A fast direct solver for the integral equations of scattering theory on planar curves with corners

Journal of Computational Physics

Volumn 231, Issue 4, 2012, Pages 1879-1899

  Bremer, James ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Corner singularities; Elliptic boundary value problems; Integral equations; Scattering theory

Indexed keywords

BOUNDARY VALUE PROBLEMS; NUMERICAL METHODS;

ACOUSTIC SCATTERING; BOUNDARY-VALUE PROBLEM; CORNER SINGULARITY; DIRECT SOLVERS; DIRICHLET PROBLEM; ELLIPTIC BOUNDARY VALUE PROBLEM; NEUMANN PROBLEM; NUMERICAL SOLUTION; PLANAR CURVES; SCATTERING THEORY;

INTEGRAL EQUATIONS;

EID: 84855197523     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2011.11.015     Document Type: Article

References (29)

1. A. Anand, J. Ovall, C. Turc, Well conditioned boundary integral equations for two-dimensional sound-hard scattering in domains with corners, J. Integral Equ. Appl. (in press).
2. Andriulli F., Tabacco A., Vecchi G. Solving the EFIE at low frequencies with a conditioning that grows only logarithmically with the number of unknowns. IEEE Trans. Antenn. Propag. 2010, 58:1614-1624.
3. Björck A. Numerical Methods for Least Squares Problems 1996, SIAM, Philadephia.
4. Bremer J. On the Nyström discretization of integral operators on planar domains with corners. Appl. Comput. Harmon. Anal. 2012, 32:45-64.
5. Bremer J., Gimbutas Z., Rokhlin V. A nonlinear optimization procedure for generalized Gaussian quadratures. SIAM J. Sci. Comput. 2010, 32:1761-1788.
6. Bremer J., Rokhlin V. Efficient discretization of Laplace boundary integral equations on polygonal domains. J. Comput. Phys. 2010, 229:2507-2525.
7. Bremer J., Rokhlin V., Sammis I. Universal quadratures for boundary integral equations on two-dimensional domains with corners. J. Comput. Phys. 2010, 229:8259-8280.
8. Bruno O.P., Ovall J.S., Turc C. A high-order integral algorithm for highly singular PDE solutions in Lipschitz domains. Computing 2009, 84:149-181.
9. Coifman R., Meyer Y. Wavelets: Calderón-Zygmund and Multilinear Operators 1997, Cambridge University Press.
10. Colton D., Kress R. Integral Equation Methods in Scattering Theory 1983, John Wiley & Sons, New York.
11. Colton D., Kress R. Inverse Acoustic and Electromagnetic Scattering Theory 1998, Springer-Verlag, New York. Second ed.
12. Englund J., Helsing J. Stress computations on perforated polygonal domains. Engrg. Anal. Boundary Elem. 2003, 26:533-546.
13. A. Gillman, P. Young, P. Martinsson, A direct solver with O(n) complexity for integral equations on one-dimensional domains. preprint available at http://arxiv.org/abs/1105.5372.
14. Gu M., Eisenstat S. Efficient algorithms for computing a strong rank-revealing QR factorization. SIAM J. Sci. Comput. 1996, 17(4):848-869.
15. Helsing J. The effective conductivity of random checkerboards. J. Comput. Phys. 2011, 230:1171-1181.
16. Helsing J. A fast and stable solver for singular integral equations on piecewise smooth curves. SIAM J. Sci. Comput. 2011, 33:153-174.
17. Helsing J., Ojala R. Corner singularities for elliptic problems: integral equations, graded meshes, quadrature, and compressed inverse preconditioning. J. Comput. Phys. 2008, 227:8820-8840.
18. Helsing J., Ojala R. Elastostatic computations on aggregates of grains with sharp interfaces, corners, and triple-junctions. J. Solids Struct. 2009, 46. (pp. 4437-3350).
19. Kolm P., Jiang S., Rokhlin V. Quadruple and octuple layer potentials in two dimensions I: Analytical apparatus. Appl. Comput. Harmonic Anal. 2003, 14:47-74.
20. Kress R. A Nyström method for boundry integral equations in domains with corners. Numerische Mathematik 1990, 58:145-161.
21. Kress R. Integral Equations 1999, Springer-Verlag, New York.
22. Martinsson P., Rokhlin V. A fast direct solver for boundary integral equations in two dimensions. J. Comput. Phys. 2006, 205.
23. Martinsson P., Rokhlin V., Tygert M. On interpolation and integration in finite-dimensional spaces of bounded functions. Commun. Appl. Math. Comput. Sci. 2006, 1:133-142.
24. Panich O. On the question of the solvability of the exterior boundary-value problems for the wave equation and Maxwell's equations. Usp. Mat. Nauk 1965, 20A:221-226.
25. Rokhlin V. Solution of acoustic scattering problems by means of second kind integral equations. Wave Motion 1983, 5:257-272.
26. Rokhlin V. Rapid solution of integral equations of scattering theory in two dimensions. J. Comput. Phys. 1990, 86:414-439.
27. Rokhlin V. Diagonal forms of translation operators for the Helmholtz equation in three dimensions. Appl. Comput. Harmonic Anal. 1993, 1:82-93.
28. Rokhlin V., Greengard L. A fast algorithm for particle simulation. J. Comput. Phys. 1987, 73:325-348.
29. Woolfe F., Liberty E., Rokhlin V., Tygert M. A fast randomized algorithm for the approximation of matrices. Appl. Comput. Harmonics Anal. 2008, 25:355-366.