High-frequency behaviour of corner singularities in Helmholtz problems

Chaumont-Frelet, T.; Nicaise, S.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1803-1845.

We analyze the singular behaviour of the Helmholtz equation set in a non-convex polygon. Classically, the solution of the problem is split into a regular part and one singular function for each re-entrant corner. The originality of our work is that the “amplitude” of the singular parts is bounded explicitly in terms of frequency. We show that for high frequency problems, the “dominant” part of the solution is the regular part. As an application, we derive sharp error estimates for finite element discretizations. These error estimates show that the “pollution effect” is not changed by the presence of singularities. Furthermore, a consequence of our theory is that locally refined meshes are not needed for high-frequency problems, unless a very accurate solution is required. These results are illustrated with numerical examples that are in accordance with the developed theory.

[1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, 10th edn. NBS (1972).

[2] J.J. Fournier and R.A. Adams, Sobolev Spaces, 2nd edn. Academic Press (2003). | MR | Zbl

[3] M. Ainsworth, Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal. 42 (2004) 553–575. | DOI | MR | Zbl

[4] M. Amara, R. Djellouli and C. Farhat, Convergence analysis of a discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of Helmholtz problems. SIAM J. Numer. Anal. 47 (2009) 1038–1066. | DOI | MR | Zbl

[5] T. Apel and S. Nicaise, The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Methods Appl. Sci. 21 (1998) 519–549. | DOI | MR | Zbl

[6] C. Bacuta, J.H. Bramble and J. Xu, Regularity estimates for elliptic boundary value problems in Besov spaces. Math. Comput. 72 (2003) 1577–1595. | DOI | MR | Zbl

[7] J.P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114 (1994) 185–200. | DOI | MR | Zbl

[8] S.N. Chandler-Wilde, D.P. Hewett, S. Langdon and A. Twigger, A high frequency boundary element method for scattering by a class of nonconvex obstacles. Numer. Math. 129 (2015) 647–689. | DOI | MR | Zbl

[9] S.N. Chandler-Wilde and S. Langdon, A Galerkin boundary element method for high frequency scattering by convex polygons. SIAM J. Numer. Anal. 45 (2007) 610–640. | DOI | MR | Zbl

[10] S.N. Chandler-Wilde, E.A. Spence, A. Gibbs and V.P. Smyshlyaev, High-Frequency Bounds for the Helmholtz Equation Under Parabolic Trapping and Applications in Numerical Analysis. Tech. report. (2017). | arXiv

[11] G. Chavent, G. Papanicolaou, P. Sacks and W.W. Symes, Inverse Problems in Wave Propagation. Springer (2012). | Zbl

[12] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. SIAM (1978). | MR | Zbl

[13] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Springer (2012). | MR | Zbl

[14] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory. SIAM (2013). | DOI | MR | Zbl

[15] M. Dauge, Elliptic boundary value problems on corner domains – smoothness and asymptotics of solutions. Vol. 1341 of Lecture Notes in Mathematics. Springer, Berlin (1988). | MR | Zbl

[16] J. Diaz, Approches analytiques et numériques de problèmes de transmision en propagation d’ondes en régime transitoire. Application au couplage fluide-structure et aux méthodes de couches parfaitement adaptées. Ph.D. thesis, ENSTA ParisTech (2005).

[17] J. Douglas, J.E. Santos, D. Sheen and L.S. Bennethum, Frequency domain treatment of one-dimensional scalar waves. Math. Model. Methods Appl. Sci. 3 (1993) 171–194. | DOI | MR | Zbl

[18] K. Du, B. Li and W. Sun, A numerical study on the stability of a class of Helmholtz problems. J. Comput. Phys. 287 (2015) 46–59. | DOI | MR | Zbl

[19] B. Engquist and A. Majda, Absorbing boundary conditions for numerical simulation of waves. Proc. Natl. Acad. Sci. USA 74 (1977) 1765–1766. | DOI | MR | Zbl

[20] S. Esterhazy and J.M. Melenk, On stability of discretizations of the Helmholtz equation, Numerical Analysis of Multiscale Problems. Vol. 83 of Lecture Notes in Computational Science and Engineering. Springer, Heidelberg (2012) 285–324. | DOI | MR | Zbl

[21] X. Feng and H. Wu, hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number. Math. Comput. 80 (2011) 1997–2024. | DOI | MR | Zbl

[22] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman (1985). | MR | Zbl

[23] P. Grisvard, Edge behaviour of the solution of an elliptic problem. Math. Nachr. 182 (1987) 281–299. | DOI | MR | Zbl

[24] P. Grisvard, Singularité en elasticité. Arch. Rational Mech. Anal. 107 (1989) 157–180. | DOI | MR | Zbl

[25] I. Harari and T.J.R. Hughes, Finite element methods for the helmholz equation in an exterior domain: model problems. Comput. Methods Appl. Mech. Eng. 87 (1991) 59–96. | DOI | MR | Zbl

[26] D.P. Hewett, S. Langdon and J.M. Melenk, A high frequency hp boundary element method for scattering by convex polygons. SIAM J. Numer. Anal. 51 (2013) 629–653. | DOI | MR | Zbl

[27] F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number. Part I: the h-version of the fem. Comput. Math. Appl. 30 (1995) 9–37. | DOI | MR | Zbl

[28] F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number. Part II: the h-p version of the fem. SIAM J. Numer. Anal. 34 (1997) 315–358. | DOI | MR | Zbl

[29] J.M. Melenk, On Generalized Finite Element Methods. Ph.D. thesis, University of Maryland (1995). | MR

[30] J.M. Melenk and S. Sauter, Convergence analysis for finite element discretizations of the helmoltz equation with Dirichlet-to-Neumann boundary conditions. Math. Comput. 79 (2010) 1871–1914. | DOI | MR | Zbl

[31] J.M. Melenk and S. Sauter, Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation. SIAM J. Numer. Anal. 49 (2011) 1210–1243. | DOI | MR | Zbl

[32] S. Nicaise, Polygonal interface problems. Vol. 39 of Methoden und Verfahren der mathematischen Physik. Peter Lang GmbH, Europäischer Verlag der Wissenschaften, Frankfurt/M. (1993). | MR | Zbl

[33] S. Nicaise, Edge elements on anisotropic meshes and approximation of the Maxwell equations. SIAM J. Numer. Anal. 39 (2001) 784–816. | DOI | MR | Zbl

[34] J.A. Nitsche, Über ein Variationsprinzip zur Lösung Dirichlet-Problemen bei Verwendung von Teilräuman die keinen Randbedingungen unteworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9–15. | DOI | MR | Zbl

[35] S. Prudhomme, F. Pascal, J.T. Oden and A. Romkes, Review of a Priori Error Estimation for Discontinuous Galerkin Methods. Tech. report, University of Texas (2000).

[36] S.A. Sauter and C. Schwab, Boundary Element Methods. Springer (2011). | DOI | MR | Zbl

[37] A.H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comput. 28 (1974) 959–962. | DOI | MR | Zbl

[38] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. | DOI | MR | Zbl

[39] J.R. Shewchuk, Triangle: engineering a 2D quality mesh generator and delaunay triangulator, Applied Computational Geometry: Towards Geometric Engineering, edited by M.C. Lin and D. Manocha. Vol. 1148 of Lecture Notes in Computer Science. Springer-Verlag (1996) 203–222. | DOI

[40] I. Singer and E. Turkel, High-order finite difference methods for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 163 (1998) 343–358. | DOI | MR | Zbl