Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 


Eliminating the pollution effect in Helmholtz problems by local subscale correction

Author: Daniel Peterseim
Journal: Math. Comp. 86 (2017), 1005-1036
MSC (2010): Primary 65N12, 65N15, 65N30
Published electronically: August 3, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a new Petrov-Galerkin multiscale method for the numerical approximation of the Helmholtz equation with large wave number $ \kappa $ in bounded domains in $ \mathbb{R}^d$. The discrete trial and test spaces are generated from standard mesh-based finite elements by local subscale correction in the spirit of numerical homogenization. The precomputation of the correction involves the solution of coercive cell problems on localized subdomains of size $ \ell H$, $ H$ being the mesh size and $ \ell $ being the oversampling parameter. If the mesh size and the oversampling parameter are such that $ H\kappa $ and $ \log (\kappa )/\ell $ fall below some generic constants and if the cell problems are solved sufficiently accurately on some finer scale of discretization, then the method is stable and its error is proportional to $ H$. Pollution effects are eliminated in this regime.

References [Enhancements On Off] (What's this?)

  • [AB14] A. Abdulle and Y. Bai, Reduced-order modelling numerical homogenization, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014), no. 2021, 20130388, 23. MR 3247814,
  • [BCWG] T. Betcke, S. N. Chandler-Wilde, I. G. Graham, S. Langdon, and M. Lindner, Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation, Numer. Methods Partial Differential Equations 27 (2011), no. 1, 31-69. MR 2743599,
  • [BP14] D. Brown and D. Peterseim, A multiscale method for porous microstructures, ArXiv e-prints, November 2014.
  • [BS00] Ivo M. Babuška and Stefan A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM Rev. 42 (2000), no. 3, 451-484 (electronic). Reprint of SIAM J. Numer. Anal. 34 (1997), no. 6, 2392-2423 [ MR1480387 (99b:65135)]. MR 1786934,
  • [BS07] L. Banjai and S. Sauter, A refined Galerkin error and stability analysis for highly indefinite variational problems, SIAM J. Numer. Anal. 45 (2007), no. 1, 37-53 (electronic). MR 2285843,
  • [BS08] Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954
  • [BY14] Randolph E. Bank and Harry Yserentant, On the $ H^1$-stability of the $ L_2$-projection onto finite element spaces, Numer. Math. 126 (2014), no. 2, 361-381. MR 3150226,
  • [Car99] Carsten Carstensen, Quasi-interpolation and a posteriori error analysis in finite element methods, M2AN Math. Model. Numer. Anal. 33 (1999), no. 6, 1187-1202. MR 1736895,
  • [CF00] C. Carstensen and S. A. Funken, Constants in Clément-interpolation error and residual based a posteriori error estimates in finite element methods, East-West J. Numer. Math. 8 (2000), no. 3, 153-175. MR 1807259
  • [CF06] Peter Cummings and Xiaobing Feng, Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations, Math. Models Methods Appl. Sci. 16 (2006), no. 1, 139-160. MR 2194984,
  • [Cia78] Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
  • [CV99] Carsten Carstensen and Rüdiger Verfürth, Edge residuals dominate a posteriori error estimates for low order finite element methods, SIAM J. Numer. Anal. 36 (1999), no. 5, 1571-1587 (electronic). MR 1706735,
  • [DGMZ12] L. Demkowicz, J. Gopalakrishnan, I. Muga, and J. Zitelli, Wavenumber explicit analysis of a DPG method for the multidimensional Helmholtz equation, Comput. Methods Appl. Mech. Engrg. 213/216 (2012), 126-138. MR 2880509,
