampling for the multiscale finite element method. Multiscale Model. Simul. 11(4), 1149–1175 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Knyazev, A.V., Neymeyr, K.: Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method. Tenth Copper Mountain Conference on Multigrid Methods (Copper Mountain, CO, 2001). Electron. Trans. Numer. Anal. 15, 38–55 (2003). (electronic)

  20. Knyazev, A.V., Neymeyr, K.: A geometric theory for preconditioned inverse iteration. III. A short and sharp convergence estimate for generalized eigenvalue problems. Special issue on accurate solution of eigenvalue problems (Hagen, 2000). Linear Algebra Appl. 358, 95–114 (2003)

  21. Knyazev, A.V., Osborn, J.E.: New a priori FEM error estimates for eigenvalues. SIAM J. Numer. Anal. 43(6), 2647–2667 (2006). (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  22. Larson, M.G.: A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems. SIAM J. Numer. Anal. 38(2), 608–625 (2000). (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK users’ guide, volume 6 of Software, Environments, and Tools. Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1998)

  24. Mehrmann, V., Miedlar, A.: Adaptive computation of smallest eigenvalues of self-adjoint elliptic partial differential equations. Numer. Linear Algebra Appl. 18(3), 387–409 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Målqvist, A., Peterseim, D.: Localization of elliptic multiscale problems. Math. Comput. 83(290), 2583–2603 (2014)

    Article  MATH  Google Scholar 

  26. Neymeyr, K.: A posteriori error estimation for elliptic eigenproblems. Numer. Linear Algebra Appl. 9(4), 263–279 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Neymeyr, K.: Solving mesh eigenproblems with multigrid efficiency. In: Numerical Methods for Scientific Computing. Variational Problems and Applications. Internat. Center Numer. Methods Eng. (CIMNE), Barcelona, pp. 176–184 (2003)

  28. Peterseim, D., Carstensen, C.: Finite element network approximation of conductivity in particle composites. Numer. Math. 124(1), 73–97 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  29. Peterseim, D.: Composite finite elements for elliptic interface problems. Math. Comput. 83(290), 2657–2674 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  30. Poincaré, H.: Sur les Equations aux Derivees Partielles de la Physique Mathematique. Am. J. Math. 12(3), 211–294 (1890)

    Article  MATH  Google Scholar 

  31. Peterseim, D., Sauter, S.: Finite elements for elliptic problems with highly varying, nonperiodic diffusion matrix. Multiscale Model. Simul. 10(3), 665–695 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  32. Sarkis, M.: Partition of unity coarse spaces and Schwarz methods with harmonic overlap. In Recent developments in domain decomposition methods (Zürich, 2001), Lect. Notes Comput. Sci. Eng., vol. 23. Springer, Berlin, pp. pages 77–94 (2002)

  33. Sauter, S.: \(hp\)-finite elements for elliptic eigenvalue problems: error estimates which are explicit with respect to \(\lambda \), \(h\), and \(p\). SIAM J. Numer. Anal. 48(1), 95–108 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  34. Strang, G., Fix, G. J.: An Analysis of the Finite Element Method, Prentice-Hall Series in Automatic Computation. Prentice-Hall Inc., Englewood Cliffs (1973)

  35. Scheichl, R., Vassilevski, P.S., Zikatanov, L.T.: Weak approximation properties of elliptic projections with functional constraints. Multiscale Model. Simul. 9(4), 1677–1699 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  36. Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory, Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2005)

  37. Xu, J., Zhou, A.: A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70(233), 17–25 (2001)

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Authors and Affiliations

  1. Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96 , Göteborg, Sweden

    Axel Målqvist

  2. Institute for Numerical Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, Wegelerstr. 6, 53115 , Bonn, Germany

    Daniel Peterseim

Authors
  1. Axel Målqvist
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  2. Daniel Peterseim
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Correspondence to Daniel Peterseim.

Additional information

A. Målqvist is supported by The Göran Gustafsson Foundation and The Swedish Research Council and D. Peterseim was partially supported by the DFG Research Center Matheon Berlin through project C33.

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Målqvist, A., Peterseim, D. Computation of eigenvalues by numerical upscaling. Numer. Math. 130, 337–361 (2015). https://doi.org/10.1007/s00211-014-0665-6

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  • DOI: https://doi.org/10.1007/s00211-014-0665-6

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