Son Young Yi

A new nonconforming mixed finite element method for linear elasticity

We have developed new nonconforming mixed finite element methods for linear elasticity with a pure traction (displacement) boundary condition based on the Hellinger–Reissner variational principle using rectangular elements. Convergence analysis yields an optimal (suboptimal) convergence rate of for the L2-error of the stress and for the displacement in the pure traction (displacement) boundary problem. However, numerical experiments have yielded optimal-order convergence rates for both stress and displacement in both problems and have shown superconvergence for the displacement at the midpoint of each element. Moreover, we observed that the optimal convergence rates are still valid for large λ.

THE CONVERGENCE OF A MULTIDIMENSIONAL, LOCALLY CONSERVATIVE EULERIAN–LAGRANGIAN FINITE ELEMENT METHOD FOR A SEMILINEAR PARABOLIC EQUATION

We study a locally conservative Eulerian–Lagrangian finite element method for approximating the solution of a semilinear parabolic equation and prove that a post-processed version of the method converges at an optimal rate as the space and time increments tend to zero.

A Study of Two Modes of Locking in Poroelasticity

In this paper, we study two modes of locking phenomena in poroelasticity: Possion locking and pressure oscillations. We first study the regularity of the solution of the Biot model to gain some insight into the cause of Poisson locking and show that the displacement gets into a divergence-free state as the Lame constant

Optimization-Based Decoupling Algorithms for a Fluid-Poroelastic System

\lambda \to \infty

An immersed interface method for a 1D poroelasticity problem with discontinuous coefficients

. We also examine the cause of pressure oscillations from an algebraic point of view when a three-field mixed finite element method is used. Based on the results of our study on the causes of the two modes of locking, we propose a new family of mixed finite elements that are free of both pressure oscillations and Poisson locking. Some numerical results are presented to validate our theoretical studies.

A lowest-order weak Galerkin method for linear elasticity

In this paper, computational algorithms for the Stokes-Biot coupled system are proposed to study the interaction of a free fluid with a poroelastic material. The decoupling strategy we employ is to cast the coupled fluid-poroelastic system as a constrained optimization problem with a Neumann type control that enforces continuity of the normal components of the stress on the interface. The optimization objective is to minimize any violation of the other interface conditions. Two numerical algorithms based on a residual updating technique are presented. One solves a least squares problem and the other solves a linear problem when the fluid velocity in the poroelastic structure is smooth enough. Both algorithms yield the minimizer of the constrained optimization problem. Some numerical results are provided to validate the accuracy and efficiency of the proposed algorithms.

A Locally Conservative Eulerian–Lagrangian Finite Difference Method for the forced KdV equation

We introduce an immersed interface method (IIM) based on a staggered grid for the 1D Biots poroelasticity model when the coefficients have discontinuities along material interfaces. The IIM uses a standard finite difference method away from the interfaces and modifies the numerical schemes near or on the interfaces to treat the irregularities using the method of undetermined coefficients. A second-order accuracy in the infinity norm has been derived for both pressure and displacement regardless of the position of the interfaces. We present some numerical results to confirm the theoretical error estimates.

Nonconforming mixed finite element methods for linear elasticity using rectangular elements in two and three dimensions

Abstract The lowest-order weak Galerkin (WG) method is considered for linear elasticity based on the displacement formulation. The new method approximates the displacement using piecewise constant vector functions both inside and on the boundary of each mesh element and its bilinear form does not require a stabilization term for the existence and uniqueness of the solution. A-priori error estimates of optimal order in the discrete H 1 - and L 2 -norms for the displacement are proved when the solution is smooth. The error estimates are independent of the Lame constant λ , thus the performance of the new method does not deteriorate as the elastic material becomes incompressible. Further, a simple post-processing technique to obtain a numerical approximation of the stress is presented. A careful error analysis reveals that the L 2 -norm error in the stress is also optimal and independent of λ . Several numerical experiments confirm the locking-free property and optimal convergence rates of the new method.

Numerical methods for unsaturated flow with dynamic capillary pressure in heterogeneous porous media

Abstract We propose a Locally Conservative Eulerian–Lagrangian Finite Difference Method (fdLCELM) for approximating the solution of the forced Korteweg–de Vries equation. The new numerical method employs an operator splitting scheme that solves the nonlinear transport equation and the linear dispersive equation sequentially. In order to conserve mass in the transport fractional step, we trace back each grid cell in time along the integral curve. The dispersive fractional step will be solved using a cell-centered finite difference method. Numerical examples are provided to confirm and illustrate the accuracy and mass conservation property of the new method. We also study the numerical stability and time evolution of various stationary solitary wave solutions in the presence of one or two bumps.