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Journal of Scientific Computing

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25-02-2017

Relaxing the CFL Condition for the Wave Equation on Adaptive Meshes

Authors: Daniel Peterseim, Mira Schedensack

Published in: Journal of Scientific Computing | Issue 3/2017

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Abstract

The Courant–Friedrichs–Lewy (CFL) condition guarantees the stability of the popular explicit leapfrog method for the wave equation. However, it limits the choice of the time step size to be bounded by the minimal mesh size in the spatial finite element mesh. This essentially prohibits any sort of adaptive mesh refinement that would be required to reveal optimal convergence rates on domains with re-entrant corners. This paper shows how a simple subspace projection step inspired by numerical homogenisation can remove the critical time step restriction so that the CFL condition and approximation properties are balanced in an optimal way, even in the presence of spatial singularities.

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Abdulle, A., Henning, P.: Localized orthogonal decomposition method for the wave equation with a continuum of scales. Math. Comput. 86(304), 549–587 (2017)

Bank, R.E., Dupont, T.: An optimal order process for solving finite element equations. Math. Comput. 36(153), 35–51 (1981). doi:10.2307/2007724 MathSciNetCrossRefMATH

Brenner, S.C.: Two-level additive Schwarz preconditioners for nonconforming finite element methods. Math. Comput. 65(215), 897–921 (1996)MathSciNetCrossRefMATH

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)CrossRefMATH

Carstensen, C., Gallistl, D., Schedensack, M.: \(L^2\) best-approximation of the elastic stress in the Arnold–Winther FEM. IMA J. Numer. Anal. 36(3), 1096–1119 (2016). doi:10.1093/imanum/drv051 MathSciNetCrossRef

Christiansen, S.H.: Foundations of finite element methods for wave equations of Maxwell type. In: Quak, E., Soomere, T. (eds.) Applied Wave Mathematics, pp. 335–393. Springer, Berlin (2009)CrossRef

Ciarlet Jr., P., He, J.: The singular complement method for 2d scalar problems. C. R. Math. Acad. Sci. Paris 336(4), 353–358 (2003). doi:10.1016/S1631-073X(03)00030-X MathSciNetCrossRefMATH

Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100(1), 32–74 (1928). doi:10.1007/BF01448839 MathSciNetCrossRefMATH

Di Pietro, D.A., Ern, A.: Mathematical aspects of discontinuous Galerkin methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69. Springer, Heidelberg (2012). doi:10.1007/978-3-642-22980-0

10.

Diaz, J., Grote, M.J.: Energy conserving explicit local time stepping for second-order wave equations. SIAM J. Sci. Comput. 31(3), 1985–2014 (2009). doi:10.1137/070709414 MathSciNetCrossRefMATH

11.

Diaz, J., Grote, M.J.: Multi-level explicit local time-stepping methods for second-order wave equations. Comput. Methods Appl. Mech. Eng. 291, 240–265 (2015). doi:10.1016/j.cma.2015.03.027 MathSciNetCrossRef

12.

Elfverson, D., Georgoulis, E.H., Målqvist, A., Peterseim, D.: Convergence of a discontinuous Galerkin multiscale method. SIAM J. Numer. Anal. 51(6), 3351–3372 (2013). doi:10.1137/120900113 MathSciNetCrossRefMATH

13.

Ern, A., Guermond, J.L.: Finite element quasi-interpolation and best approximation. ArXiv e-prints. Preprint arXiv:1505.06931 (2015)

14.

Gallistl, D., Huber, P., Peterseim, D.: On the stability of the Rayleigh–Ritz method for eigenvalues. INS Preprint No. 1527. http://peterseim.ins.uni-bonn.de/research/pub/INS1527 (2015)

15.

Gaspoz, F.D., Morin, P.: Convergence rates for adaptive finite elements. IMA J. Numer. Anal. 29(4), 917–936 (2009). doi:10.1093/imanum/drn039 MathSciNetCrossRefMATH

16.

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, pp. 185–204. Springer, NewYork (2015)

17.

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). doi:10.1137/130921520 MathSciNetCrossRefMATH

18.

Henning, P., Peterseim, D.: Oversampling for the multiscale finite element method. Multiscale Model. Simul. 11(4), 1149–1175 (2013). doi:10.1137/120900332 MathSciNetCrossRefMATH

19.

Hochbruck, M., Sturm, A.: Error analysis of a second order locally implicit method for linear Maxwell’s equations. CRC 1173-Preprint, no. 2015/1, Karlsruher Institut für Technologie. http://www.waves.kit.edu/downloads/CRC1173_Preprint_2015-1 (2015)

20.

Joly, P.: Variational methods for time-dependent wave propagation problems. In: Topics in Computational Wave Propagation. Lecture Notes Computation Science Engineering, vol. 31, pp. 201–264. Springer, Berlin (2003)

21.

Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41(6), 2374–2399 (2003)MathSciNetCrossRefMATH

22.

Målqvist, A., Peterseim, D.: Computation of eigenvalues by numerical upscaling. Numer. Math. 130(2), 337–361 (2014). doi:10.1007/s00211-014-0665-6 MathSciNetCrossRef

23.

Målqvist, A., Peterseim, D.: Localization of elliptic multiscale problems. Math. Comput. 83(290), 2583–2603 (2014). doi:10.1090/S0025-5718-2014-02868-8 MathSciNetCrossRefMATH

24.

Målqvist, A., Peterseim, D.: Generalized finite element methods for quadratic eigenvalue problems. ESAIM Math. Model. Numer. Anal. (2016). doi:10.1051/m2an/2016019

25.

Müller, F.L., Schwab, C.: Finite elements with mesh refinement for wave equations in polygons. J. Comput. Appl. Math. 283, 163–181 (2015). doi:10.1016/j.cam.2015.01.002 MathSciNetCrossRefMATH

26.

Oswald, P.: On a BPX-preconditioner for P1 elements. Computing 51(2), 125–133 (1993). doi:10.1007/BF02243847 MathSciNetCrossRefMATH

27.

Peterseim, D.: Variational multiscale stabilization and the exponential decay of fine-scale correctors. Preprint arXiv:1505.07611 (2015)

28.

Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990). doi:10.2307/2008497 MathSciNetCrossRefMATH

29.

Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Advances in Numerical Mathematics. Wiley, Hoboken (1996)MATH

30.

Wathen, A.J.: Realistic eigenvalue bounds for the Galerkin mass matrix. IMA J. Numer. Anal. 7(4), 449–457 (1987). doi:10.1093/imanum/7.4.449, http://imajna.oxfordjournals.org/content/7/4/449.abstract

Title: Relaxing the CFL Condition for the Wave Equation on Adaptive Meshes
Authors: Daniel Peterseim
Mira Schedensack
Publication date: 25-02-2017
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2017
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0394-y

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