Element Method For Interface Problems Research Articles

A locally modified second-order finite element method for interface problems and its implementation in 2 dimensions

The locally modified finite element method, which is introduced in Frei and Richter [SIAM J. Numer. Anal.52(2014) 2315–2334], is a simple fitted finite element method that is able to resolve weak discontinuities in interface problems. The method is based on a fixed structured coarse mesh, which is then refined into sub-elements to resolve an interior interface. In this work, we extend the locally modified finite element method in two space dimensions to second order using an isoparametric approach in the interface elements. Thereby we need to take care that the resulting curved edges do not lead to degenerate sub-elements. We prove optimala priorierror estimates in theL2-norm and in a discrete energy norm. Finally, we present numerical examples to substantiate the theoretical findings.

A New Immersed Finite Element Method for Two-Phase Stokes Problems Having Discontinuous Pressure

Abstract In this paper, we develop a new immersed finite element method (IFEM) for two-phase incompressible Stokes flows. We allow the interface to cut the finite elements. On the noninterface element, the standard Crouzeix–Raviart element and the P 0 {P_{0}} element pair is used. On the interface element, the basis functions developed for scalar interface problems (Kwak et al., An analysis of a broken P 1 {P_{1}} -nonconforming finite element method for interface problems, SIAM J. Numer. Anal. (2010)) are modified in such a way that the coupling between the velocity and pressure variable is different. There are two kinds of basis functions. The first kind of basis satisfies the Laplace–Young condition under the assumption of the continuity of the pressure variable. In the second kind, the velocity is of bubble type and is coupled with the discontinuous pressure, still satisfying the Laplace–Young condition. We remark that in the second kind the pressure variable has two degrees of freedom on each interface element. Therefore, our methods can handle the discontinuous pressure case. Numerical results including the case of the discontinuous pressure variable are provided. We see optimal convergence orders for all examples.

An enriched immersed finite element method for interface problems with nonhomogeneous jump conditions

This article presents the first higher degree immersed finite element (IFE) method with proven optimal convergence for elliptic interface problems with nonhomogeneous jump conditions. It also gives the first analysis for the condition numbers of the resulting systems including the optimal upper bounds with respect to the mesh size and its robustness with respect to small-cut interface elements. In this method, jump conditions are approximated optimally by basic IFE and enrichment IFE which are piecewise pth degree polynomial functions constructed by solving local Cauchy problems on interface elements. The proposed IFE method is based on a discontinuous Galerkin formulation on interface elements and a continuous Galerkin formulation on non-interface elements.

An optimally convergent higher-order finite element coupling method for interface and domain decomposition problems

We present a new, optimally accurate finite element method for interface problems that does not require matching interface grids or spatially coincident interfaces. The key idea is to enforce “extended” interface conditions through pullbacks onto the discretized interfaces. In so doing our approach circumvents the accuracy barriers prompted by polytopial approximations of the subdomains and enables high-order finite element solutions without needing more expensive curvilinear maps. Since the discrete interfaces are not required to match, the approach is also appropriate for multiphysics couplings where each subdomain is meshed independently and solved by a separate code. Error analysis reveals that the new approach is well posed and optimally convergent with respect to a broken H1 norm. Numerical examples confirm this result and also indicate optimal convergence in a broken L2 norm.

Modeling and an immersed finite element method for an interface wave equation

The electromagnetic field, which is governed by Maxwell’s equation, plays a key role in plasma simulation. In this article, we first derive the interface conditions when we rewrite the interface Maxwell’s equation, whose problem domain involves complex media such as objects of different materials, into a parabolic–hyperbolic type of interface model, which is a modified wave equation by adding a first order time derivative term due to the lossy medium. Based on the interface conditions and the existing bilinear immersed finite element space for the interface Poisson equation, we propose an immersed finite element method for the spatial discretization of this parabolic–hyperbolic interface equation on a Cartesian mesh independent of the interface. Then we use a second order finite difference method for the temporal discretization in order to develop the full discretization scheme. Compared with the unstructured body-fitting mesh which is needed by the traditional finite element method for interface problems, the Cartesian mesh independent of the interface will significantly benefit the plasma simulation in the electromagnetic field, especially the particle models (such as the particle-in-cell method). This method provides a new efficient way to study varying electromagnetic field with objects of different materials on a Cartesian mesh independent of the interface, hence builds a solid foundation for the further study on the motion of plasma in this electromagnetic field. Numerical examples are provided to demonstrate the features of the proposed method.

