Parabolic Problems with Space-Time Adaptivity

Application of adaptive dG methods and a posteriori error estimates to problems in geoscience are reviewed recently in [33]. Most of the applications of dG methods in geoscience concern reactive transport with advection [13, 62, 84] and strong permeability contrasts such as layered reservoirs [90] or vanishing and varying diffusivity posing challenges in computations [72]. The permeability in heterogeneous porous and fractured media varies over orders of magnitude in space, which results in highly variable flow field, where the local transport is dominated by advection or diffusion [85]. Accurate and efficient numerical solution of ADR equations to predict macroscopic mixing, anomalous transport of solutes and contaminants for a wide range of parameters like permeability and Péclet numbers, different flow velocities and reaction rates and reaction rates are challenging problems [85]. In order to resolve the complex flow patterns accurately, higher order time stepping methods like exponential time stepping methods are used [85].