  • [DPE12] Daniele Antonio Di Pietro and Alexandre Ern, Mathematical aspects of discontinuous Galerkin methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69, Springer, Heidelberg, 2012. MR 2882148
  • [EG12] O. G. Ernst and M. J. Gander, Why it is difficult to solve Helmholtz problems with classical iterative methods, Numerical analysis of multiscale problems, Lect. Notes Comput. Sci. Eng., vol. 83, Springer, Heidelberg, 2012, pp. 325-363. MR 3050918,
  • [EGMP13] Daniel Elfverson, Emmanuil H. Georgoulis, Axel Målqvist, and Daniel Peterseim, Convergence of a discontinuous Galerkin multiscale method, SIAM J. Numer. Anal. 51 (2013), no. 6, 3351-3372. MR 3141754,
  • [EM12] S. Esterhazy and J. M. Melenk, On stability of discretizations of the Helmholtz equation, Numerical analysis of multiscale problems, Lect. Notes Comput. Sci. Eng., vol. 83, Springer, Heidelberg, 2012, pp. 285-324. MR 3050917,
  • [FW09] Xiaobing Feng and Haijun Wu, Discontinuous Galerkin methods for the Helmholtz equation with large wave number, SIAM J. Numer. Anal. 47 (2009), no. 4, 2872-2896. MR 2551150,
  • [FW11] Xiaobing Feng and Haijun Wu, $ hp$-discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp. 80 (2011), no. 276, 1997-2024. MR 2813347,
  • [For77] Michel Fortin, An analysis of the convergence of mixed finite element methods, RAIRO Anal. Numér. 11 (1977), no. 4, 341-354, iii (English, with French summary). MR 0464543
  • [GP15] D. Gallistl and D. Peterseim, Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering, Comput. Methods Appl. Mech. Engrg. 295 (2015), 1-17. MR 3388822,
  • [GGS15] M. J. Gander, I. G. Graham, and E. A. Spence, Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?, Numer. Math. 131 (2015), no. 3, 567-614. MR 3395145,
  • [HMP14a] P. Henning, P. Morgenstern, and D. Peterseim, Multiscale partition of unity, in Meshfree Methods for Partial Differential Equations VII, M. Griebel and M. A. Schweitzer, eds., Lect. Notes Comput. Sci. Eng., vol. 100, Springer, 2014.
  • [HMP14b] Patrick Henning, Axel Målqvist, and Daniel Peterseim, A localized orthogonal decomposition method for semi-linear elliptic problems, ESAIM Math. Model. Numer. Anal. 48 (2014), no. 5, 1331-1349. MR 3264356,
  • [HP13] Patrick Henning and Daniel Peterseim, Oversampling for the multiscale finite element method, Multiscale Model. Simul. 11 (2013), no. 4, 1149-1175. MR 3123820,
  • [Het02] Ulrich Ladislas Hetmaniuk, Fictitious domain decomposition methods for a class of partially axisymmetric problems: Application to the scattering of acoustic waves, ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)-University of Colorado at Boulder. MR 2704104
  • [Het07] U. Hetmaniuk, Stability estimates for a class of Helmholtz problems, Commun. Math. Sci. 5 (2007), no. 3, 665-678. MR 2352336
  • [HMP11] R. Hiptmair, A. Moiola, and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the $ p$-version, SIAM J. Numer. Anal. 49 (2011), no. 1, 264-284. MR 2783225,
  • [HMP14c] Ralf Hiptmair, Andrea Moiola, and Ilaria Perugia, Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes, Appl. Numer. Math. 79 (2014), 79-91. MR 3191479,
  • [Hug95] Thomas J. R. Hughes, Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg. 127 (1995), no. 1-4, 387-401. MR 1365381,
  • [HFMQ98] Thomas J. R. Hughes, Gonzalo R. Feijóo, Luca Mazzei, and Jean-Baptiste Quincy, The variational multiscale method--a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg. 166 (1998), no. 1-2, 3-24. MR 1660141,
