Shuyu Sun

Symmetric and Nonsymmetric Discontinuous Galerkin Methods for Reactive Transport in Porous Media

For solving reactive transport problems in porous media, we analyze three primal discontinuous Galerkin (DG) methods with penalty, namely, symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), and incomplete interior penalty Galerkin (IIPG). A cut-off operator is introduced in DG to treat general kinetic chemistry. Error estimates in L2(H1) are established, which are optimal in h and nearly optimal in p. We develop a parabolic lift technique for SIPG, which leads to h-optimal and nearly p-optimal error estimates in the L2(L2) and negative norms. Numerical results validate these estimates. We also discuss implementation issues including penalty parameters and the choice of physical versus reference polynomial spaces.

Coupled Generalized Nonlinear Stokes Flow with Flow through a Porous Medium

In this article, we analyze the flow of a fluid through a coupled Stokes-Darcy domain. The fluid in each domain is non-Newtonian, modeled by the generalized nonlinear Stokes equation in the free flow region and the generalized nonlinear Darcy equation in the porous medium. A flow rate is specified along the inflow portion of the free flow boundary. We show existence and uniqueness of a variational solution to the problem. We propose and analyze an approximation algorithm and establish a priori error estimates for the approximation.

L 2 ( H 1 ) norm a posteriori error estimation for discontinuous Galerkin approximations of reactive transport problems

Explicita posteriori residual type error estimators in L2(H1) norm are derived for discontinuous Galerkin (DG) methods applied to transport in porous media with general kinetic reactions. They are flexible and apply to all the four primal DG schemes, namely, Oden–Babuška–Baumann DG (OBB-DG), non-symmetric interior penalty Galerkin (NIPG), symmetric interior penalty Galerkin (SIPG) and incomplete interior penalty Galerkin (IIPG). The error estimators use directly all the available information from the numerical solution and can be computed efficiently. Numerical examples are presented to demonstrate the efficiency and the effectivity of these theoretical estimators.

A Combined Mixed Finite Element and Discontinuous Galerkin Method for Miscible Displacement Problem in Porous Media

A combined method consisting of the mixed finite element method for flow and the discontinuous Galerkin method for transport is introduced for the coupled system of miscible displacement problem. A “cut-off” operator M is introduced in the discontinuous Galerkin formular in order to make the combined scheme converge. Optimal error estimates in L 2(H 1) for concentration and in L ∞(L 2) for velocity are derived.

A Locally Conservative Finite Element Method Based on Piecewise Constant Enrichment of the Continuous Galerkin Method

This paper presents a locally conservative finite element method based on enriching the approximation space of the continuous Galerkin method with elementwise constant functions. The proposed method has a smaller number of degrees of freedom than the discontinuous Galerkin method. Numerical examples on coupled flow and transport in porous media are provided to illustrate the advantages of this method. We also present a theoretical analysis of the method and establish optimal convergence of numerical solutions.

Coupling Discontinuous Galerkin and Mixed Finite Element Discretizations using Mortar Finite Elements

Discontinuous Galerkin (DG) and mixed finite element (MFE) methods are two popular methods that possess local mass conservation. In this paper we investigate DG-DG and DG-MFE domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface. The subdomain grids need not match, and the mortar grid may be much coarser, giving a two-scale method. Convergence results in terms of the fine subdomain scale

MHD Mixed Convective Boundary Layer Flow of a Nanofluid through a Porous Medium due to an Exponentially Stretching Sheet

h

A finite element method for the numerical solution of the coupled Cahn-Hilliard and Navier-Stokes system for moving contact line problems

and the coarse mortar scale

Two-Phase Fluid Simulation Using a Diffuse Interface Model with Peng--Robinson Equation of State

H

Analysis of Discontinuous Galerkin Methods for Multicomponent Reactive Transport Problems

are established for both types of couplings. In addition, a nonoverlapping parallel domain decomposition algorithm is developed, which reduces the coupled system to an interface mortar problem. The properties of the interface operator are analyzed.