Modified nodal cubic spline collocation for three-dimensional variable coefficient second order partial differential equations

• Published in: Numerical algorithms Vol. 64; no. 2; pp. 349 - 383
• Main Authors: Bialecki, Bernard, Karageorghis, Andreas
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.10.2013; Springer Nature B.V
• Subjects: Algebra; Algorithms; Boundary conditions; Coefficients; Collocation; Computer Science; Cubes; Diffusion rate; Dirichlet problem; Fast Fourier transformations; Mathematical analysis; Mathematical models; Nodes; Numeric Computing; Numerical Analysis; Original Paper; Partial differential equations; Poisson equation; Splines; Theory of Computation; Nodal spline collocation; Fast Fourier transforms; 65N35; 65N22; Matrix decomposition algorithm
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-012-9669-4

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 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 . For the general variable coefficient diffusion-dominated case, the system is solved using the preconditioned biconjugate gradient stabilized method.