A projected multigrid method for the solution of non-linear finite element problems on adaptively refined grids

This thesis describes the formulation and application of an adaptive multigrid method for the efficient solution of nonlinear elliptic and parabolic partial differential equations and systems. A continuous Galerkin finite-element method is combined with locally adaptive mesh refinement and an optimal multigrid solver to achieve this efficiency. The novel contribution of this work lies in the manner in which these two techniques are combined. In particular the multigrid solver provides a natural and simple method of handling grid points that are not fully connected, so called hanging nodes. This allows for a straightforward adaptive gridding scheme that does not need to take any special measures to repair these hanging nodes for a standard element-by-element implementation of the finite element assembly process. Specifically, on each element, only the usual finite element basis functions are required, even in the vicinity of hanging nodes. Furthermore the standard multigrid full approximation scheme (FAS) may be applied with only minor modifications to account for the presence of the hanging nodes. A wide cross-section of nonlinear elliptic and parabolic problems are used to demonstrate the performance of the proposed algorithm, which is shown to provide optimal accuracy at an optimal computational cost.