Analysis of a spectral-galerkin approximation to the helmholtz equation in exterior domains

SIAM Journal on Numerical Analysis

Volumn 45, Issue 5, 2007, Pages 1954-1978

  Shen, Jie ^a   Wang, Li Lian ^b  

^a PURDUE UNIVERSITY   (United States)

^b NANYANG TECHNOLOGICAL UNIVERSITY   (Singapore)

Author keywords

Error analysis; Helmholtz equation; Spectral Galerkin; Unbounded domain; Wave scattering

Indexed keywords

A PRIORI ESTIMATES; ASYMPTOTIC BEHAVIORS; BOUNDED DOMAINS; DIRICHLET; DISCRETE PROBLEMS; ERROR BOUNDS; EXTERIOR DOMAINS; GALERKIN APPROXIMATIONS; HANKEL FUNCTIONS; NEUMANN; NUMERICAL EXAMPLES; SPECTRAL-GALERKIN; UNBOUNDED DOMAIN; UNBOUNDED DOMAINS; WAVE NUMBERS; WAVE SCATTERING;

ERROR ANALYSIS; ERRORS; GALERKIN METHODS; HELMHOLTZ EQUATION; MATHEMATICAL OPERATORS; NAVIER STOKES EQUATIONS; NONLINEAR EQUATIONS; NUMERICAL ANALYSIS;

ASYMPTOTIC ANALYSIS;

EID: 55549088559     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/060665737     Document Type: Article

References (31)

1. M. ABRAMOWITZ AND I. A. STEGUN, EDS., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, 1984. Reprint of the 1972 edition, Selected Government Publications.
2. M. AINSWORTH, Discrete dispersion relation for hp-version finite element approximation at high wave number, SIAM J. Numer. Anal., 42 (2004), pp. 553-575.
3. M. AINSWORTH, Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods, J. Comput. Phys., 198 (2004), pp. 106-130.
4. I. M. BABUŠKA AND S. A. SAUTER, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM J. Numer. Anal., 34 (1997), pp. 2392-2423.
5. O. P. BRUNO AND F. REITICH, Numerical solution of diffraction problems: A method of variation of boundaries, J. Opt. Soc. Amer. A, 10 (1993), pp. 1168-1175.
6. O. P. BRUNO AND F. REITICH, Numerical solution of diffraction problems: A method of variation of boundaries. II. Finitely conducting gratings, Padé approximants, and singularities, J. Opt. Soc. Amer. A, 10 (1993), pp. 2307-2316.
7. C. CANUTO, M. Y. HUSSAINIv, A. QUARTERONI, AND T. A. ZANG, Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1987.
8. 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), pp. 610-640.
9. S. N. CHANDLER-WILDE AND P. MONK, Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces, SIAM J. Math. Anal., 37 (2005), pp. 598-618.
10. 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.
11. L. DEMKOWICZ AND F. IHLENBURG, Analysis of a coupled finite-infinite element method for exterior Helmholtz problems, Numer. Math., 88 (2001), pp. 43-73.
12. J. DOUGLAS, J. E. SANTOS, D. SHEEN, AND L. S. BENNETHUM, Frequency domain treatment of one-dimensional scalar waves, Math. Models Methods Appl. Sci., 3 (1993), pp. 171-194.
13. M. J. GROTE AND J. B. KELLER, On nonreflecting boundary conditions, J. Comput. Phys., 122 (1995), pp. 231-243.
14. B. GUO AND L. WANG, Jacobi approximations in non-uniforrnly Jacobi-weighted Sobolev spaces, J. Approx. Theory, 128 (2004), pp. 1-41.
15. I. HARARI AND T. J. R. HUGHES, Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains, Comput. Methods Appl. Mech. Engrg., 97 (1992), pp. 103-124.
16. F. IHLENBURG, Finite Element Analysis of Acoustic Scattering, Appl. Math. Sci. 132, Springer- Verlag, New York, 1998.
17. F. IHLENBURG AND I. BABUŠKA, Finite element solution of the Helmholtz equation with high wave number, part: The h-version of FEM, Comput. Math. Appl., 30 (1995), pp. 9-37.
18. 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), pp. 315-358.
19. S. LANGDON AND S. N. CHANDLER-WILDE, A wavenumber independent boundary element method for an acoustic scattering problem, SIAM J. Numer. Anal., 43 (2006), pp. 2450-2477.
20. J.-L. LIONS AND E. MAGENES, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York, 1972 (translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181).
21. J. M. MELENK, On Generalized Finite Element Methods, Ph.D. thesis, The University of Maryland, College Park, MD, 1995.
22. P. MONK, Finite Element Methods for Maxwell's Equations, Numer. Math. Sci. Comput., Oxford University Press, New York, 2003.
23. J.-C. NÉDÉLEC, Acoustic and Electromagnetic Equations, Appl. Math. Sci. vol. 144, Springer- Verlag, New York, 2001.
24. D. P. NICHOLLS AND J. SHEN, A stable, high-order method for two-dimensional boundedobstacle scattering, SIAM J. Sci. Comput., 28 (2006), pp. 1398-1419.
25. D. P. NICHOLLS AND F. REITICH, Stability of high-order perturbative methods for the computation of Dirichlet-Neumann operators, J. Comput. Phys., 170 (2001), pp. 276-298.
26. D. P. NICHOLLS AND F. REITICH, Shape deformations in rough surface scattering: Improved algorithms, J. Opt. Soc. Amer. A, 21 (2004), pp. 606-621.
27. L. RAYLEIGH, On the dynamical theory of gratings, Proc. R. Soc. Lond. Ser. A, 79 (1907), pp. 399-416.
28. J. SHEN, Efficient spectral-Galerkin method I. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), pp. 1489-1505.
29. J. SHEN, Efficient spectral-Galerkin methods III: Polar and cylindrical geometries, SIAM J. Sci. Comput., 18 (1997), pp. 1583-1604.
30. J. SHEN AND L.-L. WANG, Spectral approximation of the Helmholtz equation with high wave numbers, SIAM J. Numer. Anal., 43 (2005), pp. 623-644.
31. G. N. WATSON, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, UK, 1966.