Description:

The numerical methods for PDE using overlapping grids have become increasingly popular in the recent years. Such grids are constructed by splitting the computational domain into simple overlapping parts and then meshing the parts independently. One advantage of this method is that the construction of a complicated grid for the whole domain is reduced to the construction of a simple and easy to handle (in some cases, even structured) grid in each of the subdomains. Different discretization methods can be used in each subdomain. In addition, parts of the mesh can be removed and new mesh can be added during the computations without the need to re-mesh the entire domain. Such splitting naturally leads to computational structures and parallelizable algorithms. By suitable grid extension, discretizations based on nonoverlapping, nonmatching grids can be reduced to discretizations by overlapping grids.

In the context of finite element methods on nonmatching grids, such techniques naturally fall into the category of domain decomposition methods. The most popular among these techniques are the ones based on the Lagrange multipliers and mortar methods. For overlapping or nonoverlapping nonmatching grids, they usually lead to nonconforming discretizations with certain disadvantages at the theoretical level and at the implementation level.

In the proposed research, we use a completely different technique, introduced first by Y. Huang and J. Xu (Math. Comp., 2001), by using a partition of unity method. The main advantage of this method is that the global discrete space created by gluing local approximation spaces has optimal approximation properties and is a conforming space. In addition, this new approach is problem-independent. For elliptic boundary value problems, optimal error estimates can be proved for minimal overlapping sizes of multiple subdomains in any spatial dimensions.

Following the ideas of Huang and Xu, and together with Zikatanov, we proposed a finite element method for overlapping grids for the Stokes problem. We prove that, for "glued" mini-type spaces, the discrete inf-sup condition holds with a constant independent of the overlapping size of the subdomains, provided the overlapped regions have a certain property, (JNM, 2005). Under minimal assumption for the partition of unity functions, together with Jiguang Sun, we proved that the collection of all degrees of freedom corresponding to subdomains remains an independent set of degrees of freedom, (Adv. App. Comp. Math., 2006). The results are also valid for multiple subdomains and any spatial dimension. In a future research, we hope to extend this method to mixed methods and/or other discretization of non-elliptic PDEs or systems. Since the method is in some sense problem-independent, a successful study of it is expected to have many important applications in science and engineering such as composite materials and subsurface flows in environmental applications.

Collaborators:

Jinchao Xu, Pennsylvania State University

Ludmil Zikatanov, Pennsylvania State

Jiguang Sun, Delaware State University