Partition of unity methods for heterogeneous domain decomposition

In many applications, mathematical and numerical models involve simultaneously more than one single phenomenon. In this situation different equations are used in possibly overlapping subregions of the domain in order to approximate the physical model and obtain an efficient reduction of the computational cost. The coupling between the different equations must be carefully handled to guarantee accurate results. However in many cases, since the geometry of the overlapping subdomains is neither given a-priori nor characterized by coupling equations, a matching relation between the different equations is not available; see, e.g. Degond and Jin (SIAM J Numer Anal 42(6):2671–2687, 2005), Gander et al. (Numer Algorithm 73(1):167–195, 2016) and references therein. To overcome this problem, we introduce a new methodology that interprets the (unknown) decomposition of the domain by associating each subdomain to a partition of unity (membership) function. Then, by exploiting the feature of the partition of unity method developed in Babuska and Melenk (Int J Numer Methods Eng 40:727–758, 1996) and Griebel and Schweitzer (SIAM J Sci Comput 22(3):853–890, 2000), we define a new domain-decomposition strategy that can be easily embedded in infinite-dimensional optimization settings. This allows us to develop a new optimal control methodology that is capable to design coupling mechanisms between the different approximate equations. Numerical experiments demonstrate the efficiency of the proposed framework.