DEVELOPMENTS AND APPLICATIONS OF THE TECHNIQUE OF THE COMPOSITE MESH

Abstract

The technique of the Composite Mesh has been introduced in the 90's, as it can be seen in papers by the authors in the IV WCCM held in Buenos Aires. Since then some developments and applications have been made in several kinds of problems. The technique -inspired in the mixtures theory of composite material- is based on the utilization of two finite element meshes of different element size sharing the whole domain. Each mesh is endowed with a participation factor. The composite mesh may be used for two different aims. One of them is to provide a posteriori error estimates, and the other one is to reduce the discretization errors thus leading to improved numerical solutions. For the first case, it is shown that the residues at the connecting points of both meshes serve as a posteriori error estimates being able to detect the regions where mesh adaptation should be conducted. This aspect has been tested in different kind of problems and compared with other error estimates. The composite mesh may be used also to provide an improved numerical solution. This is performed by a suitable choice of the participation factors. Important reduction in the dis- cretization errors has been obtained for a wide class of problems, including those with border singularities. This improved solutions are obtained without increasing the computational cost as the problem size to be solved is of the same order of that for the fine element mesh. Applications have been made to elliptic and parabolic problems in regular domains or in the presence of border singularities. Problems from structural mechanics (elastic analysis of plain stress and plates), as well as catalytic chemical reactors have also been addressed. It can be mentioned that the technique of the composite mesh has been also applied in the frame of multigrid solving, showing interesting reductions in the solution errors, and making use of the fact that mesh with different size are available in the process. In this paper we make a review of the advances in this technique and present some results on different application areas. We include some results both on error estimation and on solution improvement