Goal oriented adaptivity for coupled flow and transport problems with applications in oil reservoir simulations

Goal oriented adaptivity has become an important and widely used technique for solving partial differential equations at a reduced computational cost. The technique is based on a computable estimate of the error in a linear functional of interest, and an adaptive strategy that improves the solution, for instance by refining the mesh locally according to the error estimator.In this paper, we extend this theory to a multi-physical setting involving a coupled set of equations. Here adaptive algorithms become even more crucial since we do not only need estimators that indicate in which part of the domain the solution needs to be improved, but also which equation contributes most to the error, and thus, needs to be solved more accurately.We derive a goal oriented error estimator for a system of coupled pressure and transport equations that serves as a basic model for oil reservoir simulation. We utilize a 2D model to present extensive numerical results, where we explain in detail how the adaptive algorithm works in practice.