On the robustness of hierarchical multilevel splittings for discontinuous Galerkin rotated bilinear FE problems

In this paper we present a new framework for multilevel preconditioning of large sparse systems of linear algebraic equations arising from the interior penalty discontinuous Galerkin approximation of second-order elliptic boundary value problems. Though the focus is on a particular family of rotated bilinear non-conforming (Rannacher-Turek) finite elements in two space dimensions (2D) the proposed rather general setting is neither limited to this particular choice of elements nor to 2D problems Under the assumption that the finest partition is a result of multilevel refinement of a given coarse mesh, we develop a novel concept for the hierarchical splitting of the unknowns. Next we show how the constant in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality can be estimated locally in this setting. Two basic problems are studied: the scalar elliptic equation and the Lamé system of linear elasticity. This part of the presented theoretical results is in the spirit of algebraic multilevel iteration (AMLI) methods. One innovative contribution of this work is the construction of robust methods for problems with large jumps (several orders of magnitude) in the PDE coefficients that can only be resolved on the finest finite element mesh. In the well-established theory of hierarchical basis multilevel methods one basic assumption is that the PDE coefficients are smooth functions on the elements of the coarsest mesh partition. The presented numerical study of the CBS constant shows a well expressed robustness of the developed hierarchical multilevel splitting with respect to coefficient jumps between elements on the finest mesh.