Multilevel additive Schwarz preconditioner for nonconforming mortar finite element methods

Mortar elements form a family of special non-overlapping domain decomposition methods which allows the coupling of different triangulations across subdomain boundaries. We discuss and analyze a multilevel preconditioner for mortar finite elements on nonmatching triangulations. The analysis is carried out within the abstract framework of additive Schwarz methods. Numerical results show a performance of our preconditioner as predicted by the theory. Our condition number estimate depends quadratically on the number of refinement levels. 1. Introduction. Domain decomposition techniques for the numerical solution of partial differential equations have been analyzed extensively and used successfully. Large problems are decomposed into a set of smaller ones by breaking up the given domain into subdomains. Then, with parallel computation in mind, flexible methods are established by using characteristic properties of the given partial differential equation. In this paper, we consider two different aspects of domain decomposition. Within the framework of mortar finite element methods, we use a dis-cretization scheme based on the coupling of nonmatching triangulations along the interfaces. Then in order to construct an efficient solver for the resulting linear system of equations, we introduce and analyze a special multilevel Schwarz method. We will use the mortar finite elements introduced by Bernardi, et al. in [3, 4]. A characteristic feature of mortar methods is that subdomain meshes may be constructed separately in each subdomain and are, in general, nonmatching along the interfaces. This is in contrast to traditional domain decomposition methods, where a globally conforming triangulation is used. Mortar finite elements therefore provide a more flexible approach than standard conforming formulations. Often, mortar methods are recommended when the splitting into subdomains is somehow prescribed for physical or geometrical reasons. Then, for each subdomain an optimal approximation scheme can be chosen, involving the choice of the triangulation as well as the discretization. The strong condition of pointwise continuity across the interfaces is replaced by a weaker one resulting in a nonconforming approximation scheme. In spite thereof, mortar finite elements provide the same accuracy as standard conforming finite elements. In Section 2, we briefly recall the standard mortar formulation. An introduction of our multi-level preconditioner can be found in Section 3. In Section 4, we establish our main result, which is an upper bound for the condition number of the additive Schwarz operator in terms of the refinement level. In Section 5, we discuss some aspects of the implementation of the method. Finally in Section …