Collection of abstracts 13th GAMM-Seminar Kiel on Numerical Treatment of Multi-Scale Problems January 24th to 26th, 1997.

Saturday, January 25th, 1997

Spectral Elements for Advection-Diffusion Equationss

We would like to describe a way to approximate boundary-value problems by spectral elements, which is particularly suited for advection-diffusion equations, where the second-order diffusive terms are largely dominated by the first-order transport terms.

First of all, we need a good solver for the case of the single domain (i.e. the usual square in 2-D or the cube in 3-D). We consider the technique suggested in [1], where the equation is collocated at a set of points obtained by shifting upwind (with respect to the direction of the flux individuated by the advective terms) the standard Legendre nodes. This procedure is interesting for two reasons: it provides {\it stabilized} approximated solutions (not affected by spurious oscillations) and allows the construction of very effective finite-differences preconditioners, giving a fast solution of the systems by an iterative solver, with a number of iterations not depending on the polynomial degree and the size of the advective terms.

The second issue concerns the implementation of the spectral elements. An iterative domain-decom\-po\-si\-tion algorithm reduces the global solution to a sequence of subproblems in each single domain. The technique we adopted is standard. In practice, we impose Dirichlet boundary conditions at the interfaces, and update the interface values according to the error of the normal derivatives of the multidomain solution across the interface sides. A speed up of the convergence is obtained by a suitable tridiagonal preconditioner applied to the values at the nodes of each interface side. This is the crucial part of the implementation. We present a preconditioning matrix capable to handle advection dominated equations, resulting in a convergence behavior which seems to be independent, as pointed out by numerical experiments, of both the polynomial degree and the size of the advective terms. To this purpose a special set of grid points has to be taken on each interface side. These nodes are located upwind with respect to the direction of the tangential component of the flux along the interface.

The method has been successfully experimented and in [2] the whole procedure is explained and supplied with a lot of test cases, including the solution of incompressible Navier-Stokes equations.

[1] Funaro D.: A New Scheme for the Approximation of Advection-Diffusion Equations by Collocation, SIAM J. Numer. Anal., Vol.30, n.6(1993), pp.1664-1676.
[2] Funaro D.: Spectral Elements for Transport Dominated Equations, to appear.