Do Y. Kwak

Gwanghyun Jo, Do Y. Kwak

We develop a numerical scheme for a two-phase immiscible flow in heterogeneous porous media using a structured grid finite element method, which have been successfully used for the computation of various physical applications involving elliptic equations \cite{li2003new, li2004immersed, chang2011discontinuous, chou2010optimal, kwak2010analysis}. The proposed method is based on the implicit pressure-explicit saturation procedure. To solve the pressure equation, we use an IFEM based on the Rannacher-Turek \cite{rannacher1992simple} nonconforming space, which is a modification of the work in \cite{kwak2010analysis} where `broken' $P_1$ nonconforming element of Crouzeix-Raviart \cite{crouzeix1973conforming} was developed. For the Darcy velocity, we apply the mixed finite volume method studied in \cite{chou2003mixed, kwak2010analysis} on the basis of immersed finite element method (IFEM). In this way, the Darcy velocity of the flow can be computed cheaply (locally) after we solve the pressure equation. The computed Darcy velocity is used to solve the saturation equation explicitly. Thus the whole procedure can be implemented in an efficient way using a structured grid which is independent of the underlying heterogeneous porous media. Numerical results show that our method exhibits optimal order convergence rates for the pressure and velocity variables, and suboptimal rate for saturation.

Do Y. Kwak, Juho Lee

In recent years, the immersed finite element methods (IFEM) introduced in \cite{Li2003}, \cite{Li2004} to solve elliptic problems having an interface in the domain due to the discontinuity of coefficients are getting more attentions of researchers because of their simplicity and efficiency. Unlike the conventional finite element methods, the IFEM allows the interface cut through the interior of the element, yet after the basis functions are altered so that they satisfy the flux jump conditions, it seems to show a reasonable order of convergence. In this paper, we propose an improved version of the $P_1$ based IFEM by adding the line integral of flux terms on each element. This technique resembles the discontinuous Galerkin (DG) method, however, our method has much less degrees of freedom than the DG methods since we use the same number of unknowns as the conventional $P_1$ finite element method. We prove $H^1$ and $L^2$ error estimates which are optimal both in order and regularity. Numerical experiments were carried out for several examples, which show the robustness of our scheme.

Do Y. Kwak, Sangwon Jin, Dae H. Kyeong

We develop a new finite element method for solving planar elasticity problems involving of heterogeneous materials with a mesh not necessarily aligning with the interface of the materials. This method is based on the `broken' Crouzeix-Raviart $P_1$-nonconforming finite element method for elliptic interface problems \cite{Kwak-We-Ch}. To ensure the coercivity of the bilinear form arising from using the nonconforming finite elements, we add stabilizing terms as in the discontinuous Galerkin (DG) method \cite{Arnold-IP},\cite{Ar-B-Co-Ma},\cite{Wheeler}. The novelty of our method is that we use meshes independent of the interface, so that the interface may cut through the elements. Instead, we modify the basis functions so that they satisfy the Laplace-Young condition along the interface of each element. We prove optimal $H^1$ and divergence norm error estimates. Numerical experiments are carried out to demonstrate that the our method is optimal for various Lamè parameters $μ$ and $λ$ and locking free as $λ\to\infty$.

Seungwoo Lee, Do Y. Kwak, Imbo Sim

We consider the approximation of eigenvalue problems for elasticity equations with interface. This kind of problems can be efficiently discretized by using immersed finite element method (IFEM) based on Crouzeix-Raviart P1-nonconforming element. The stability and the optimal convergence of IFEM for solving eigenvalue problems with interface are proved by adapting spectral analysis methods for the classical eigenvalue problem. Numerical experiments demonstrate our theoretical results.

Seungwoo Lee, Do Y. Kwak, Imbo Sim

We consider the approximation of elliptic eigenvalue problem with an immersed interface. The main aim of this paper is to prove the stability and convergence of an immersed finite element method (IFEM) for eigenvalues using Crouzeix-Raviart $P_1$-nonconforming approximation. We show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem. We analyze the IFEM for elliptic eigenvalue problem with an immersed interface and derive the optimal convergence of eigenvalues. Numerical experiments demonstrate our theoretical results.

Do Y. Kwak, K. T. Wee

We study some numerical methods for solving second order elliptic problem with interface. We introduce an immersed interface finite element method based on the `broken' $P_1$-nonconforming piecewise linear polynomials on interface triangular elements having edge averages as degrees of freedom. This linear polynomials are broken to match the homogeneous jump condition along the interface which is allowed to cut through the element. We prove optimal orders of convergence in $H^1$ and $L^2$-norm. Next we propose a mixed finite volume method in the context introduced in \cite{Kwak2003} using the Raviart-Thomas mixed finite element and this `broken' $P_1$-nonconforming element. The advantage of this mixed finite volume method is that once we solve the symmetric positive definite pressure equation(without Lagrangian multiplier), the velocity can be computed locally by a simple formula. This procedure avoids solving the saddle point problem. Furthermore, we show optimal error estimates of velocity and pressure in our mixed finite volume method. Numerical results show optimal orders of error in $L^2$-norm and broken $H^1$-norm for the pressure, and in $H(\Div)$-norm for the velocity.