Preconditioning cubic collocation method for elliptic equations

Analysis on eigenvalues for preconditioning cubic spline collocation method of elliptic equations

In the work of solving a uniformly elliptic differential equations Au := − u+ a1ux + a2uy + a0u = f in the unit square with boundary conditions by the C1-cubic spline collocation method, one may need to investigate efficient preconditioning techniques. For this purpose, using the generalized field of values argument, we show the uniform bounds of the eigenvalues of the preconditioned matrix whe...

Preconditioning techniques in Chebyshev collocation method for elliptic equations

When one approximates elliptic equations by the spectral collocation method on the Chebyshev-Gauss-Lobatto (CGL) grid, the resulting coefficient matrix is dense and illconditioned. It is known that a good preconditioner, in the sense that the preconditioned system becomes well conditioned, can be constructed with finite difference on the CGL grid. However, there is a lack of an efficient solver...

A Collocation-i/ '-Galerkin Method for Some Elliptic Equations

A collocation-//"'-Galerkin method is defined for some elliptic boundary value problems on a rectangle. The method uses tensor products of discontinuous piecewise polynomial spaces and collocation based on Jacobi points with weight function >c2(l x)2. Optimal order of L2 rates of convergence is established for the approximation solution. A numerical example which confirms these results is prese...

23 . Indirect Method of Collocation : 2 nd Order Elliptic Equations

ADI Methods for Cubic Spline Collocation Discretizations of Elliptic PDE

This paper presents the formulation, the analysis and the implementation of Alternating Direction Implicit (ADI) methods for solving the linear system of algebraic equations that arise from the discretization of multidimensional linear elliptic Partial Diierential Equations (PDEs). The theoretical analysis is carried out for a board class of PDE problems. Numerical experiments connrm the theore...