A direct solver for variable coefficient elliptic PDEs discretized via a composite spectral collocation method

Journal of Computational Physics

Volumn 242, Issue , 2013, Pages 460-479

  Martinsson, P G ^a  

^a UNIVERSITY OF COLORADO   (United States)

Author keywords

Direct solver; Fast direct solver; Helmholtz equation; Spectral method

Indexed keywords

DEGREES OF FREEDOM (MECHANICS); ITERATIVE METHODS; NUMERICAL METHODS;

DIRECT SOLVERS; ELLIPTIC PDES; FAST DIRECT SOLV; HELMHOLTZ'S EQUATIONS; PRE-COMPUTATION; SPECTRAL COLLOCATION METHOD; SPECTRAL METHODS; VARIABLE COEFFICIENT ELLIPTIC PROBLEMS; VARIABLE COEFFICIENTS;

PROBLEM SOLVING;

EID: 84875344012     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2013.02.019     Document Type: Article

References (38)

1. Ainsworth M., Wajid H.A. Dispersive and dissipative behavior of the spectral element method. SIAM Journal on Numerical Analysis 2009, 47(5):3910-3937.
2. Ballard G., Demmel J., Holtz O., Schwartz O. Minimizing communication in numerical linear algebra. SIAM Journal on Matrix Analysis and Applications 2011, 32(3):866-901.
3. Yu Chen, A fast, direct algorithm for the Lippmann-Schwinger integral equation in two dimensions, Adv. Comput. Math. 16(2-3) (2002) 175-190, Modeling and computation in optics and electromagnetics.
4. Yu Chen, Total wave based fast direct solver for volume scattering problems, arXiv:1302.2101.
5. Duan Ran, Rokhlin Vladimir High-order quadratures for the solution of scattering problems in two dimensions. J. Comput. Physics 2009, 228(6):2152-2174.
6. I.S. Duff, A.M. Erisman, and J.K. Reid, Direct methods for sparse matrices, Oxford, 1989.
7. Ethridge F., Greengard L. A new fast-multipole accelerated poisson solver in two dimensions. SIAM Journal on Scientific Computing 2001, 23(3):741-760.
8. George A. Nested dissection of a regular finite element mesh. SIAM J. on Numerical Analysis 1973, 10:345-363.
9. A. Gillman, Fast direct solvers for elliptic partial differential equations, Ph.D. thesis, Applied Mathematics, University of Colorado at Boulder, 2011.
10. Gillman Adrianna, Young Patrick, Martinsson Per-Gunnar A direct solver o(n) complexity for integral equations on one-dimensional domains. Frontiers of Mathematics in China 2012, 7:217-247, 10.1007/s11464-012-0188
11. Graham I.G., Sloan I.H. Fully discrete spectral boundary integral methods for helmholtz problems on smooth closed surfaces in {minus 45 degree rule}mathbbr̂3. Numerische Mathematik 2002, 92(2):289-323.
12. Grasedyck Lars, Hackbusch Wolfgang Construction and arithmetics of H-matrices. Computing 2003, 70(4):295-334.
13. Greengard Leslie, Gueyffier Denis, Martinsson Per-Gunnar, Rokhlin Vladimir Fast direct solvers for integral equations in complex three-dimensional domains. Acta Numer. 2009, 18:243-275.
14. Haldenwang P., Labrosse G., Abboudi S., Deville M. Chebyshev 3-d spectral and 2-d pseudospectral solvers for the helmholtz equation. Journal of Computational Physics 1984, 55(1):115-128.
15. Haney M., Snieder R., Ampuero J.-P., Hofmann R. Spectral element modeling of fault-plane reflections arising from fluid pressure distributions. Geophys. J. Int. 2007, 170(2):933-951.
16. Hesthaven J.S., Dinesen P.G., Lynov J.P. Spectral collocation time-domain modeling of diffractive optical elements. Journal of Computational Physics 1999, 155(2):287-306.
17. Ho K.L., Greengard L. A fast direct solver for structured linear systems by recursive skeletonization. SIAM Journal on Scientific Computing 2012, 34(5):2507-2532.
