Fuzzy Domain Decomposition: a new perspective on heterogeneous DD methods

In a wide variety of physical problems, the complexity of the physics involved is such that it is necessary to develop approximations, because the complete physical model is simply too costly. Sometimes however the complete model is essential to capture all the physics, and often this is only in part of the domain of interest. One can then use heterogeneous domain decomposition techniques: if we know a priori where an approximation is valid, we can divide the computational domain into subdomains in which a particular approximation is valid and the topic of heterogeneous domain decomposition methods is to find the corresponding coupling conditions to insure that the overall coupled solution is a good approximation of the solution of the complete physical model. For an overview of such techniques, see [9, 10] and references therein. However, there are many physical problems where it is not a priori knownwhere which approximation is valid. In such problems, one needs to track the domain of validity of a particular approximation, and this is usually not an easy task. An example of such a method is the -method, see [4, 1]. In this contribution, we introduce a new formalism for heterogeneous domain decomposition, which is not based on a sharp decomposition into subdomains where different models are valid. The main idea relies on the notion of Fuzzy Sets introduced by Zadeh [12] in 1965. The Fuzzy Set Theory relaxes the notion of belonging to a set through membership functions to (fuzzy) sets that account for partially belonging to a set. In the context of heterogeneous domain decomposition, this could be useful if one assumes that the computational domain can be decomposed into fuzzy sets that form a partition of the domain in a sense that needs to be specified. Once such a partition is given, one can compute the solution of the coupled problem using the membership functions. Note that the membership functions can depend on space and time and therefore can take into account a change in the validity domain of a particular approximation.We show here that this technique leads to an excellent coupling strategy for the 1D advection dominated diffusion problem. Such a domain decomposition method would be able, in principle, to take into account part of the domain where none of the available approximations are valid under the assumption that a combination of them is a good enough approximation there. On the assumption u= u1+u2: The idea to use fuzzy set theory came from an assumption that arose in some specific coupling methods (see below). We formulate it here for a generic partial differential equation of the form