A Locally Conservative Enriched Galerkin Approximation and User-Friendly Efficient Solver for Elliptic and Parabolic Problems

- 발표자 : 이영주 교수 (Texas State University-San Marcos) - 초록 : We present and analyze an enriched Galerkin finite element method (EG) to solve elliptic and parabolic equations with jump coefficients. The EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with the piecewise constant functions, which can be considered as an additional penalty stabilization. The method is shown to be locally and globally conservative, while keeping lower degree of freedoms in comparisons with discontinuous Galerkin finite element methods (DG). Moreover, we present and analyze a fast, effective and user-friendly EG solver simpler than DG and whose cost is roughly that of CG and can handle {an arbitrary} order of approximations. A number of numerical tests in two and three dimensions are presented to confirm our theoretical results as well as to demonstrate the advantages of the EG when coupled with transport.

Overcoming Locking in Poroelasticity

- 발표자 : 이선영 교수 (University of Texas at El Paso) - 초록 : The Biot’s consolidation model describes the interaction between the fluid flow and deformation in an elastic porous material. There is an extensive body of literature on finite element methods for the Biot model. It has been well known, however, that standard Galerkin finite element methods produce unstable and oscillatory behavior of the fluid pressure for a certain range of material parameters and the stabilization of pressure oscillations has been a subject of extensive research in recent years.In this talk, we discuss two modes of locking in poroelaticity: pressure oscillations and Poisson (volume) locking. First, we re-examine the cause of pressure oscillations in the three-field mixed finite element method from an algebraic point of view. We then investigate a possible Poisson locking in poroelasticity when a Lame constant λ approaches ∞ by studying the regularity of the solution of the Biot model. Based on this study, we propose new finite element methods that are free of locking. We discuss the existence and uniqueness theory and prove a-priori error estimates for the approximate solutions. Some numerical results are presented to confirm our theoretical results.

A Four-Field Mixed Finite Element Method for the Biot Model and its Solution Algorithms

- 발표자 : 이선영 교수 (University of Texas at El Paso) - 초록 : In this talk, we propose a four-field mixed finite element method for the 2D Biot’s consolidation model of poroelasticity. The method is based on coupling two mixed finite element methods for each subproblem: the standard mixed finite element method for the flow subproblem and the Hellinger-Reissner formulation for the mechanical subproblem. Optimal a-priori error estimates are proved for both semi-discrete and fully discrete problems. In solving the coupled system, the two subproblems can be solved either simultaneously in a fully coupled scheme or sequentially in a loosely coupled scheme. We formulate four iteratively coupled methods, known as drained, undrained, fixed-strain, and fixed-stress splits, in which the diffusion operator is separated from the elasticity operator and the two subproblems are solved in a staggered way while ensuring convergence of the solution. We prove a-priori convergence results for each iterative coupling scheme. We also explore an efficient preconditioning method for the saddle point system resulting from the fully coupled method. The proposed preconditoner is a block diagonal preconditioner based on the Schur complement. Some numerical results are provided to show the efficiency of the preconditioner when applied to a poroelasticty problem in a layered medium.

Auxiliary Space Preconditioner for the Linear Elasticity Equation with Weakly Imposed Symmetry Problems

- 발표자 : 이영주 교수 (Texas State University-San Marcos) - 초록 : In this talk, we develop a fictitious space preconditioner for the linear system of equations arising from the mixed finite element discretization of the linear elasticity equations with weakly imposed symmetry. The preconditioner consists of a fast Poisson solver, and d copies of (vector) H(div) solvers (such as HX-precoditioner) where d is the space dimension. We show that the preconditioner is uniform with respect to the mesh size and parameters in the equation. This preconditioner also gives an efficient solver for the pseudo-stress formulation of the Stokes equation.