An immersed finite element space and its approximation capability

This article discusses an immersed finite element (IFE) space introduced for solving a second‐order elliptic boundary value problem with discontinuous coefficients (interface problem). The IFE space is nonconforming and its partition can be independent of the interface. The error estimates for the interpolation of a function in the usual Sobolev space indicate that this IFE space has an approximation capability similar to that of the standard conforming linear finite element space based on body‐fit partitions. Numerical examples of the related finite element method based on this IFE space are provided. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 338–367, 2004     

A selective immersed discontinuous Galerkin method for elliptic interface problems

This article proposes a selective immersed discontinuous Galerkin method based on bilinear immersed finite elements (IFE) for solving second‐order elliptic interface problems. This method applies the discontinuous Galerkin formulation wherever selected, such as those elements around an interface or a singular source, but the regular Galerkin formulation everywhere else. A selective bilinear IFE space is constructed and applied to the selective immersed discontinuous Galerkin method based on either the symmetric or nonsymmetric interior penalty discontinuous Galerkin formulation. The new method can solve an interface problem by a rectangular mesh with local mesh refinement independent of the interface even if its geometry is nontrivial. Meanwhile, if desired, its computational cost can be maintained very close to that of the standard Galerkin IFE method. It is shown that the selective bilinear IFE space has the optimal approximation capability expected from piecewise bilinear polynomials. Numerical examples are provided to demonstrate features of this method, including the effectiveness of local mesh refinement around the interface and the sensitivity to the penalty parameters. Copyright © 2013 John Wiley & Sons, Ltd.     

Approximation capabilities of immersed finite element spaces for elasticity Interface problems

We construct and analyze a group of immersed finite element (IFE) spaces formed by linear, bilinear, and rotated Q₁ polynomials for solving planar elasticity equation involving interface. The shape functions in these IFE spaces are constructed through a group of approximate jump conditions such that the unisolvence of the bilinear and rotated Q₁ IFE shape functions are always guaranteed regardless of the Lamé parameters and the interface location. The boundedness property and a group of identities of the proposed IFE shape functions are established. A multi‐point Taylor expansion is utilized to show the optimal approximation capabilities for the proposed IFE spaces through the Lagrange type interpolation operators.     

The convergence of the bilinear and linear immersed finite element solutions to interface problems

This article analyzes the error in both the bilinear and linear immersed finite element (IFE) solutions for second‐order elliptic boundary problems with discontinuous coefficients. The discontinuity in the coefficients is supposed to happen across general curves, but the mesh of the IFE methods can be allowed not to align with the curve of discontinuity. It has been shown that the bilinear and linear IFE solutions converge to the exact solution under the usual assumptions about the meshes and regularity.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 312–330 2012     

Linear and bilinear immersed finite elements for planar elasticity interface problems

This article is to discuss the linear (which was proposed in  and ) and bilinear immersed finite element (IFE) methods for solving planar elasticity interface problems with structured Cartesian meshes. Basic features of linear and bilinear IFE functions, including the unisolvent property, will be discussed. While both methods have comparable accuracy, the bilinear IFE method requires less time for assembling its algebraic system. Our analysis further indicates that the bilinear IFE functions are guaranteed to be applicable to a larger class of elasticity interface problems than linear IFE functions. Numerical examples are provided to demonstrate that both linear and bilinear IFE spaces have the optimal approximation capability, and that numerical solutions produced by a Galerkin method with these IFE functions for elasticity interface problem also converge optimally in both L²$L^2$ and semi-H¹$H^1$ norms.     

Quadratic immersed finite element spaces and their approximation capabilities

This paper discusses a class of quadratic immersed finite element (IFE) spaces developed for solving second order elliptic interface problems. Unlike the linear IFE basis functions, the quadratic IFE local nodal basis functions cannot be uniquely defined by nodal values and interface jump conditions. Three types of one dimensional quadratic IFE basis functions are presented together with their extensions for forming the two dimensional IFE spaces based on rectangular partitions. Approximation capabilities of these IFE spaces are discussed. Finite element solutions based on these IFE for representative interface problems are presented to further illustrate capabilities of these IFE spaces. Dedicated to the 60th birthday of Charles A. Micchelli Mathematics subject classifications (2000) 65N15, 65N30, 65N50, 65Z05. Yanping Lin: Supported by NSERC. Weiwei Sun: This work was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (project CityU 1141/01P).     

