The accurate solution of poisson's equation by expansion in chebyshev polynomials

Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

“…However, this cost can be significantly reduced by using a discrete version of "separation of variables" -the matrix decomposition/diagonalization method [15,8,24]. To this end, we consider the following generalized eigenvalue problems:…”

“…In particular, we shall extend most of the solution techniques in [24] for usual PDEs to fractional PDEs. More precisely, for separable fractional PDEs, we shall apply the matrix diagonalization methods (i.e., discrete separation of variables) [15,8,24] to reduce a (discrete) multi-dimensional problem to a sequence of (discrete ) one-dimensional problems or to a diagonal system with a total cost of just a few N d+1 flops (d is the space dimension); for non-separable fractional PDEs, we shall apply a preconditioned BICGSTAB iterative method using (i) a related fractional separable problem with constant-coefficients as preconditioned, and (ii) a fast matrix-free algorithm for the matrix-vector multiplication, so that the total cost is still O(N d+1 ). Thus, the cost of a spectral method for fractional PDEs is essentially the same as that for usual PDEs.…”

Improved condition number for spectral methods

“…For iterative methods an effective preconditioning is necessary (see Phillips et al [17] or Heinrichs [8], [9]). A direct solver based on an ADI-factorization technique is proposed by Haidvogel and Zang [7], where the tau method has been used for discretization.…”

A finite difference method for 3D incompressible flows in cylindrical coordinates

“…Because of the geometry and of the used grid, the N discrete independent systems, one for each mode, arising from the Fourier expansion are in tensor form and are therefore solved by means of a partial diagonalization in the axial direction [20]. For the scalar Poisson equation and the m = 0 modes of the momentum equation, this technique leads to the resolution of a set of independent tridiagonal systems.…”