Domain Decomposition Methods for Reservoir Simulation Problems

Ilya Mishev and Sergey Nepomnyaschikh

We outline some of the specific requirements reservoir simulation problems pose and investigate in more details the influence of the jumps of the coefficients and the size of the timesteps.  We propose approaches to construct efficient Domain Decomposition preconditioning for such problems. We assume that the coefficients are smooth in some subdomains and have jumps on interfaces between these subdomains. In this situation, it is natural to split the domain into subdomains with smooth coefficients. The suggested domain decomposition methods use the Additive Schwarz method (ASM) framework that consists of three components:

1) local preconditioners in subdomains;
2) extension operators;
3) interface preconditioners.

We consider three algorithms. The first algorithm, proposed by Bramble, Pasciak, and Vassilev (BPV), is computationally very cheap and easy to implement, but efficiency is not very high, especially for elliptic problems. We consider an optimal algorithm with respect to the rate of convergence and the arithmetic cost, but a practical implementation of this method is rather complicated.  The third suggested algorithm (BPV+) is a compromise between the first two.