A posteriori error estimator for a multiscale hybrid mixed method for Darcy's flows

This article presents a computable and efficient procedure for a posteriori error estimations of approximate solutions of Darcy's flows given by a Multiscale Hybrid Mixed method. Based on a partition of the domain by polytopal macro subregions, this is a strategy for efficient simulations of problems with strongly varying solutions. The flux approximations interacting with the skeleton (subregion boundaries) are strongly constrained by a given trace space. Whilst the dimension of the piecewise polynomial trace space is expected to be as low as possible, the small-scale features of the solution are supposed to be accurately resolved by completely independent local stable mixed solvers, the trace variable playing the role of Neumann boundary data. As the method already gives an equilibrated global H(div)-conforming flux approximation, the methodology for the error estimation only requires a potential reconstruction. In addition to usual residual errors and indicators associated with the potential reconstruction, the estimation also takes into account the effect of discretizations of practical nonhomogeneous Dirichlet and Neumann boundary conditions. Based on the proposed a posteriori error estimator, h-adaptive algorithms are constructed to guide a proper choice of the trace space. The performance of the error estimator and the adaptive scheme is numerically investigated through a set of illustrating test problems.