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This article concerns finite element approximations of a quasi-static poroelasticity model in displacementpressure formulation, which describes the dynamics of poro-elastic materials under an applied mechanical force on the boundary. To better describe the multiphysics process of deformation and diffusion for poroelastic materials, we first present a reformulation of the original model by introducing two pseudo-pressures. We then propose a time-stepping algorithm that decouples the reformulated partial differential equation (PDE) problem at each time step into two sub-problems: one of which is a generalized Stokes problem for the displacement vector field (of the solid network of the poroelastic material) along with one pseudopressure field, and the other is a diffusion problem for the other pseudo-pressure field (of the solvent of the material). To make this multiphysics approach feasible numerically, two critical issues must be resolved: the first one is the uniqueness of the generalized Stokes problem and the other is to find a good boundary condition for the diffusion equation, so that it also becomes uniquely solvable. To address the first issue, we discover certain conserved quantities for the PDE solution that provide ideal candidates for a needed boundary condition for the pseudo-pressure field. The solution to the second issue is to use the generalized Stokes problem to generate a boundary condition for the diffusion problem. A practical advantage of the time-stepping algorithm allows one to use any convergent Stokes solver (and its code) together with any convergent diffusion equation solver (and its code) to solve the poroelasticity model. In this article, the Taylor-Hood mixed finite element method combined with the P1 -conforming finite element method is used as an example to demonstrate the viability of the proposed multiphysics approach. It is proved that the solutions of the fully discrete finite element methods fulfill a discrete energy law, which mimics the differential energy law satisfied by the PDE solution and converges optimally in the energy norm. Moreover, it is showed that the proposed formulation also has a built-in mechanism to overcome so-called 'locking phenomenon' associated with the numerical approximations of the poroelasticity model. Numerical experiments are presented to show the performance of the proposed approach and methods, and to demonstrate the absence of 'locking phenomenon' in our numerical experiments. This article also presents a detailed PDE analysis for the poroelasticity model, especially it is proved that this model converges to the well-known Biot's consolidation model from soil mechanics as the constrained specific storage coefficient tends to zero. As a result, the proposed approach and methods are robust under such a limit process. [ABSTRACT FROM AUTHOR]

This paper develops and analyses two fully discrete interior penalty discontinuous Galerkin (IP-DG) methods for the Allen–Cahn equation, which is a nonlinear singular perturbation of the heat equation and originally arises from phase transition of binary alloys in materials science, and its sharp interface limit (the mean curvature flow) as the perturbation parameter tends to zero. Both fully implicit and energy-splitting time-stepping schemes are proposed. The primary goal of the paper is to derive sharp error bounds which depend on the reciprocal of the perturbation parameter ∈ (also called 'interaction length') only in some lower polynomial order, instead of exponential order, for the proposed IP-DG methods. The derivation is based on a refinement of the nonstandard error analysis technique first introduced in Feng & Prohl (2003, Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows. Numer. Math., 2003, 94, 33–65). The centrepiece of this new technique is to establish a spectrum estimate result in totally discontinuous DG finite element spaces with the help of a similar spectrum estimate result in the conforming finite element spaces which was established in Feng & Prohl. As a nontrivial application of the sharp error estimates, they are used to establish convergence, and the rates of convergence of the zero-level sets of the fully discrete IP-DG solutions to the classical and generalized mean curvature flow. Numerical experiment results are also presented to gauge the theoretical results and the performance of the proposed fully discrete IP-DG methods. [ABSTRACT FROM AUTHOR]

This paper presents a heterogeneous finite element method for a fluid–solid interaction problem. The method, which combines a standard finite element discretization in the fluid region and a mixed finite element discretization in the solid region, allows the use of different meshes in fluid and solid regions. Both semi‐discrete and fully discrete approximations are formulated and analysed. Optimal order a priori error estimates in the energy norm are shown. The main difficulty in the analysis is caused by the two interface conditions which describe the interaction between the fluid and the solid. This is overcome by explicitly building one of the interface conditions into the finite element spaces. Iterative substructuring algorithms are also proposed for effectively solving the discrete finite element equations. [ABSTRACT FROM PUBLISHER]