Jiangguo Liu

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.

Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity

This paper presents a family of weak Galerkin finite element methods (WGFEMs) for Darcy flow computation. The WGFEMs are new numerical methods that rely on the novel concept of discrete weak gradients. The WGFEMs solve for pressure unknowns both in element interiors and on the mesh skeleton. The numerical velocity is then obtained from the discrete weak gradient of the numerical pressure. The new methods are quite different than many existing numerical methods in that they are locally conservative by design, the resulting discrete linear systems are symmetric and positive-definite, and there is no need for tuning problem-dependent penalty factors. We test the WGFEMs on benchmark problems to demonstrate the strong potential of these new methods in handling strong anisotropy and heterogeneity in Darcy flow.

A matched interface and boundary method for solving multi-flow Navier-Stokes equations with applications to geodynamics

We have developed a second-order numerical method, based on the matched interface and boundary (MIB) approach, to solve the Navier-Stokes equations with discontinuous viscosity and density on non-staggered Cartesian grids. We have derived for the first time the interface conditions for the intermediate velocity field and the pressure potential function that are introduced in the projection method. Differentiation of the velocity components on stencils across the interface is aided by the coupled fictitious velocity values, whose representations are solved by using the coupled velocity interface conditions. These fictitious values and the non-staggered grid allow a convenient and accurate approximation of the pressure and potential jump conditions. A compact finite difference method was adopted to explicitly compute the pressure derivatives at regular nodes to avoid the pressure-velocity decoupling. Numerical experiments verified the desired accuracy of the numerical method. Applications to geophysical problems demonstrated that the sharp pressure jumps on the clast-Newtonian matrix are accurately captured for various shear conditions, moderate viscosity contrasts and a wide range of density contrasts. We showed that large transfer errors will be introduced to the jumps of the pressure and the potential function in case of a large absolute difference of the viscosity across the interface; these errors will cause simulations to become unstable.

A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods

This paper presents a comparative study on the newly introduced weak Galerkin finite element methods (WGFEMs) with the widely accepted discontinuous Galerkin finite element methods (DGFEMs) and the classical mixed finite element methods (MFEMs) for solving second-order elliptic boundary value problems. We examine the differences, similarities, and connection among these methods in scheme formulations, implementation strategies, accuracy, and computational cost. The comparison and numerical experiments demonstrate that WGFEMs are viable alternatives to MFEMs and hold some advantages over DGFEMs, due to their properties of local conservation, normal flux continuity, no need for penalty factor, and definiteness of discrete linear systems.

Mathematical modeling for intracellular transport and binding of HIV-1 Gag proteins

This paper presents a modeling study for the intracellular trafficking and trimerization of the HIV-1 Gag proteins. A set of differential equations including initial and boundary conditions is used to characterize the transport, diffusion, association and dissociation of Gag monomers and trimers for the time period from the initial production of Gag protein monomers to the initial appearance of immature HIV-1 virions near the cell membrane (the time duration Ta). The existence and stability of the steady-state solution of the initial boundary value problems provide a quantitative characterization of the tendency and equilibrium of Gag protein movement. The numerical simulation results further demonstrate Gag trimerization near the cell membrane. Our calculations of Ta are in good agreement with published experimental data. Sensitivity analysis of Ta to the model parameters indicates that the timing of the initial appearance of HIV-1 virions on the cell membrane is affected by the diffusion and transport processes. These results provide important information and insight into the Gag protein transport and binding and HIV-1 virion formation.

On Application of the Weak Galerkin Finite Element Method to a Two-Phase Model for Subsurface Flow

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 the WGFEM, whereas the saturation is solved by a finite volume method. The numerical velocity obtained from solving the Darcy equation by the WGFEM is locally conservative and has continuous normal components across element interfaces. This ensures accuracy and robustness of the finite volume solver for the saturation equation. Numerical experiments on benchmarks demonstrate that the combined methods can handle very well two-phase flow problems in high-contrast heterogeneous porous media.

A Comparative Study of Locally Conservative Numerical Methods for Darcy's Flows

Abstract This paper presents a comparative study on locally mass-conservative numerical methods for Darcys flows. The classical mixed finite element method (MFEM) is compared with the newly developed discontinuous finite volume method (DFVM) with and without weak over-penalization (WOP). These numerical methods are tested on three representative problems in porous media flows. In particular, locality, accuracy of numerical solutions, computational costs, and implementation issues are examined. The study indicates that the discontinuous finite volume methods could be viable alternatives to the classical mixed finite element method for Darcys flows.

The Lowest-order Weak Galerkin Finite Element Method for the Darcy Equation on Quadrilateral and Hybrid Meshes

Abstract This paper investigates the lowest-order weak Galerkin finite element method for solving the Darcy equation on quadrilateral and hybrid meshes consisting of quadrilaterals and triangles. In this approach, the pressure is approximated by constants in element interiors and on edges. The discrete weak gradients of these constant basis functions are specified in local Raviart–Thomas spaces, specifically R T 0 for triangles and unmapped R T [ 0 ] for quadrilaterals. These discrete weak gradients are used to approximate the classical gradient when solving the Darcy equation. The method produces continuous normal fluxes and is locally mass-conservative, regardless of mesh quality, and has optimal order convergence in pressure, velocity, and normal flux, when the quadrilaterals are asymptotically parallelograms. Implementation is straightforward and results in symmetric positive-definite discrete linear systems. We present numerical experiments and comparisons with other existing methods.

Modeling HIV-1 viral capsid nucleation by dynamical systems

There are two stages generally recognized in the viral capsid assembly: nucleation and elongation. This paper focuses on the nucleation stage and develops mathematical models for HIV-1 viral capsid nucleation based on six-species dynamical systems. The Particle Swarm Optimization (PSO) algorithm is used for parameter fitting to estimate the association and dissociation rates from biological experiment data. Numerical simulations of capsid protein (CA) multimer concentrations demonstrate a good agreement with experimental data. Sensitivity and elasticity analysis of CA multimer concentrations with respect to the association and dissociation rates further reveals the importance of CA trimer-of- dimers in the nucleation stage of viral capsid self- assembly.

DarcyLite: A Matlab Toolbox for Darcy Flow Computation☆

Abstract DarcyLite is a Matlab toolbox for numerical simulations of flow and transport in 2-dim porous media. This paper focuses on the finite element (FE) methods and the corresponding code modules in DarcyLite for solving the Darcy equation. Specifically, four major types of finite element solvers, the continuous Galerkin (CG), the discontinuous Galerkin (DG), the weak Galerkin (WG), and the mixed finite element methods (MFEM), are examined. Furthermore, overall design and implementation strategies in DarcyLite are discussed. Numerical results are included to demonstrate the usage and performance of this toolbox.