Unity Methods in Physics and Geometry

1. Introduction. In this article we consider a class of adaptive multilevel domain decomposition-like algorithms, built from a combination of adaptive multilevel finite element, domain decomposition, and partition of unity methods. These algorithms have several interesting features such as very low communication requirements, and they inherit a simple and elegant approximation theory framework from partition of unity methods. They are also very easy to use with highly complex sequential adaptive finite element packages, requiring little or no modification of the underlying sequential finite element software. The parallel algorithm can be implemented as a simple loop which starts off a sequential local adaptive solve on a collection of processors simultaneously. We first review the Partition of Unity Method (PUM) of Babuška and Melenk in Section 2, and outline the PUM approximation theory framework. In Section 3, we describe a variant we refer to here as the Parallel Partition of Unity Method (PPUM), which is a combination of the Partition of Unity Method with the parallel adaptive algorithm from [4]. We then derive two global error estimates for PPUM, by exploiting the PUM analysis framework it inherits, and by employing some recent local estimates of Xu and Zhou [22]. We then discuss a duality-based variant of PPUM in Section 4 which is more appropriate for certain applications, and we derive a suitable variant of the PPUM approximation theory framework. Our implementation of PPUM-type algorithms using the FEtk and MC software packages is described in Section 5. We then present a short numerical example in Section 6 involving the Einstein constraints arising in gravitational wave models.