The Mortar Element Method with Overlapping Subdomains

The mortar element methods were introduced in [BMP94] for non overlapping domain decompositions in order to couple different variational approximations in different subdomains. In the finite element context, one important advantage of the mortar element methods is that it allows for using structured grids in subdomains thus fast solvers [AAH98]. The resulting methods are nonconforming but still yield optimal approximations. The literature on the mortar element methods is growing numerous see [AMW99] and reference therein. In this paper, we shall discuss the case of overlapping subdomains, with meshes constructed in an independent manner in each subdomain. As pointed by F. Hecht, J.L. Lions, and O. Pironneau, [LP99, HLP99] such a situation can occur if the domain of computation is a scene constructed by Constructive Solid Geometry as usual in Image Synthesis and Virtual Reality : each object of the scene is described by set operations on primitive shapes like cubes, cylinders, spheres and cones. With VRML (the language of VR), the objects may be described as unions of more elementary objects with primitive shapes, which are never intersected, so it is not possible to construct a global mesh. Each simple object must have its individual mesh. In [LP99, HLP99], many algorithms (including algorithms from control theory) for this situation are proposed, and cover more general cases than overlapping subdomains (domain with holes for example). We also note that independent of the development of the mortar methods, overlapping domain decomposition with non matching grids has been used for finite difference discretizations in the engineering community : these methods are often referred to as the chimera methods see [CH90, SB87], To our knowledge, mortar methods with overlapping subdomains have been proposed first by Y. Kuznetsov [Kuz97] who focused on iterative solvers with Lagrange multipliers. For two overlapping subdomains, the mortar method has been analyzed by X.C. Cai and M. Dryja and M. Sarkis [CDS99] in two dimensions. They have considered two subdomains, with non matching grids and piecewise linear Lagrange finite elements. In particular, they have considered the case when the overlapping parameter is 0, (two rectangular subdomains for a L shaped domain). They have also proposed iterative solvers and preconditioners for the linear systems arising from the mortar discretization. In this paper, we generalize their method in two dimensions, with more than two subdomains. We shall see that technical difficulties arise when the boundary of two