Abstract: In this talk, we discuss our recent efforts on developing efficient open-access computational tools for poroelasticity. In particular, we examine (1) the newly added Matlab modules to our code package DarcyLite; (2) our new efforts on C++ modules for deal.II package. Several new finite element solvers have been implemented. These include utilizing the novel Arbogast-Correa elements within the weak Galerkin framework to solve Darcy, elasticity, and poroelasticity problems. Numerical simulations will be presented.

Abstract: In this paper, we study linearly first and second order in time, uniquely solvable and unconditionally energy stable numerical schemes to approximate the phase field model of solid-state dewetting problems based on the novel approach SAV (scalar auxiliary variable), a new developed efficient and accurate method for a large class of gradient flows. The schemes are based on the first order Euler method and the second order backward differential formulas(BDF2) for time discretization, and finite element methods for space discretization. It is shown that the schemes are unconditionally stable and the discrete equations are uniquely solvable for all time steps. we present some numerical experiments to validate the stability and accuracy of the proposed schemes.

Abstract: A novel numerical scheme including time and spatial discretization is presented for the coupled Cahn-Hilliard and Navier-Stokes equations in this paper. Variable densities and viscosities are considered in the numerical scheme. By introducing an intermediate velocity in both Cahn-Hilliard equation and the momentum equation, the scheme can keep the discrete energy law. A splitting method based on the pressure stabilization is implemented to solve the Navier-Stokes equation, while the stabilization approach or convex splitting method is used for the Cahn-Hilliard equation. This novel scheme is totally decoupled, linear, unconditionally energy stable for two-phase incompressible flow diffuse interface model. Numerical results demonstrate the validation, accuracy, robustness and discrete energy law of the proposed scheme in this paper.

Abstract: In this paper, we firstly study numerical methods for gas flow simulation in dual-continuum porous media. Typical methods for oil flow simulation in dual-continuum porous media cannot be used straightforward to this kind of simula-tion due to the artificial mass loss caused by the compressibility and the non-robustness caused by the non-linear source term. To avoid these two problems, corrected numerical methods are proposed using mass balance equations and lo-cal linearization of the non-linear source term. The improved numerical methods are successful for the computation of gas flow in the double-porosity double-permeability porous media. After this improvement, temporal advancement for each time step includes three fractional steps: i) advance matrix pressure and frac-ture pressure using the typical computation; ii) solve the mass balance equation system for mean pressures; iii) correct pressures in i) by mean pressures in ii). Numerical results show that mass conservation of gas for the whole domain is guaranteed while the numerical computation is robust.

Abstract: We study the numerical approximation of the incompressible miscible displacement equations on general quadrilateral grids in two dimensions. The flow equation is discretized by multipoint flux mixed finite element method and the transport equation is approximated by discontinuous Galerkin method. First-order convergence for velocity in $L^{\infty}(L^2)$ and concentration in $L^2(H^1)$ is derived. A numerical example is presented to support the theoretical analysis.