Convergence analysis for finite element discretizations of the helmholtz equation with dirichlet-to-neumann boundary conditions

Mathematics of Computation

Volumn 79, Issue 272, 2010, Pages 1871-1914

  Melenk, J M ^a   Sauter, S ^b  

^a VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

^b UNIVERSITY OF ZURICH   (Switzerland)

Author keywords

Convergence; Helmholtz equation at high wave number; Hp finite elements; Stability

Indexed keywords

EID: 77956609030     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-10-02362-8     Document Type: Article

References (46)

1. M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Applied Mathematics Series 55. National Bureau of Standards, U.S. Department of Commerce, 1972.
2. M. Ainsworth. Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal., 42(2):553-575, 2004.
3. I. Babuška, A. Craig, J. Mandel, and J. Pitkäranta. Efficient preconditioning for the p version finite element method in two dimensions. SIAM J. Numer. Anal., 28(3):624-661, 1991.
4. I. Babuška and S. Sauter. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers. SIAM, J. Numer. Anal., 34(6):2392-2423, 1997.
5. I. Babuška and M. Suri. The optimal convergence rate of the p-version of the finite element method. SIAM J. Numer. Anal., 24:750-776, 1987.
6. I. M. Babuška, F. Ihlenburg, E. T. Paik, and S. A. Sauter. A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Comp. Meth. Appl. Mech. Eng., 128:325-359, 1995.
7. L. Banjai and S. Sauter. A Refined Galerkin Error and Stability Analysis for Highly Indefinite Variational Problems. SIAM J. Numer. Anal., 45(1):37-53, 2007.
8. S. Brenner and L. Scott. The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, 1994.
9. A. Buffa and P. Monk. Error estimates for the ultra weak variational formulation of the Helmholtz equation. Math. Mod. Numer. Anal., pages 925-940, 2008.
10. O. Cessenat and B. Desprès. Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation. SIAM J. Numer. Anal., 35:255-299, 1998.
11. O. Cessenat and B. Desprès. Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation. J. Computational Acoustics, 11:227-238, 2003.
12. S. Chandler-Wilde and P. Monk. Wave-number-explicit bounds in time-harmonic scattering. SIAM J. Numer. Anal., pages 1428-1455, 2008.
13. P. Ciarlet. The finite element method for elliptic problems. North-Holland, 1987.
14. L. Demkowicz. Polynomial exact sequences and projection-based interpolation with applications to Maxwell's equations. In D. Boffi, F. Brezzi, L. Demkowicz, L. Durán, R. Falk, and M. Fortin, editors, Mixed Finite Elements, Compatibility Conditions, and Applications, volume 1939 of Lectures Notes in Mathematics. Springer-Verlag, 2008.
15. A. Deraemaeker, I. Babǔska, and P. Bouillard. Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimesnions. Int. J. Numer. Meth. Eng., 46(4), 1999.
16. T. Eibner and J. Melenk. An adaptive strategy for hp-FEM based on testing for analyticity. Computational Mechanics, 39:575-595, 2007.
17. C. Farhat, I. Harari, and U. Hetmaniuk. A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comp. Meth. Appl. Mech. Eng., 192:1389-1419, 2003.
18. C. Farhat, R. Tezaur, and P. Weidemann-Goiran. Higher-order extensions of discontinuous Galerkin method for mid-frequency Helmholtz problems. Int. J. Numer. Meth. Eng., 61, 2004.
19. C. Gittelson, R. Hiptmair, and I. Perugia. Plane wave discontinuous Galerkin methods. Technical Report NI07088-HOP, Isaac Newton Institute Cambridge, Cambridge, UK, 2007.
20. I. S. Gradshteyn and I. Ryzhik. Table of Integrals, Series, and Products. Academic Press, New York, London, 1965.
21. B. Guo and J. Zhang. Stable and compatible polynomial extensions in three dimensions and applications to the p and h-p finite element method. SIAM J. Numer. Anal., 47(2):1195-1225, 2009.
22. I. Harari. Reducing spurious dispersion, anisotropy and reflection in finite element analysis of time-harmonic acoustics. Comput. Methods Appl. Mech. Engrg., 140(1-2):39-58, 1997.
23. I. Harari. Finite element dispersion of cylindrical and spherical acoustic waves. Comput. Methods Appl. Mech. Engrg., 190(20-21):2533-2542, 2001.
24. I. Harari and D. Avraham. High-order finite element methods for acoustic problems. J. Com-put. Acoust., 5(1):33-51, 1997.
25. I. Harari and T. Hughes. Finite element methods for the Helmholtz equation in an exterior domain: Model problems. Computer Methods in Applied Mechanics and Engineering, 87:59-96, North Holland, 1991.
26. T. Huttunen and P. Monk. The use of plane waves to approximate wave propagation in anisotropic media. J. Computational Mathematics, 25:350-367, 2007.
27. F. Ihlenburg. Finite Element Analysis of Acousting Scattering. Springer, New York, 1998.
28. F. Ihlenburg and I. Babǔska. Finite element solution to the Helmholtz equation with high wave number. Part II: The h-p version of the FEM. Siam J. Num. Anal., 34(1):315-358, 1997.
29. W. McLean. Strongly Elliptic Systems and Boundary Integral Equations. Cambridge, Univ. Press, 2000.
30. J. Melenk. hp finite element methods for singular perturbations, volume 1796 of Lecture Notes in Mathematics. Springer-Verlag, 2002.
31. J. Melenk. hp-interpolation of nonsmooth functions and an application to hp a posteriori error estimation. SIAM J. Numer. Anal., 43:127-155, 2005.
32. J. Melenk and S. Sauter. Wave-number explicit convergence analysis for finite element discretizations of the Helmholtz equation. Technical Report 09/2009, Inst. für Mathematik, Univ. Zürich, 2009.
33. J. M. Melenk. On Generalized Finite Element Methods. Ph.D. thesis, University of Maryland at College Park, 1995.
34. R. Muñoz-Sola. Polynomial liftings on a tetrahedron and applications to the hp-version of the finite element method in three dimensions. SIAM J. Numer. Anal., 34(1):282-314, 1997.
35. J. C. Nédélec. Acoustic and Electromagnetic Equations. Springer, New York, 2001.
36. A. Oberai and P. Pinsky. A numerical comparison of finite element methods for the Helmholtz equation. J. Comput. Acoust., 8(1):211-221, 2000.
37. F. Olver. Asymptotics and Special Functions. A K Peters, Natick, 1997.
38. E. Perrey-Debain, O. Laghrouche, and P. Bettess. Plane-wave basis finite elements and boundary elements for three-dimensional wave scattering. Phil. Trans. R. Soc. London A, 362:561-577, 2004.
39. S. Sauter. A refined finite element convergence theory for highly indefinite Helmholtz problems. Computing, 78(2):101-115, 2006.
40. A. Schatz. An obeservation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comp., 28:959-962, 1974.
41. C. Schwab. p- and hp-Finite Element Methods. Oxford University Press, 1998.
42. E. Stein. Singular integrals and differentiability properties of functions. Princeton University Press, 1970.
43. M. Stojek. Least-squares Trefftz-type elements for the Helmholtz equation. Int. J. Numer. Meth. Engr., 41:831-849, 1998.
44. R. Tezaur and C. Farhat. Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems. Int. J. Numer. Meth. Engr., 66:796-815, 2006.
45. H. Triebel. Interpolation Theory, Function Spaces, Differential Operators. Johann Ambrosius Barth, 2 edition, 1995.
46. G. N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press, 1922.