A Posteriori Error Estimation of $hp$-$dG$ Finite Element Methods for Highly Indefinite Helmholtz Problems
Abstract
In this paper, we will consider an $hp$-finite elements discretization of a highly indefinite Helmholtz problem by some ${dG}$-formulation which is based on the ultra-weak variational formulation by Cessenat and Deprés. We will introduce an a posteriori error estimator and derive reliability and efficiency estimates which are explicit with respect to the wavenumber and the discretization parameters $h$ and $p$. In contrast to the conventional conforming finite element method for indefinite problems, the ${dG}$-formulation is unconditionally stable and the adaptive discretization process may start from a very coarse initial mesh. Numerical experiments will illustrate the efficiency and robustness of the method.
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References
1.
J. D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, Amsterdam, 2005.
2.
M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley, New York, 2000.
3.
I. Babuška, F. Ihlenburg, T. Strouboulis, and S. K. Gangaraj, A posteriori error estimation for finite element solutions of Helmholtz' equation I. The quality of local indicators and estimators, Internat. J. Numer. Methods Engrg., 40 (1997), pp. 3443--3462.
4.
I. Babuška, F. Ihlenburg, T. Strouboulis, and S. K. Gangaraj, A posteriori error estimation for finite element solutions of Helmholtz' equation II. Estimation of the pollution error, Internat. J. Numer. Methods Engrg., 40 (1997), pp. 3883--3900.
5.
I. Babuška and W. C. Rheinboldt, A-posteriori error estimates for the finite element method, Internat. J. Numer. Methods Engrg., 12 (1978), pp. 1597--1615.
6.
I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), pp. 736--754.
7.
A. Bonito and R. H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal., 48 (2010), pp. 734--771.
8.
A. Buffa and P. Monk, Error estimates for the ultra weak variational formulation of the Helmholtz equation, M2AN Math. Model. Numer. Anal., 42 (2008), pp. 925--940.
9.
O. Cessenat and B. Després, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), pp. 255--299.
10.
O. Cessenat and B. Després, Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation, J. Comput. Acoust., 11 (2003), pp. 227--238.
11.
S. N. Chandler-Wilde, I. G. Graham, S. Langdon, and E. A. Spence, Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering, Acta Numer., 21 (2012), pp. 89--305.
12.
P. Cummings and X. Feng, Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations, Math. Models Methods Appl. Sci., 16 (2006), pp. 139--160.
13.
B. Després, Sur une formulation variationnelle de type ultra-faible, C. R. Acad. Sci. Paris Sér. I Math., 318 (1994), pp. 939--944.
14.
R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, New York, 1993.
15.
D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Springer, Heidelberg, 2012.
16.
W. Dörfler and S. Sauter, A posteriori error estimation for highly indefinite Helmholtz problems, Comput. Methods Appl. Math., 13 (2013), pp. 333--347.
17.
S. Esterhazy and J. M. Melenk, On stability of discretizations of the Helmholtz equation, in Numerical Analysis of Multiscale Problems, I. Graham, T. Hou, O. Lakkis, and R. Scheichl, eds., Lect. Notes Comput. Sci. Eng. 83, Springer, Berlin, 2012, pp. 285--324.
18.
X. Feng and H. Wu, Discontinuous Galerkin methods for the Helmholtz equation with large wave number, SIAM J. Numer. Anal., 47 (2009), pp. 2872--2896.
19.
X. Feng and H. Wu, $hp$-discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp., 80 (2011), pp. 1997--2024.
20.
X. Feng and Y. Xing, Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp., 82 (2013), pp. 1269--1296.
21.
C. Gittelson, R. Hiptmair, and I. Perugia, Plane wave discontinuous Galerkin methods: Analysis of the h-version, Internat. J. Numer. Methods Engrg., 43 (2009), pp. 297--331.
22.
M. Grigoroscuta-Strugaru, M. Amara, H. Calandra, and R. Djellouli, A modified discontinuous Galerkin method for solving efficiently Helmholtz problems, Commun. Comput. Phys., 11 (2012), pp. 335--350.
23.
I. Harari, A survey of finite element methods for time-harmonic acoustics, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 1594--1607.
24.
