Abstract
In this paper, we extend the work of Daripa et al. [14–16,7] to a larger class of elliptic problems in a variety of domains. In particular, analysis-based fast algorithms to solve inhomogeneous elliptic equations of three different types in three different two-dimensional domains are derived. Dirichlet, Neumann and mixed boundary value problems are treated in all these cases. Three different domains considered are: (i) interior of a circle, (ii) exterior of a circle, and (iii) circular annulus. Three different types of elliptic problems considered are: (i) Poisson equation, (ii) Helmholtz equation (oscillatory case), and (iii) Helmholtz equation (monotone case). These algorithms are derived from an exact formula for the solution of a large class of elliptic equations (where the coefficients of the equation do not depend on the polar angle when written in polar coordinates) based on Fourier series expansion and a one-dimensional ordinary differential equation. The performance of these algorithms is illustrated for several of these problems. Numerical results are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Averbuch, M. Israeli and L. Vozovoi, A fast Poisson solver of arbitrary order accuracy in rectangular regions, SIAM J. Sci. Comput. 19 (1998) 933–952.
L. Badea and P. Daripa, On fast direct elliptic solver by modified Fourier method, Numer. Algorithms 15 (1997) 287–313.
L. Badea and P. Daripa, On a domain embedding method using the optimal distributed control and a fast algorithm, submitted.
B. Bialecki, A fast domain decomposition Poisson solver on a rectangle for Hermite bicubic orthogonal spline collocation, SIAM J. Numer. Anal. 30 (1993) 425–434.
L. Borges and P. Daripa, A parallel solver for singular integrals, in: Proc. of PDPTA'99–Internat. Conf. on Parallel and Distributed Processing Techniques and Applications, Vol. III, Las Vegas, Nevada, 28 June-1 July 1999, pp. 1495–1501.
L. Borges and P. Daripa, A parallel version of a fast algorithm for singular integral transforms, Numer. Algorithms 23 (2000) 71–96.
L. Borges and P. Daripa, A fast parallel algorithms for the Poisson equation on a disk, J. Comput. Phys. 169 (2001) 151–192.
E. Braverman, M. Israeli, A. Averbuch and L. Vozovoi, A fast 3D Poisson solver of arbitrary order accuracy, J. Comput. Phys. 144 (1998) 109–136.
T.F. Chan and D.C. Resasco, A domain-decomposed fast Poisson solver on a rectangle, SIAM J. Sci. Statist. Comput. 8 (1987) S14–S26.
T.F. Chan and F. Saied, A comparison of elliptic solvers for general two-dimensional regions, SIAM J. Sci. Statist. Comput. 6 (1985) 742–760.
S.C. Chang, Solution of elliptic PDEs by fast Poisson solvers using a local relaxation factor, J. Comput. Phys. 67 (1986) 91–123.
P. Daripa, On applications of a complex variable method in compressible flows, J. Comput. Phys. 88 (1990) 337–361.
P. Daripa, A fast algorithm to solve nonhomogeneous Cauchy-Riemann equations in the complex plane, SIAM J. Sci. Statist. Comput. 13 (1992) 1418–1432.
P. Daripa, A fast algorithm to solve the Beltrami equation with applications to quasiconformal mappings, J. Comput. Phys. 106 (1993) 355–365.
P. Daripa and D. Mashat, Singular integral transforms and fast numerical algorithms, Numer. Algorithms 18 (1998) 133–157.
P. Daripa and D. Mashat, An efficient numerical method for quasiconformal mappings of doubly connected domains, Numer. Algorithms 18 (1998) 159–178.
R. Dautray and J.L. Lions, eds., Analyse Mathématique et Calcul Numérique, I, Modèles Physiques (Masson, Paris, 1987).
M.D. Greenberg, Application of Green's Functions in Science and Engineering (Prentice-Hall, Englewood Cliffs, NJ, 1971).
L. Greengard and J.Y. Lee, A direct adaptive Poisson solver of arbitrary order accuracy, J. Comput. Phys. 125 (1996) 415–424.
L. Greengard and V. Rokhlin, A new version of the fast multipole method for the Laplace equation in three dimensions, Acta Numerica (1997) 229–269.
R.W. Hockney, A fast direct solution of Poisson's equation using Fourier analysis, J. Assoc. Comput. Math. 12 (1965) 95–113.
E. Houstis, R. Lynch and J. Rice, Evaluation of numerical methods for ellipitic partial differential equations, J. Comput. Phys. 27 (1978) 323–350.
A. McKenney, L. Greengard and A. Mayo, A fast Poisson solver for complex geometries, J. Comput. Phys. 118 (1995) 348–355.
D. O'Leary, and O. Widlund, Capacitance, matrix methods for the Helmholtz equation on general three-dimensional regions, Math. Comp. 33 (1979) 849–879.
A. Onana, S.V. Kwankam and E. Zoue, A fast Poisson solver, Period. Math. Hungar. 28 (1994) 89–101.
L.A. Pipes and L.R. Harvill, Applied Mathematics for Engineers and Physicists, 3rd ed. (McGraw-Hill, New York, 1970).
J. Rice, E. Houstis and R. Dyksen, A population of linear, second order, elliptic partial differential equations on rectangular domains. I, II, Math. Comp. 36 (1981) 475–484.
J. Shen, Efficient Galerkin methods III: Polar and cylindrical geometries, SIAM J. Sci. Comput. 18 (1997) 1583–1604.
A.S.L. Shieh, Fast Poisson solvers on general two-dimensional regions for the Dirichlet problem, Numer. Math. 31 (1978) 405–429.
G. Sköllermo, A Fourier method for the numerical solution of Poisson's equation, Math. Comp. 29 (1975) 697–711.
P.N. Swarztrauber, The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Rev. 19 (1977) 491–501.
P.N. Swarztrauber and R.A. Sweet, A direct method for the discrete Poisson equation on a disc, SIAM J. Numer. Anal. 10 (1973) 900–907.
C. Temperton, On the FACR(ℓ) algorithm for the discrete Poisson equation, J. Comput. Phys. 34 (1980) 315–329.
K. Yosida, Lectures on Differential and Integral Equations (Dover, New York, 1991).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Badea, L., Daripa, P. A Fast Algorithm for Two-Dimensional Elliptic Problems. Numerical Algorithms 30, 199–239 (2002). https://doi.org/10.1023/A:1020176803736
Issue Date:
DOI: https://doi.org/10.1023/A:1020176803736