Abstract
We formulate a fourth order modified nodal cubic spline collocation scheme for variable coefficient second order partial differential equations in the unit cube subject to nonzero Dirichlet boundary conditions. The approximate solution satisfies a perturbed partial differential equation at the interior nodes of a uniform \(N\times N\times N\) partition of the cube and the partial differential equation at the boundary nodes. In the special case of Poisson’s equation, the resulting linear system is solved by a matrix decomposition algorithm with fast Fourier transforms at a cost \(O(N^3\log N)\). For the general variable coefficient diffusion-dominated case, the system is solved using the preconditioned biconjugate gradient stabilized method.
Similar content being viewed by others
References
Berikelashvili, G., Gupta, M.M., Mirianashvili, M.: Convergence of fourth order compact difference scheme for three-dimensional convection-diffusion equations. SIAM J. Numer. Anal. 45, 443–455 (2007)
Bialecki, B., Fairweather, G., Karageorghis, A.: Matrix decomposition algorithm for modified spline collocation for Helmholtz problems. SIAM J. Sci. Comput. 24, 1733–1753 (2003)
Bialecki, B., Fairweather, G., Karageorghis, A.: Optimal superconvergent one step nodal cubic spline collocation methods. SIAM J. Sci. Comput. 27, 575–598 (2005)
Bialecki, B., Fairweather, G., Karageorghis, A.: Matrix decomposition algorithms for elliptic boundary value problems: a survey. Numer. Algorithms 56, 253–295 (2011)
Bialecki, B., Wang, Z.: Modified nodal cubic spline collocation for elliptic equations. Numer. Methods Partial Differ. Equ. 28, 1817–1839 (2012)
de Boor, C.: A Practical Guide to Splines, Revised Edition, Applied Mathematical Sciences, vol. 27. Springer-Verlag, New York (2001)
Bjontegaard, T., Maday, Y., Ronquist, E.M.: Fast tensor-product solvers: partially deformed three-dimensional domains. J. Sci. Comput. 39, 28–48 (2009)
Boisvert, R.F.: A fourth-order-accurate Fourier method for the Helmholtz equation in three dimensions. ACM Trans. Math. Softw. 13, 221–234 (1987)
Boisvert, R.F.: Algorithm 651: algorithm HFFT–high-order fast-direct solution of the Helmholtz equation. ACM Trans. Math. Softw. 13, 235–249 (1987)
Ge, L., Zhang, J.: Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients. J. Comput. Appl. Math. 143, 9–27 (2002)
Gupta, M.M., Zhang, J.: High accuracy multigrid solution of the 3D convection-diffusion equation. Appl. Math. Comput. 113, 249–274 (2000)
Hadjidimos, A., Houstis, E., Rice, J.R., Vavalis, E.: Iterative line cubic spline collocation methods for elliptic partial differential equations in several dimensions. SIAM J. Sci. Comput. 14, 715–734 (1993)
Hadjidimos, A., Houstis, E.N., Rice, J.R., Vavalis, E.: Analysis of iterative line spline collocation methods for elliptic partial differential equations. SIAM J. Matrix Anal. Appl. 21, 508–521 (1999)
Pozo, R., Remington, K.: Fast three-dimensional elliptic solvers on distributed network cluster. In: Joubert, G.R., et al. (eds.) Parallel Computing: Trends and Applications, pp. 201–208. Elsevier, Amsterdam (1994)
Samarski, A.A., Nikolaev, E.S.: Numerical Methods for Grid Equations, Volume II, Iterative Methods. Birkhauser (1989)
Tsompanopoulou, P., Vavalis, E.: ADI methods for cubic spline collocation discretizations of elliptic PDEs. SIAM J. Sci. Comput. 19, 341–363 (1998)
Tsompanopoulou, P., Vavalis, E.: Alternating direction implicit spline collocation methods for elliptic partial differential equations. Tech. report 95-3, Department of Mathematics, University of Crete, Heraklion, Greece (1995, in Greek)
van der Vorst, H.A.: BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644 (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bialecki, B., Karageorghis, A. Modified nodal cubic spline collocation for three-dimensional variable coefficient second order partial differential equations. Numer Algor 64, 349–383 (2013). https://doi.org/10.1007/s11075-012-9669-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-012-9669-4