12. Mortar Projection in Overlapping Composite Mesh Difference Methods

We study experimentally the effect of the mortar projection in an overlapping composite mesh difference method for two-dimensional elliptic problems. In [CDS99], an overlapping mortar element method was proposed. This method has several desirable properties. For example, the discretisation is consistent, the accuracy is of optimal order and the error is independent of the size of the overlap, as well as the ratio of the mesh sizes. However, a major disadvantage of the method is that it needs weights in the bilinear form. The artificially introduced piecewise constant weights make the scheme consistent, but at the same time make it impossible to use fast solvers for the subdomain problems. On the other hand, the composite mesh difference method (CMDM) [Sta77, CMS00, GC99] does not need any weights, and its accuracy is also of optimal order if used with higher order interface interpolations. For example, the 2D bicubic or modified 1D cubic interface interpolation [GC99] is needed if one uses P1 or Q1 finite elements for the interior of the subdomains. But if the computationally more efficient low order interpolation is used on the interfaces, it may lead to a local inconsistent discretisation, resulting in an error that depends on the size of the overlap. The goal of this paper is to take the mortar approach, drop the weights and compare its results to the non-mortar methods. Of course, in an ideal scheme, which is yet to be discovered, the accuracy should be of optimal order and the error be independent of the size of the overlap and the ratio of mesh sizes. In order to be able to use fast solvers for the subdomain problems, it is also desirable not to have weights in the discretisation on the overlapping parts of subdomains.