A Mortar Finite Element Method for Plate Problems

In the paper we discuss two versions of mortar finite element methods applied to clamped plate problems. The problems are approximated by the nonconforming Morley and Adini element methods in each subregion into which the original region of the discussed problems have been partitioned. On the interfaces between subdomains and at crosspoints of subregions some continuity conditions are imposed. The main results of the paper are the proof of the solvability of the discrete problems and their error bounds. The mortar method is a domain decomposition method that allow us to use discretizations of different type with independent discretizations parameters in nonoverlapping subdomains, see e.g. [BMP94], [BM97], [BB99] for a general presentation of the mortar method in the two and three dimensions for elliptic boundary value problems of second order. In the paper mortar element methods for the locally nonconforming discretizations of the clamped plate problems are discussed. In [Lac98] there are formulated results for mortar method with nonconforming discrete Kirchoff triangle elements (DKT) for a similar problem while in [Bel97] the mortar method for the biharmonic problem is analyzed in the case of local spectral discretizations. The paper is based on the results which are obtained in the PhD thesis of the author, see [Mar99b], cf. also [Mar99a]. This paper is concerned with the mortar method where locally in the subdomains the nonconforming Adini and Morley plate finite elements are used. We restrict ourselves to the geometrically conforming version of the mortar method, i.e. the local substructures form a coarse triangulation. We first introduce independent local discretizations for the two discussed elements in each subdomain. The 2-D triangulations of two neighboring subregions do not necessarily match on their common interface, cf. Figure 1. The mortar technique for nonconforming plate elements which is discussed here requires the continuity of the solution at the vertices of subdomains and that the solution on two neighboring subdomains satisfies two mortar conditions of the L type on their common interface. The form of these conditions depends on the local discretization methods and in some cases these conditions combine interpolants defined locally on interfaces. It follows from the fact that the respective traces of local functions also depend on the values of respective degrees of freedom at interior nodal points. We give error bounds for the both mortar methods. The results obtained in this paper can be generalized to analogous mortar discretizations of simply supported plate problems.