  • [HS07] T. J. R. Hughes and G. Sangalli, Variational multiscale analysis: the fine-scale Green's function, projection, optimization, localization, and stabilized methods, SIAM J. Numer. Anal. 45 (2007), no. 2, 539-557. MR 2300286,
  • [Mål11] Axel Målqvist, Multiscale methods for elliptic problems, Multiscale Model. Simul. 9 (2011), no. 3, 1064-1086. MR 2831590,
  • [Mel95] Jens Markus Melenk, On generalized finite-element methods, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)-University of Maryland, College Park. MR 2692949
  • [MIB96] Ch. Makridakis, F. Ihlenburg, and I. Babuška, Analysis and finite element methods for a fluid-solid interaction problem in one dimension, Math. Models Methods Appl. Sci. 6 (1996), no. 8, 1119-1141. MR 1428148,
  • [MP14b] Axel Målqvist and Daniel Peterseim, Localization of elliptic multiscale problems, Math. Comp. 83 (2014), no. 290, 2583-2603. MR 3246801,
  • [MP14a] Axel Målqvist and Daniel Peterseim, Computation of eigenvalues by numerical upscaling, Numer. Math. 130 (2015), no. 2, 337-361. MR 3343928,
  • [MPS13] J. M. Melenk, A. Parsania, and S. Sauter, General DG-methods for highly indefinite Helmholtz problems, J. Sci. Comput. 57 (2013), no. 3, 536-581. MR 3123557,
  • [MS10] J. M. Melenk and S. Sauter, Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp. 79 (2010), no. 272, 1871-1914. MR 2684350,
  • [MS11] J. M. Melenk and S. Sauter, Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation, SIAM J. Numer. Anal. 49 (2011), no. 3, 1210-1243. MR 2812565,
  • [Pet15] D. Peterseim, Variational multiscale stabilization and the exponential decay of fine-scale correctors, ArXiv e-prints, 1505.07611, 2015, to appear, DOI 10.1007/978-3-319-41640-3_11.
  • [PS14] D. Peterseim and R. Scheichl, Rigorous numerical upscaling at high contrast, ArXiv e-prints, 1601.06549, Comput. Methods Appl. Math., 2016, DOI 10.1515/cmam-2016-0022.
  • [RHP08] G. Rozza, D. B. P. Huynh, and A. T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics, Arch. Comput. Methods Eng. 15 (2008), no. 3, 229-275. MR 2430350,
  • [Sau06] S. A. Sauter, A refined finite element convergence theory for highly indefinite Helmholtz problems, Computing 78 (2006), no. 2, 101-115. MR 2255368,
  • [Sch74] Alfred H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28 (1974), 959-962. MR 0373326
  • [Szy06] Daniel B. Szyld, The many proofs of an identity on the norm of oblique projections, Numer. Algorithms 42 (2006), no. 3-4, 309-323. MR 2279449,
  • [TF06] Radek Tezaur and Charbel Farhat, Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems, Internat. J. Numer. Methods Engrg. 66 (2006), no. 5, 796-815. MR 2219901,
  • [Wu14] Haijun Wu, Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I: Linear version, IMA J. Numer. Anal. 34 (2014), no. 3, 1266-1288. MR 3232452,
  • [ZMD] J. Zitelli, I. Muga, L. Demkowicz, J. Gopalakrishnan, D. Pardo, and V. M. Calo, A class of discontinuous Petrov-Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D, J. Comput. Phys. 230 (2011), no. 7, 2406-2432. MR 2772923,

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N12, 65N15, 65N30

Retrieve articles in all journals with MSC (2010): 65N12, 65N15, 65N30

Additional Information

Daniel Peterseim
Affiliation: Rheinische Friedrich-Wilhelms-Universität Bonn, Institute for Numerical Simulation, Wegelerstr. 6, 53115 Bonn, Germany

Keywords: Pollution effect, finite element, multiscale method, numerical homogenization
Received by editor(s): November 27, 2014
Received by editor(s) in revised form: October 17, 2015
Published electronically: August 3, 2016
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society