Browsing by Author "Masri, Rami"

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    Analysis of discontinuous Galerkin schemes for flow and transport problems
    (2022-04-21) Masri, Rami; Riviere, Beatrice
    We formulate and theoretically analyze interior penalty discontinuous Galerkin (dG) methods for flow and transport problems. In particular, the analyses of dG formulations for (1) non-linear convection diffusion equations, (2) incompressible Navier– Stokes equations, (3) Cahn–Hilliard–Navier–Stokes equations, and (4) elliptic and parabolic problems with a Dirac line source are presented. First, we formulate a new locally implicit dG method for nonlinear convection diffusion equations, show that this scheme yields a less restrictive constraint on the time step, and prove optimal error estimates. This formulation is motivated by applications to coupled systems of solute transport and blood flow where it is combined with a Runge–Kutta dG scheme to simulate these systems in one dimensional vessel networks. The second scheme we analyze is a pressure correction dG scheme for the incompressible Navier–Stokes equations in two and three dimensional domains. Studying this scheme is motivated by its ability to efficiently simulate flow in large-scale complex computational domains. We show unconditional stability, unique solvability, and convergence of the discrete velocity by obtaining error estimates. The derivation of these error estimates requires the development of several tools including new lifting operators. Further, optimal error estimates in the L2 norm for velocity are obtained via introducing dual Stokes problems. To complete this analysis, we also show convergence of the discrete pressure. The pressure correction dG approach is extended to the Cahn–Hilliard–Navier– Stokes equations within a decoupled splitting framework. We show that the proposed scheme is uniquely solvable. The discrete mass conservation, the energy dissipation, and the L∞ stability of the order parameter, are established. We prove optimal a priori error estimates in the broken gradient norm. Using multiple duality arguments, we obtain an optimal error estimate in the L2 norm. The stability proofs and error analysis are based on induction arguments without any regularization of the potential function. The third class of problems we consider are elliptic and parabolic problems with a Dirac line source. Such problems are used to couple one dimensional flow models in blood vessels to three dimensional models in tissues. The analysis of such problems is challenging since the gradient of the true solution is singular. We propose dG discretizations of such problems and prove convergence in the global L 2 norm. For the elliptic problem, we show convergence in weighted energy norms. In addition, we show almost optimal local error estimates in the L 2 and energy norms in domains excluding the line. For the parabolic problem, we establish global error estimates for the semidiscrete formulation and for the fully discrete backward Euler dG discretization.
  • Thumbnail Image
    Item
    Derivation and Numerical Simulation of Oxygen Transport in Blood Vessels
    (2019-09-09) Masri, Rami; Riviere, Beatrice
    Modeling and simulating the transport of oxygen in blood provides critical insight on the planning of cardiovascular surgeries. Mathematical simulation provides a quantitative angle on the understanding of changes in hemodynamics. Due to the complexity of the cardio- vascular circulation, this is a computationally challenging task. Further, oxygen transport is coupled to the velocity field of blood. Thus, the numerical solution of the transport equation requires either the specification or the computation of the velocity field of blood. The latter approach is expensive when the three-dimensional Navier Stokes equations are considered, and the a-priori specification of the velocity does not account for changes in the velocity field. To counteract these difficulties, we propose a model reduction of the convection diffusion equation of oxygen in a compliant vessel with varying radius. We ob- tain a one-dimensional equation coupled to the reduced model of blood flow. We employ discontinuous Galerkin methods to efficiently solve the resulting system in one vessel. We show stability of the proposed numerical scheme for a general nonlinear convection diffusion equation. We verify the model using the method of manufactured solutions. We extend the numerical method to a bifurcation of vessels, and we simulate oxygen transport in a three vessel network
  • 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.