Ariel Almendral : On the convergence of multigrid methods for PDEs in mathematical finance
The traditional analysis of multigrid methods for second order elliptic and
parabolic problems is based on the assumption that the diffusion term of the equation is
uniformly elliptic. This condition is not satisfied by the Black and Scholes equation arising
in mathematical finance.
The purpose of this presentation is to show that the Multigrid V-cycle for the one-dimensional
Black and Scholes equation converges in a time dependent energy norm. We extend the standard multigrid
analysis for uniformly elliptic problems to this particular degenerate parabolic equation.
In particular, we prove that the Jacobi smoother, used as a point smoother, satisfies
suitable ``approximation properties''. Furthermore, the multigrid algorithm is shown to be an
order optimal method for this problem.