Coupled space-time discontinuous Galerkin method for dynamical modeling in porous media

This thesis deals with coupled space-time discontinuous Galerkin methods for the modeling of dynamical phenomena in fluid saturated porous media. The numerical scheme consists of finite element discretizations in the spatial and in the temporal domain simultaneously. In particular, two major classes of approaches have been investigated. The first one is the so-called time-discontinuous Galerkin (DGT) method, consisting of discontinuous polynomials in the temporal domain but continuous ones in space. A natural upwind f1ux treatment is introduced to enforce the continuity condition at discrete time levels. The proposed numerical approach is suitable for solving first-order time-dependend equations. For the second-order equations, an Embedded Velocity Integration (EVI) technique is developed to degenerate a second-order equation into a first-order one. The resulting first-order differential equation with the primary variable in rate term (velocity) can in turn be solved by the time-discontinuous Galerkin method efficiently. Applications concerning both the first- and second-order differential equations as weil as wave propagation problems in porous materials are investigated. The other one is the coupled space-time discontinuous Galerkin (DGST) method, in which neither the spatial nor the temporal approximations pocesses strong continuity. Spatial fluxes combined with flux-weighted constraints are employed to enforce the interelement consistency in space, while the consistency in the time domain is enforced by the temporal upwind flux investigated in the DGT method. As there exists no coupling between the spatial and temporal fluxes, various flux treatments in space and in time are employed independently. The resulting numerical scheme is able to capture the steep gradients or even discontinuities. Applications concerning the single-phase flow within the porous media are presented.