19 Marzo, 2012 14:30 oclock

A Cartesian ghost cell multigrid solver for elliptic equations with discontinuous coefficients in arbitrary domains.

Armando Coco, Dipartimento di Matematica ed Informatica Università di Catania

presso presso Aula Seminari F. Saleri VI Piano MOX- Dipartimento di Matematica, Politecnico di Milano

Elliptic equations with jumping coefficients across a possibly moving) one-codimensional interface arise in several applications, such as the steady-state diffusion problem in two
materials with different diffusion coefficients, the incompressible Navier-Stokes equation for fluids with different density or viscosity,
porous-media equations, electrostatic problems, charge transport in semiconductors, crystal growth, and many others.
Cartesian grid methods are an effective alternative to boundary-fitted grid methods, since the complex task of the grid generation process at each time step is avoided.
Furthermore, iterative schemes may be preferred because a good initial guess (the solution at the previous time step) is provided at each time step.
In this talk we present a multigrid method to solve the elliptic equation with discontinuous coefficient in an arbitrary domain, identified by a level set function. The method is based on ghost-cell technique and finite difference discretization on a regular Cartesian grid. The structure of ghost points is complex and elimination of discrete boundary conditions from the system is hard to perform. In addition, a simple Gauss-Seidel scheme for the whole system does not converge. Therefore, in order to provide a good smoother for the multigrid approach, we relax the whole problem introducing a fictitious time and looking for the steady-state solution. The convergence factor of the multigrid is close to the one predicted by the Local Fourier Analysis for inner relaxations, and it does not
depend on the geometry nor on the jump in the coefficient. Preliminary study on adaptive grids and on application to lava flow are presented.