18. Hoffman A.J., Martin M.S., Rose D.J. Complexity bounds for regular finite difference and finite element grids. SIAM J. Numer. Anal. 1973, 10:364-369.
19. Kopriva D.A. A staggered-grid multidomain spectral method for the compressible navierstokes equations. Journal of Computational Physics 1998, 143(1):125-158.
20. Kwan Y.Y., Shen J. An efficient direct parallel spectral-element solver for separable elliptic problems. Journal of Computational Physics 2007, 225(2):1721-1735.
21. Sabine Le Borne, Lars Grasedyck, and Ronald Kriemann, Domain-decomposition based H-LU preconditioners, Domain decomposition methods in science and engineering XVI, Lect. Notes Comput. Sci. Eng., vol. 55, Springer, Berlin, 2007, pp. 667-674. MR 2334161.
22. P.G. Martinsson, A high-order accurate discretization scheme for variable coefficient elliptic pdes in the plane with smooth solutions, Jan. 2011, arXiv:1101.3383.
23. Martinsson P.G., Rokhlin V. A fast direct solver for boundary integral equations in two dimensions. J. Comp. Phys. 2005, 205(1):1-23.
24. Mehdizadeh O.Z., Paraschivoiu M. Investigation of a two-dimensional spectral element method for helmholtz equation. Journal of Computational Physics 2003, 189(1):111-129.
25. Melenk J.M., Sauter S. Convergence analysis for finite element discretizations of the helmholtz equation with dirichlet-to-neumann boundary conditions. Math. Comp 2010, 79(272):1871-1914.
26. Patera A.T. A spectral element method for fluid dynamics: Laminar flow in a channel expansion. Journal of Computational Physics 1984, 54(3):468-488.
27. Pfeiffer H.P., Kidder L.E., Scheel M.A., Teukolsky S.A. A multidomain spectral method for solving elliptic equations. Computer physics communications 2003, 152(3):253-273.
28. J. Poulson, B. Marker, J.R. Hammond, N.A. Romero, and R. van de Geijn, Elemental: A new framework for distributed memory dense matrix computations, ACM Trans. Math. Software, Note: to appear (2010).
29. Pozrikides C. Finite and spectral element methods 2005, Taylor & Francis.
30. Schmitz P.G., Ying L. A fast direct solver for elliptic problems on general meshes in 2d. Journal of Computational Physics 2012, 231(4):1314-1338.
31. Shen J., Wang L.L. Spectral approximation of the helmholtz equation with high wave numbers. SIAM journal on numerical analysis 2005, 43(2):623-644.
32. Shen J., Wang L.L. Analysis of a spectral-galerkin approximation to the helmholtz equation in exterior domains. SIAM Journal on Numerical Analysis 2007, 45(5):1954-1978.
33. Starr Page, Rokhlin Vladimir, II On the numerical solution of two-point boundary value problems. Comm. Pure Appl. Math. 1994, 47(8):1117-1159.
34. Trefethen L.N. Spectral methods in matlab 2000, SIAM, Philadelphia.
35. Jianlin Xia, Shivkumar Chandrasekaran, Ming Gu, and Xiaoye S. Li, Superfast multifrontal method for large structured linear systems of equations, SIAM J. Matrix Anal. Appl. 31(3) (2009) 1382-1411. MR 2587783 (2011c:65072).
36. Xia Jianlin, Chandrasekaran Shivkumar, Gu Ming, Li Xiaoye S. Fast algorithms for hierarchically semiseparable matrices. Numerical Linear Algebra with Applications 2010, 17(6):953-976.
37. Yang B., Hesthaven J.S. Multidomain pseudospectral computation of maxwell's equations in 3-d general curvilinear coordinates. Applied Numerical Mathematics 2000, 33(14):281-289.
38. Zhu C., Qin G., Zhang J. Implicit chebyshev spectral element method for acoustics wave equations. Finite Elements in Analysis and Design 2011, 47(2):184-194.