Browsing by Author "Shen, Boqian"

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source
    (EDP Sciences, 2023) Masri, Rami; Shen, Boqian; Riviere, Beatrice
    The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L2 norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L2 norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L2 norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L2 in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.
  • Thumbnail Image
    Item
    Numerical Methods for Two-phase Flow in Rigid and Deformable Porous Media
    (2022-04-22) Shen, Boqian; Riviere, Beatrice
    The thesis focuses on developing numerical schemes for two-phase flow in rigid and deformable porous media problems. We present a stable and efficient sequential Discontinuous Galerkin (DG) method for solving the linear poroelasticity equations, which characterize two-phase flow within a deformable porous media. More precisely, we approximate the pressure of the wetting phase, the pressure of the non-wetting phase, and the displacement of the solid skeleton in three dimensions by a high-order interior penalty discontinuous Galerkin (IPDG) spatial discretization combined with a backward Euler discretization in time. The proposed work is based on previous developments in single fluid flow in deformable porous media. The numerical scheme solves the coupled equations sequentially while keeping each equation implicitly with respect to its unknown. The equations are fully decoupled in this sequential approach, which significantly reduces the computational cost compared to the implicit and iterative approaches. Numerical experiments show the convergence of the scheme is optimal. Finally, we apply the sequential DG scheme to a variety of physical problems with realistic data including common benchmarks, heterogeneous porous media with discontinuous permeability, porosity and capillary pressure, and porous media subjected to load. The second part of this thesis proposes an adaptive hybrid numerical scheme for solving two-phase flow in rigid porous media problems. The spatial discretization for transport phenomena problem in heterogeneous porous media requires locally mass conservative methods, such as finite volume methods and discontinuous Galerkin methods. The numerical scheme uses discontinuous Galerkin methods in regions of interest where high accuracy is needed and uses finite volume methods in the rest of the domain. The proposed schemes take advantage of the high accuracy of the discontinuous Galerkin method due to its local mesh adaptivity and local choice of polynomial degree. Finite volume methods are only first-order accurate but computationally efficient and robust for general geometries with structured mesh. We develop adaptive indicators to dynamically identify the regions of each method. By using such an adaptive indicator we are able to find the optimal balance between accuracy and computational cost.