I. Harari and T. J. R. Hughes, Galerkin/least-squares finite element methods for the reduced wave equation with nonreflecting boundary conditions in unbounded domains, Comput. Methods Appl. Mech. Engrg., 98 (1992), pp. 411--454.
25.
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), pp. 264--284.
26.
R. H. W. Hoppe and N. Sharma, Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the Helmholtz equation, IMA J. Numer. Anal., 33 (2013), pp. 898--921.
27.
P. Houston, D. Schötzau, and T. P. Wihler, Energy norm a posteriori error estimation of $hp$-adaptive discontinuous Galerkin methods for elliptic problems, Math. Models Methods Appl. Sci., 17 (2007), pp. 33--62.
28.
T. Huttunen and P. Monk, The use of plane waves to approximate wave propagation in anisotropic media, J. Comput. Math., 25 (2007), pp. 350--367.
29.
F. Ihlenburg, Finite Element Analysis of Acousting Scattering, Springer, New York, 1998.
30.
F. Ihlenburg and I. Babuška, Finite element solution to the Helmholtz equation with high wave number. Part II: The h-p version of the FEM, SIAM J. Numer. Anal., 34 (1997), pp. 315--358.
31.
S. Irimie and P. Bouillard, A residual a posteriori error estimator for the finite element solution of the Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 190 (2001), pp. 4027--4042.
32.
J. D. Jackson, Classical Electrodynamics, 2nd ed., Wiley, New York, 1975.
33.
R. Kellogg, Interpolation Between Subspaces of a Hilbert Space, Tech. report BN-719, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland at College Park, College Park, 1971.
34.
J. M. Melenk, On Generalized Finite Element Methods, Ph.D. thesis, University of Maryland at College Park, 1995.
35.
J. M. Melenk, $hp$-interpolation of nonsmooth functions and an application to $hp$-a posteriori error estimation, SIAM J. Numer. Anal., 43 (2005), pp. 127--155.
36.
J. M. Melenk, A. Parsania, and S. Sauter, Generalized dG methods for highly indefinite Helmholtz problems based on the ultra-weak variational formulation, J. Sci. Comput., 57 (2013), pp. 536--581.
37.
J. M. Melenk and S. Sauter, Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary condition., Math. Comp., 79 (2010), pp. 1871--1914.
38.
J. M. Melenk and S. Sauter, Wave-number explicit convergence analysis for Galerkin discretizations of the Helmholtz equation, SIAM J. Numer. Anal., 49 (2011), pp. 1210--1243.
39.
J. M. Melenk and B. I. Wohlmuth, On residual-based a posteriori error estimation in $hp$-FEM, Adv. Comput. Math., 15 (2001), pp. 311--331 (2002).
40.
P. Monk and D.-Q. Wang, A least-squares method for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 121--136.
41.
R. H. Nochetto, K. G. Siebert, and A. Veeser, Theory of adaptive finite element methods: An introduction, in Multiscale, Nonlinear and Adaptive Approximation, Springer, Berlin, 2009, pp. 409--542.
42.
J. T. Oden, S. Prudhomme, and L. Demkowicz, A posteriori error estimation for acoustic wave propagation problems, Arch. Comput. Methods Engrg., 12 (2005), pp. 343--389.
43.
S. Sauter and J. Zech, A Posteriori Error Estimation of $hp$-dG Finite Element Methods for Highly Indefinite Helmholtz Problems (extended version), \newblock arXiv:1407.1430v2 [math.NA], 2014.
44.
A. H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp., 28 (1974), pp. 959--962.
45.
C. Schwab, $p$- and $hp$-Finite Element Methods, Clarendon Press, New York, 1998.
46.
R. Verfürth, A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, Oxford, UK, 2013.
47.
H. 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), pp. 1266--1288.
48.
J. Zech, A Posteriori Error Estimation of $hp$-DG Finite Element Methods for Highly Indefinite Helmholtz Problems, M.S. thesis, Instiut für Mathematik, Unversität Zürich, 2014; also available online from \newblock https://www.math.uzh.ch/compmath/index.php?id=zech.
49.
L. Zhu and H. Wu, Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part II: $hp$ version, SIAM J. Numer. Anal., 51 (2013), pp. 1828--1852.
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History
Submitted: 23 June 2014
Accepted: 24 March 2015
Published online: 29 October 2015
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