A partially penalty immersed Crouzeix-Raviart finite element method for interface problems

The elliptic equations with discontinuous coefficients are often used to describe the problems of the multiple materials or fluids with different densities or conductivities or diffusivities. In this paper we develop a partially penalty immersed finite element (PIFE) method on triangular grids for anisotropic flow models, in which the diffusion coefficient is a piecewise definite-positive matrix. The standard linear Crouzeix-Raviart type finite element space is used on non-interface elements and the piecewise linear Crouzeix-Raviart type immersed finite element (IFE) space is constructed on interface elements. The piecewise linear functions satisfying the interface jump conditions are uniquely determined by the integral averages on the edges as degrees of freedom. The PIFE scheme is given based on the symmetric, nonsymmetric or incomplete interior penalty discontinuous Galerkin formulation. The solvability of the method is proved and the optimal error estimates in the energy norm are obtained. Numerical experiments are presented to confirm our theoretical analysis and show that the newly developed PIFE method has optimal-order convergence in the L^{2} norm as well. In addition, numerical examples also indicate that this method is valid for both the isotropic and the anisotropic elliptic interface problems.

Superconvergence of immersed finite element methods for interface problems

In this article, we study superconvergence properties of immersed finite element methods for the one dimensional elliptic interface problem. Due to low global regularity of the solution, classical superconvergence phenomenon for finite element methods disappears unless the discontinuity of the coefficient is resolved by partition. We show that immersed finite element solutions inherit all desired superconvergence properties from standard finite element methods without requiring the mesh to be aligned with the interface. In particular, on interface elements, superconvergence occurs at roots of generalized orthogonal polynomials that satisfy both orthogonality and interface jump conditions.

Advances of finite element analysis for elliptic, parabolic and Maxwell interface equations

The development of finite element methods for interface problems in the recent two decades is reviewed with an emphasis on elliptic, parabolic and Maxwell interface problems. We summarize some important results, e.g., regularity, variational forms, discretization schemes, numerical quadrature, interface approxima-fition, and convergence analysis. The state of the art for these problems is described along with the history of developments. Finally, we outline some applications governed by the PDE systems with interfaces.

Immersed Finite Element Method for Interface Problems with Algebraic Multigrid Solver

AbstractThis article is to discuss the bilinear and linear immersed finite element (IFE) solutions generated from the algebraic multigrid solver for both stationary and moving interface problems. For the numerical methods based on finite difference formulation and a structured mesh independent of the interface, the stiffness matrix of the linear system is usually not symmetric positive-definite, which demands extra efforts to design efficient multigrid methods. On the other hand, the stiffness matrix arising from the IFE methods are naturally symmetric positive-definite. Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and linear IFE methods for both stationary and moving interface problems. The numerical examples demonstrate the features of the proposed algorithms, including the optimal convergence in both L2and semi-H1norms of the IFE-AMG solutions, the high efficiency with proper choice of the components and parameters of AMG, the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems, and the relationship between the cost and the moving interface location.

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.

The h–p version of the finite element method for problems with interfaces

AbstractRegularities of the solutions of interface problems in two dimensions are described in the frame of the weighted Sobolev spaces and countably normed spaces. Based upon the regularity of solutions the geometric meshes and the distribution of polynomial degrees are properly designed so that the h–p version of the finite element method for interface problems can lead to the exponential rate of convergence. Numerical results on an elliptic equation with interfaces are presented. The optimal mesh factor, optimal degree factors, and optimal layer factors of the geometric mesh in neighbourhoods of singular points having varied intensities are discussed from both theoretical and practical point of view.

Discussion of “ Finite Element Method for Interface Problems ” by Santosh K. Arya and Gilbert A. Hegemier (February, 1982)

Discussion of “ <i>Finite Element Method for Interface Problems</i> ” by Santosh K. Arya and Gilbert A. Hegemier (February, 1982)