A posteriori-based, local multilevel mesh refinement for the Darcy porous media flow problem

Abstract

In this work we develop an a posteriori-based adaptive local multilevel mesh refinement strategy for the Darcy porous media flow problem discretized with the finite volume method. Our approach is based on coupling the FIC “flux interface correction” method [1] and the equilibrated flux a posteriori error estimate [2]. The choice of the FIC method is motivated by the fact that it is well-adapted to problems discretized in conservative form since that the FIC provides corrections by balancing fluxes computed from both coarse and fin grids across the interface. The equilibrated flux a posteriori error estimate provides a guaranteed upper bound on the total error in the fluxes and takes advantage of the conservative scheme in such a way that the estimate can be easily coded and cheaply evaluated. We use this a posteriori estimate to detect automatically the zone of interest where the error is dominant in order to defined the local sub-grids. Numerous numerical experiments on practical problems illustrate the performance of our methodology. A comparison with a classical h-adaptive method is also carried out in this work.