Multilevel preconditioning for mixed problems in three dimensions

This thesis aims to provide a rigorous theoretical examination of multilevel preconditioning schemes for some discrete variational problems in three dimensions, which involve the differential operators div and curl. For instance, such kinds of problems occur in the dual formulation of (.) The investigations are set in a finite element framework relying on regularly refined simplical meshes and finite element schemes for which a canonical construction on the basis of differential forms is provided. The design of multilevel decompositions of these spaces requires a separate treatment of the nontrivial kernels of the relevant differential operators. The author presents some nodal splittings and establishes their asymptoticly uniform stability with respect to the depth of refinement. The proof relies on classical algebraic multilevel theory applied to quotient spaces and duality techniques. New approximation and extension theorems for Nédélec spaces play a pivotal role Finally, the result confirms that some iterative procedures for the solution of discrete saddle point problems achive optimal complexity if fitted with the new multilevel preconditioners. Applications to problems arising from Maxwell's equations are straightforward.