Solution of the three-dimensional Helmholtz equation using ... - cerfacs

Motivations
Wave propagation modelling
Solution method components: Iterative method and multigrid
Solution strategy
Numerical experiments
Perspectives and conclusions
Classical GMRES / Restarted GMRES
Multigrid
Classical GMRES
The solution x m of Ax = b is sought in the Krylov subspace:
}
x 0 + K m(A, r 0 ) = x 0 + Span
{r 0 , Ar 0 , A 2 r 0 , ..., A m−1 r 0 = x 0 + Span {V m} ,
minimizing ||r m || 2 = ||b − Ax m || 2 , r 0 : initial residual.
Arnoldi’s relation: AV m = V m+1 ¯Hm .
Convergence reached in n (dim(A)) iterations at most.
O(nm 2 ) complexity.
Practicable GMRES variants
Restart GMRES: Alternative in memory ,CPU to the classical
GMRES.
Principle: restart GMRES(m), m small, up to the convergence.
FGMRES: Enables preconditioner to vary at each iteration.
12/27 Multigrid for geophysics applications