Immersed finite element methods for parabolic equations with moving interface

This article presents three Crank‐Nicolson‐type immersed finite element (IFE) methods for solving parabolic equations whose diffusion coefficient is discontinuous across a time dependent interface. These methods can use a fixed mesh because IFEs can handle interface jump conditions without requiring the mesh to be aligned with the interface. These methods will be compared analytically in the sense of accuracy and computational cost. Numerical examples are provided to demonstrate features of these three IFE methods. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Inf‐sup stability of Petrov‐Galerkin immersed finite element methods for one‐dimensional elliptic interface problems

In this article, we analyze the Petrov‐Galerkin immersed finite element method (PG‐IFEM) when applied to one‐dimensional elliptic interface problems. In the PG‐IFEM (T. Hou, X. Wu and Y. Zhang, Commun. Math. Sci., 2 (2004), 185‐205, and S. Hou and X. Liu, J. Comput. Phys., 202 (2005), 411‐445), the classic immersed finite element (IFE) space was taken as the trial space while the conforming linear finite element space was taken as the test space. We first prove the inf‐sup condition of the PG‐IFEM and then show the optimal error estimate in the energy norm. We also show the optimal estimate of the condition number of the stiffness matrix. The results are extended to two dimensional problems in a special case.     

A partially penalized P1/CR immersed finite element method for planar elasticity interface problems

We propose a partially penalized P₁/CR immersed finite element (IFE) method with midpoint values on edges as degrees of freedom for CR elements to solve planar elasticity interface problems. Optimal approximation errors in L² norm and H¹ semi‐norm are obtained for the P₁/CR IFE spaces. Moreover, by adding some stabilization terms on the edges of interface elements, we derive an optimal error estimate for the P₁/CR IFE method. Our method differs from the method with average values on edges as degrees of freedom for P₁/CR elements in Qin et al.'s study, where no approximation theoretical result was presented. Numerical examples confirm our theoretical results.     

A second order isoparametric finite element method for elliptic interface problems

A second order isoparametric finite element method (IPFEM) is proposed for elliptic interface problems. It yields better accuracy than some existing second-order methods, when the coefficients or the flux across the immersed curved interface is discontinuous. Based on an initial Cartesian mesh, a mesh optimization strategy is presented by employing curved boundary elements at the interface, and an incomplete quadratic finite element space is constructed on the optimized mesh. It turns out that the number of curved boundary elements is far less than that of the straight one, and the total degree of freedom is almost the same as the uniform Cartesian mesh. Numerical examples with simple and complicated geometrical interfaces demonstrate the efficiency of the proposed method.     

基于Crouzeix-Raviart元的界面浸入有限元方法及其收敛性分析

本文对具间断系数的二阶椭圆界面问题提出一种浸入有限元方法(theimmersed finite element method), 即在界面单元上采用依赖于界面的线性多项式空间离散, 而在非界面单元上采用Crouzeix-Raviart非协调元离散. 论证表明, 该方法具有对界面问题解的最优L²-模和H¹-模收敛精度.     

Analysis of optimal preconditioners for CutFEM

In this article, we consider a class of unfitted finite element methods for scalar elliptic problems. These so-called CutFEM methods use standard finite element spaces on a fixed unfitted triangulation combined with the Nitsche technique and a ghost penalty stabilization. As a model problem we consider the application of such a method to the Poisson interface problem. We introduce and analyze a new class of preconditioners that is based on a subspace decomposition approach. The unfitted finite element space is split into two subspaces, where one subspace is the standard finite element space associated to the background mesh and the second subspace is spanned by all cut basis functions corresponding to nodes on the cut elements. We will show that this splitting is stable, uniformly in the discretization parameter and in the location of the interface in the triangulation. Based on this we introduce an efficient preconditioner that is uniformly spectrally equivalent to the stiffness matrix. Using a similar splitting, it is shown that the same preconditioning approach can also be applied to a fictitious domain CutFEM discretization of the Poisson equation. Results of numerical experiments are included that illustrate optimality of such preconditioners for the Poisson interface problem and the Poisson fictitious domain problem.     

