Spectral Collocation Scheme Research Articles

The wave equation in generalized coordinates

This work introduces a spectral collocation scheme for the viscoelastic wave equation transformed from Cartesian to generalized coordinates. Both the spatial derivatives of field variables and the metrics of the transformation are calculated by the Chebychev pseudospectral method. The technique requires a special treatment of the boundary conditions, which is based on 1-D characteristics normal to the boundaries. The numerical solution of Lamb’s problem requires two 1-D stretching transformations for each Cartesian direction. The results show excellent agreement between the elastic numerical and analytical solutions, demonstrating the effectiveness of the differential operator and boundary treatment. Another example, requiring 1-D transformations, tests the propagation of a Rayleigh wave around a corner of the numerical mesh. Two‐dimensional transformations adapt the grid to topographic features: a syncline, and an anticlinal structure formed with fine layers.

Stabilization of spectral methods by finite element bubble functions

We show that a standard spectral collocation scheme can be stabilized by adding extra trial/test functions with local support (bubbles). Applications are given to a scalar advection—diffusion problem and to the Stokes problem.

Boundary and interface conditions within a finite element preconditioner for spectral methods

The performances of a finite element preconditioner in the iterative solution of spectral collocation schemes for elliptic boundary value problems is investigated. It is shown how to make the preconditioner cheap by ADI iterations and how to take advantage of the finite element properties in enforcing Neumann and interface conditions in the spectral schemes.

A multidomain spectral collocation method for the Stokes problem

We propose a multidomain spectral collocation scheme for the approximation of the two-dimensional Stokes problem. We show that the discrete velocity vector field is exactly divergence-free and we prove error estimates both for the velocity and the pressure.