Fast direct solution of the Helmholtz equation with a perfectly matched layer or an absorbing boundary condition

International Journal for Numerical Methods in Engineering

Volumn 57, Issue 14, 2003, Pages 2007-2025

  Heikkola, Erkki ^a   Rossi, Tuomo ^a   Toivanen, Jari ^a  

^a UNIVERSITY OF JYVÄSKYLÄ   (Finland)

Author keywords

Absorbing boundary conditions; Fast direct solver; Finite element discretization; Helmholtz equation; Perfectly matched layer

Indexed keywords

BOUNDARY CONDITIONS; FINITE ELEMENT METHOD; LINEAR EQUATIONS;

FINITE-ELEMENT DISCRETIZATION;

WAVE EFFECTS;

MATHEMATICAL METHOD;

EID: 0042424860     PISSN: 00295981     EISSN: None     Source Type: Journal    
DOI: 10.1002/nme.752     Document Type: Article

References (39)

1. Bayliss A, Gunzburger M, Turkel E. Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM Journal on Applied Mathematics, 1982; 42:430-451.
2. Engquist B, Majda A. Absorbing boundary conditions for the numerical simulation of waves. Mathematics of Computation 1977; 31:629-651.
3. Givoli D. Nonreflecting boundary conditions. Journal of Computational Physics 1991; 94:1-29.
4. Keller JB, Givoli D. Exact nonreflecting boundary conditions. Journal of Computational Physics 1989; 82: 172-192.
5. MacCamy RC, Marin SP. A finite element method for exterior interface problems. International Journal of Mathematics and Mathematical Sciences 1980; 3:311-350.
6. Gerdes K. A summary of infinite element formulations for exterior Helmholtz problems. Computer Methods in Applied Mechanics and Engineering 1998; 164:95-105.
7. Shirron JJ, Babuška I. A comparison of approximate boundary conditions and infinite element methods for exterior Helmholtz problems. Computer Methods in Applied Mechanics and Engineering 1998; 164:121-139.
8. Bérenger JP. A perfectly matched layer for the absorption of electromagnetic waves. Journal of Computational Physics 1994; 114:185-200.
9. Bérenger JP. Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. Journal of Computational Physics 1996; 127:363-379.
10. Hu FQ. On absorbing boundary conditions for linearized Euler equations by a perfectly matched layer. Journal of Computational Physics 1996; 129:201-219.
11. Qi Q, Geers TL. Evaluation of the perfectly matched layer for computational acoustics. Journal of Computational Physics 1998; 139:166-183.
12. Turkel E, Yefet A. Absorbing PML boundary layers for wave-like equations. Applied Numerical Mathematics 1998; 27:533-557.
13. Harari I, Slavutin M, Turkel E. Analytical and numerical studies of a finite element PML for the Helmholtz equation. Journal of Computational Acoustics 2000; 8:121-137.
14. Lassas M, Somersalo E. Analysis of the PML equations in general convex geometry. Proceedings of the Royal Society, Edinburgh Section A 2001; 131:1183-1207.
15. Tsynkov SV, Turkel E. A cartesian perfectly matched layer for the Helmholtz equation. In Absorbing Boundaries and Layers, Domain Decomposition Methods, Applications to Large Scale Computers. Tourrette L, Halpern L (eds), Nova Science Publishers, Inc.: New York, 2001; 279-309.
16. Collino F, Monk P. Optimizing the perfectly matched layer. Computer Methods in Applied Mechanics and Engineering 1998; 164:157-171.
17. Collino F, Monk P. The perfectly matched layer in curvilinear coordinates. SIAM Journal on Scientific Computing 1998; 19:2061-2090.
18. Bamberger A, Joly P, Roberts JE. Second-order absorbing boundary conditions for the wave equation: a solution for the corner problem. SIAM Journal on Numerical Analysis 1990; 27:323-352.
19. Ihlenburg F. Finite Element Analysis of Acoustic Scattering. Springer: New York, 1998.
20. Babuška I, Sauter SA. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?. SIAM Review 2000; 42:451-484.
21. Gerdes K, Ihlenburg F. On the pollution effect in FE solutions of the 3D-Helmholtz equation. Computer Methods in Applied Mechanics and Engineering 1999; 170:155-172.
22. Turkel E. Numerical difficulties solving time harmonic systems. In Multiscale Computational Methods in Chemistry and Physics. Brandt A, Bernhole J, Binder K (eds), IOS Press/Ohmsha: Amsterdam, 2001; 319-337.
23. Swarztrauber PN. The methods of cyclic reduction. Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle. SIAM Review 1977; 19:490-501.
24. Rossi T, Toivanen J. A nonstandard cyclic reduction method, its variants and stability. SIAM Journal on Matrix Analysis and Applications 1999; 20:628-645.
25. Rossi T, Toivanen J. A parallel fast direct solver for block tridiagonal systems with separable matrices of arbitrary dimension. SIAM Journal on Scientific Computing 1999; 20:1778-1796.
26. Vassilevski PS. Fast algorithm for solving a linear algebraic problem with separable variables. Comptes Rendus de L'Academie Bulgare des Science 1984; 37:305-308.
27. Buzbee BL, Dorr FW, George JA, Golub GH. The direct solution of the discrete Poisson equation on irregular regions. SIAM Journal on Numerical Analysis 1971; 8:722-736.
28. Ernst OG. A finite-element capacitance matrix method for exterior Helmholtz problems. Numerische Mathematik 1996; 75:175-204.
29. Heikkola E, Kuznetsov YA, Lipnikov KN. Fictitious domain methods for the numerical solution of three-dimensional acoustic scattering problems. Journal of Computational Acoustics 1999; 7:161-183.
30. Heikkola E, Kuznetsov YA, Neittaanmäki P, Toivanen J. Fictitious domain methods for the numerical solution of two-dimensional scattering problems. Journal of Computational Physics 1998; 145:89-109.
31. Heikkola E, Rossi T, Toivanen J. A parallel fictitious domain method for the three-dimensional Helmholtz equation. SIAM Journal on Scientific Computing 2003; 24:1567-1588.
32. Tezaur R, Macedo A, Farhat C. Iterative solution of large-scale acoustic scattering problems with multiple right hand-sides by a domain decomposition method with Lagrange multipliers. International Journal for Numerical Methods in Engineering 2001; 51:1175-1193.
33. Petropoulos PG. On the termination of the perfectly matched layer with local absorbing boundary conditions. Journal of Computational Physics 1998; 143:665-673.
34. Kuznetsov YA. Computational methods in subspaces. In Computational Processes and Systems, Marchuk GI (ed.), No. 2. Nauka: Moscow, 1985; 265-350.
35. Banegas A. Fast Poisson solvers for problems with sparsity. Mathematics of Computation 1978; 32:441-446.
36. Kuznetsov YA, Matsokin AM. Partial solution of systems of linear algebraic equations. In Numerical Methods in Linear Algebra (Proc. Third Sem. Methods of Numer. Appl. Math., Novosibirsk, 1978), Marchuk GI, Lions JL (eds), Akad. Nauk SSSR Sibirsk. Otdel. Vychisl. Tsentr: Novosibirsk, 1978; 62-89 (In Russian).
37. Babuška I, Ihlenburg F, Paik ET, Sauter SA. A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Computer Methods in Applied Mechanics and Engineering 1995; 128:325-359.
38. Crawford CR. Reduction of a band-symmetric generalized eigenvalue problem. Communication of the Association for Computing Machinery 1973; 16:41-44.
39. Wilkinson J. The Algebraic Eigenvalue Problem. Oxford Science Publications: Oxford, 1965.