A finite element method for interface problems in domains with smooth boundaries and interfaces

This paper is concerned with the analysis of a finite element method for nonhomogeneous second order elliptic interface problems on smooth domains. The method consists in approximating the domains by polygonal domains, transferring the boundary data in a natural way, and then applying a finite element method to the perturbed problem on the approximate polygonal domains. It is shown that the error in the finite element approximation is of optimal order for linear elements on a quasiuniform triangulation. As such the method is robust in the regularity of the data in the original problem.     

半线性椭圆问题Petrov-Galerkin逼近及亏量迭代

本文考虑了二阶半线性椭圆问题的Petrov-Galerkin逼近格式,用双二次多项式空间作为形函数空间,用双线性多项式空间作为试探函数空间,证明了此逼近格式与标准的二次有限元逼近格式有同样的收敛阶.并且根据插值算子的逼近性质,进一步证明了半线性有限元解的亏量迭代序列收敛到Petrov-Galerkin解.     

一类非定常对流占优扩散问题差分-流线扩散法的后处理

讨论了一类非定常对流占优扩散方程的差分-流线扩散格式(FDSD),利用插值后处理技术,提高了特殊网格下该FDSD格式在双线性元空间的精度,从而按L∞(L2(Ω) 模达到最优．     

Some remarks on the flux-free finite element method for immiscible two-fluid flows

We give some theoretical considerations on the the flux-free finite element method for the generalized Stokes interface problem arising from the immiscible two-fluid flow problems. In the flux-free finite element method, the flux constraint is posed as another Lagrange multiplier to keep the zero-flux on the interface. As a result, the mass of each fluid is expected to be preserved at every time step. We first study the effect of discontinuous coefficients (viscosity and density) on the error of the standard finite element approximations very carefully. Then, the analysis is extended to the flux-free finite element method.     

Convergence of a finite element scheme for the two-dimensional time-dependent Schrödinger equation in a long strip

This paper addresses the finite element method for the two-dimensional time-dependent Schrödinger equation on an infinite strip by using artificial boundary conditions. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying the Crank-Nicolson scheme in time and a bilinear or quadratic finite element approximation in space. This scheme, by a rigorous analysis, has been proved to be unconditionally stable and convergent, and its convergence order has also been obtained. Finally, two numerical examples are given to verify the accuracy of the scheme.     

抛物型界面问题的变网格有限元方法

本文针对抛物型界面问题,提出了一种线性三角形变网格有限元方法.其主要思路是针对空间变量采用有限元离散,对时间变量采用差分离散,但是不同时刻的有限元剖分网格可以不同.在不引入Ritz投影这一传统分析工具的情况下,得到了最优误差估计结果,使得证明过程更加简洁.给出的数值算例验证了理论分析的正确性.     

MODIFIED MORLEY ELEMENT METHOD FOR A FOURTH ORDER ELLIPTIC SINGULAR PERTURBATION PROBLEM

This paper proposes a modified Morley element method for a fourth order ellipticsingular perturbation problem. The method also uses Morley element or rectangle Morleyelement, but linear or bilinear approximation of finite element functions is used in the lowerpart of the bilinear form. It is shown that the modified method converges uniformly in theperturbation parameter.     

有限元近似可计算的误差界

本文研究了有限元近似可计算的误差界，利用“二次插值过渡”方法，获得二维线性、双线性有限元和三维三线性有限元的新的插值常数估计值．理论分析和数值实验表明该结果是有效的，发展了P．Arbenz等人的工作．