An Analysis of broken P1P_1-Nonconforming Finite Element Method For Interface Problems

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

A stabilized P1P_1 immersed finite element method for the interface elasticity problems

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 Lam\`e parameters $\mu$ and $\lambda$ and locking free as $\lambda\to\infty$.Comment: Submitted to M2an on May 18 2015. Added a new author (Dae H. Kyeong

Immersed finite element method for eigenvalue problems in elasticity

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.Comment: 17 pages, 11 figures, 1 tabl

An IMPES scheme for a two-phase flow in heterogeneous porous media using a structured grid

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

Immersed Finite Element Method for Eigenvalue Problem

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

Residual-based a Posteriori Error Estimate for Interface Problems: Nonconforming Linear Elements

In this paper, we study a modified residual-based a posteriori error estimator for the nonconforming linear finite element approximation to the interface problem. The reliability of the estimator is analyzed by a new and direct approach without using the Helmholtz decomposition. It is proved that the estimator is reliable with constant independent of the jump of diffusion coefficients across the interfaces, without the assumption that the diffusion coefficient is quasi-monotone. Numerical results for one test problem with intersecting interfaces are also presented

A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problems

In this paper, a stabilized extended finite element method is proposed for Stokes interface problems on unfitted triangulation elements which do not require the interface align with the triangulation. The velocity solution and pressure solution on each side of the interface are separately expanded in the standard nonconforming piecewise linear polynomials and the piecewise constant polynomials, respectively. Harmonic weighted fluxes and arithmetic fluxes are used across the interface and cut edges (segment of the edges cut by the interface), respectively. Extra stabilization terms involving velocity and pressure are added to ensure the stable inf-sup condition. We show a priori error estimates under additional regularity hypothesis. Moreover, the errors {in energy and $L^2$ norms for velocity and the error in $L^2$ norm for pressure} are robust with respect to the viscosity {and independent of the location of the interface}. Results of numerical experiments are presented to {support} the theoretical analysis.Comment: 36 page

Finite Element Methods for Interface Problems: Robust Residual-Based A Posteriori Error Estimates

For elliptic interface problems, this paper studies residual-based a posteriori error estimations for various finite element approximations. For the conforming and the Raviart-Thomas mixed elements in two-dimension and for the Crouzeix-Raviart nonconforming and the discontinuous Galerkin elements in both two- and three-dimensions, the global reliability bounds are established with constants independent of the jump of the diffusion coefficient. Moreover, we obtain these estimates with no assumption on the distribution of the diffusion coefficient

A new cubic nonconforming finite element on rectangles

A new nonconforming rectangle element with cubic convergence for the energy norm is introduced. The degrees of freedom (DOFs) are defined by the twelve values at the three Gauss points on each of the four edges. Due to the existence of one linear relation among the above DOFs, it turns out the DOFs are eleven. The nonconforming element consists of P_2\oplus \Span\{x^3y-xy^3\}. We count the corresponding dimension for Dirichlet and Neumann boundary value problems of second-order elliptic problems. We also present the optimal error estimates in both broken energy and L_2(\O) norms. Finally, numerical examples match our theoretical results very well

A nonconforming immersed finite element method for elliptic interface problems

A new immersed finite element (IFE) method is developed for second-order elliptic problems with discontinuous diffusion coefficient. The IFE space is constructed based on the rotated Q1 nonconforming finite elements with the integral-value degrees of freedom. The standard nonconforming Galerkin method is employed in this IFE method without any penalty stabilization term. Error estimates in energy and L2 norms are proved to be better than $O(h\sqrt{|\log h|})$ and $O(h^2|\log h|)$, respectively, where the logarithm factors reflect jump discontinuity. Numerical results are reported to confirm our analysis