Serge.Goossens@cs.kuleuven.ac.be

Serge Goossens

Katholieke Universiteit Leuven

Department of Computer Science

Celestijnenlaan 200A

B-3001 Heverlee

BELGIUM

Abstract

The main topic addressed in this talk is the use of the mortar projection in an overlapping composite mesh difference method. We summarise the overlapping nonmatching mortar method described by Cai et al. This method has several desirable properties: it has a full theory, it is consistent, the accuracy is of optimal order and the error is independent of the size of the overlap. The disadvantage of this method is that it needs weights, which makes it impossible to use fast solvers for the subdomain problems. The Composite Mesh Difference Method (CMDM) has a full theory, and the accuracy is of optimal order. The main advantage of this method is that no weights are necessary and therefore it allows for fast subdomain solvers. The disadvantage is that low order interpolation may lead to a local inconsistent discretisation, resulting in an error that depends on the size of the overlap. We discus the easy to implement combination of bilinear interpolation with the standard five-point stencil, showing the disadvantages of a CMDM due to the local inconsistency when the low order interpolaton is used.

The goal is to take the mortar approach, drop the weights and compare its results to the non-mortar methods. In the ideal scheme the accuracy has to be of optimal order and the error has to be independent of the size of overlap. For the purpose of preconditioning and the use of fast solvers it is desirable not to have weights in the discretisation equations on the overlapping parts of the domain.

The standard stencils (corresponding to P1 or Q1 finite elements on a uniform mesh) with bicubic interpolation have these properties. But these schemes have several disadvantages. First of all a bicubic interpolation formula uses 16 points, which is a lot. Second, they fit in the CMDM theory, but this theory requires a large overlap since the interpolation constant is quite large, i.e. 25/16 in 2D. The numerical results show that there is no dependency on the amount of overlap since the scheme is fully consistent. The modified stencil with 1D cubic interpolation proposed by Goossens and Cai also fits in the CMDM theory. In this case the interpolation constant is smaller, i.e. 5/4 for 1D cubic interpolation. Consequently the theory calls for less overlap. The scheme is fully consistent so the numerical results show that there is no dependency on the amount of overlap.

The P1 and Q1 finite element discretisations on a uniform mesh can be considered as finite difference stencils for which the local truncation error is second order. The mortar projection can be used for the interpolation. These schemes satisfy all the assumptions made in the CMDM theory and the resulting method is second order. However we need to consider 2 interpolations for the mortar projection. The first interpolation is the actual projection from the master to the slave side of the mortar on the interface. In the case of overlapping nonmatching grids we also need to compute the master side of the mortar, which requires evaluating the P1 or Q1 finite element function. This boils down to linear interpolation. As a result for P1 and Q1 finite elements a linear interpolation is done in the direction normal on interface.

Based on our experience with bilinear interpolation we can estimate the effect of doing linear interpolation in the direction normal on interface. This interpolation gives rise to an extra term in the bound on the error in the extended subdomain just as in the error bound for the P1 stencil with bilinear interpolation. A final point we are considering is the dependency on the overlap. A large overlap may be required since the mortar projection does not satisfy the maximum principle (an example will be shown). We will present numerical results illustrating the influence of the size of the overlap both on the accuracy of the discretisation and on the performance of the iterative solver.