Wavelets preconditioning of a two phases
flow problem
MICHEL BELLIARD, PATRICK GIROUD, HASSIB SELMI
The subject of the research is the development of a
wavelets preconditioner to solve the energy balance
in the background of the finite element method with
embedded meshes.
For each time step, we have a linear system to solve
in specific enthalpy variable. At the n+1 iteration
number, this write
LnHn+1=fn
with
- Ln=(M)/(Deltat)+An,
- M is the mass matrix,
- An is the matrix of the discretization of the differential
operator in the finite element framework,
- Hn+1 is the vector representing the enthalpy,
- and fn the source vector.
The domain of resolution is a rectangle and we impose
a Dirichlet condition on a part of the boundary.
We begin with preconditioning the system with Schauder
wavelets which take care of
the Dirichlet boundary conditions.
We have first inverting a coarse matrix with the Choleski
direct method and secondly, for the details, we have two strategy:
- preconditioning with a diagonal 2-2j,
- or with the inverse of the diagonal of Ln in the
wavelet basis.