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- Bernard Bialecki, Graeme Fairweather, Andreas Karageorghis
- Numerical Algorithms
- 2010

We provide an overview of matrix decomposition algorithms (MDAs) for the solution of systems of linear equations arising when various discretization techniques are applied in the numerical solutionâ€¦ (More)

- Zhou-Ming Lou, Bernard Bialecki, Graeme Fairweather
- Numerische Mathematik
- 1998

where here and throughout the paper, âˆ† denotes the Laplacian, Î© = (0, 1)Ã— (0, 1), âˆ‚Î© is the boundary of Î©, andâˆ‚/âˆ‚n is the outer normal derivative on âˆ‚Î©. In linear elasticity,u represents the Airyâ€¦ (More)

- Bernard Bialecki, Graeme Fairweather, Andreas Karageorghis
- SIAM J. Scientific Computing
- 2002

We consider the solution of various boundary value problems for the Helmholtz equation in the unit square using a nodal cubic spline collocation method and modifications of it which produce optimalâ€¦ (More)

- Bernard Bialecki, Graeme Fairweather, Andreas Karageorghis
- SIAM J. Scientific Computing
- 2005

We formulate new optimal (fourth) order one step nodal cubic spline collocation methods for the solution of various elliptic boundary value problems in the unit square. These methods are constructedâ€¦ (More)

- Abeer Ali Abushama, Bernard Bialecki
- Numerical Algorithms
- 2006

We formulate a modified nodal cubic spline collocation scheme for the solution of the biharmonic Dirichlet problem on the unit square. We prove existence and uniqueness of a solution of the schemeâ€¦ (More)

- Bernard Bialecki, Andreas Karageorghis
- SIAM J. Scientific Computing
- 2000

A Legendre spectral Galerkin method is presented for the solution of the biharmonic Dirichlet problem on a square. The solution and its Laplacian are approximated using the set of basis functionsâ€¦ (More)

- Bernard Bialecki, Graeme Fairweather
- SIAM J. Scientific Computing
- 1995

Fast direct methods are presented for the solution of linear systems arising in highorder, tensor-product orthogonal spline collocation applied to separable, second order, linear, elliptic partial diâ€¦ (More)

- Bernard Bialecki
- SIAM J. Scientific Computing
- 1994

- Bernard Bialecki, Andreas Karageorghis
- Numerical Algorithms
- 2004

A spectral collocation method with collocation at the Legendre Gauss points is discussed for solving the Helmholtz equation âˆ’Î”u+Îº(x,y)u=f(x,y) on a rectangle with the solution u subject toâ€¦ (More)

- Bernard Bialecki, Ryan I. Fernandes
- SIAM J. Scientific Computing
- 2006