This paper presents studies on applying the novel weak Galerkin finite element method (WGFEM) to a two-phase model for subsurface flow, which couples the Darcy equation for pressure and a transport equation for saturation in a nonlinear manner. The coupled problem is solved in the framework of operator decomposition. Specifically, the Darcy equation is solved by theWGFEM, whereas the saturation...

In this study prediction of the steady-state flow of branched polymer melts in pipe geometry with finite volume method is presented. Our analysis in this study revealed that;for normal-stress tqq , the XPP model can predict this tensor unlike the other viscoelastic models such as PTT or Gieskus which can not predict tqq...

In this paper, an H1-Galerkin mixed finite element method is proposed and analyzed for parabolic partial differential equations with nonselfadjoint elliptic parts. Compared to the standard H1-Galerkin procedure, C1-continuity for the approximating finite dimensional subspaces can be relaxed for the proposed method. Moreover, it is shown that the finite element approximations have the same rates...

1. Poisson Equation 1 2. Outline of Topics 3 2.1. Finite Difference Method 3 2.2. Finite Element Method 3 2.3. Finite Volume Method 3 2.4. Sobolev Spaces and Theory on Elliptic Equations 3 2.5. Iterative Methods: Conjugate Gradient and Multigrid Methods 3 2.6. Nonlinear Elliptic Equations 3 2.7. Fast Multiple Method 3 3. Physical Examples 4 3.1. Gauss’ law and Newtonian gravity 4 3.2. Electrost...

We apply in this paper the weak Galerkin method to the second order parabolic differential equations based on a discrete weak gradient operator. We establish both the continuous time and the discrete time weak Galerkin finite element schemes, which allow using the totally discrete functions in approximation space and the finite element partitions of arbitrary polygons with certain shape regular...

Stability analysis algorithms coupled with a robust steady state solver are used to understand the behavior of the 2D model problem of thermal convection in a 8 : 1 differentially heated cavity. The system is discretized using a Galerkin=Least Squares Finite Element formulation, and solved to steady state on parallel computers using a fully coupled Newton method and iterative linear solvers. An...

We study the solution of linear systems resulting from the discreitization of unsteady diffusion equations with stochastic coefficients. In particular, we focus on those linear systems that are obtained using the so-called stochastic Galerkin finite element method (SGFEM). These linear systems are usually very large with Kronecker product structure and, thus, solving them can be both timeand co...

Given a P1 conforming or nonconforming Galerkin finite element method (GFEM) solution ph, which approximates the exact solution p of the diffusion-reaction equation −∇ · K∇p+ αp = f with full tensor variable coefficient K, we evaluate the approximate flux uh to the exact flux u = −K∇p by a simple but physically intuitive formula over each finite element. The flux is sought in the continuous (in...

The Discontinuous Petrov-Galerkin (DPG) method for high frequency wave propagation problems is discussed. The DPG method offers uniform pre-asymptotic stability for any wavenumber. This allows for a fully automatic adaptive hp algorithm, that can start from very coarse meshes. Moreover DPG always delivers a Hermitian positive definite system, suggesting the use of the Conjugate Gradient algorit...

In this work, we present high-resolution solutions for viscoelastic 4:1 planar contraction flow problems using a transient finite element method based on the fractional step method (FSM) and stabilization techniques (DEVSS-G/DG) with linear equal-order interpolation function. The Oldroyd-B model was used as the constitutive equation. A parallel multi-frontal algorithm was implemented to enhance...