Abstract
Two research subjects in geosciences which lately underwent significant progress are treated in this review. In the first part we focus on one key ingredient for the numerical approximation of the Darcy flow problem, namely the discretization of diffusion terms on general polygonal/polyhedral meshes. We present different schemes and discuss in detail their fundamental numerical properties such as stability, consistency, and robustness. The second part of the paper is devoted to error control and adaptivity for model geosciences problems. We present the available a posteriori estimates guaranteeing the maximal overall error and show how the different error components can be identified. These estimates are used to formulate adaptive stopping criteria for linear and nonlinear solvers, time step choice adjustment, and adaptive mesh refinement. Numerical experiments illustrate such entirely adaptive algorithms.