
A Monotone Multigrid Method Based on Higher Order BSplines
Angela Kunoth (Univ. Bonn)
For the efficient numerical solution of elliptic variational
inequalities on closed convex sets, monotone multigrid methods
based on piecewise linear finite elements have been investigated
over the past decades. In these methods, the appropriate approximation
of the obstacle on coarser grids is essential, which is achieved by
considering point values in the piecewise linear case but which does
not give rise to admissible obstacles in the case of higher order
basis functions.
On the other hand, there are a number of problems which
profit from higher order approximations, among these the
problem of prizing American options, formulated as a parabolic
free boundary value problem.
Here a monotone multigrid method will be presented for discretizations
in terms of Bsplines of arbitrary order to solve elliptic
variational inequalities. In order to maintain monotonicity
(upper bound) and quasioptimality (lower bound) of the coarse grid
corrections for the equivalent linear complementary problem, an optimized
coarse grid correction algorithm based on Bspline
evaluation coefficients will be introduced. This algorithm is of optimal
complexity of the degrees of freedom of the coarse grid, yielding
optimal multigrid complexity for the resulting monotone multigrid method.

