An Analysis of a Broken $P_1$-Nonconforming Finite Element Method for Interface Problems

Kwak, Dae‐Young; Wee, Kye T.; Chang, Kwang Sung

doi:10.1137/080728056

Cited by 80 publications

(81 citation statements)

References 30 publications

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“…However, when the basis functions are modified so that they satisfy the interface conditions, they seem to work well [10], [20], [21]. These methods were extended to the case of Crouzeix-Raviart P 1 nonconforming finite element method [12] by Kwak et al [18], and to the problems with nonzero jumps in [7]. Some related works on interface problems can be found in [5], [16], [17], [19], [22], [23], [26].…”

Section: Introductionmentioning

confidence: 99%

A Modified $P_1$ -- Immersed Finite Element Method

Kwak¹,

Lee²

2015

Int. J. of Pure and Appl. Math.

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In recent years, the immersed finite element methods (IFEM) introduced in [20], [21] 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 to 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.

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Section: Introductionmentioning

confidence: 99%

A Modified $P_1$ -- Immersed Finite Element Method

Kwak¹,

Lee²

2015

Int. J. of Pure and Appl. Math.

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“…From these assumptions, we know that the solution u(x) ∈ H 2 (Ω i ) for i = 1, 2. There are many applications of such an interface problem; see, e.g., the works of Sutton and Balluffi (1995), Zienkiewicz and Taylor (2000), Li and Ito (2006) or Kwak and Chang (2010) and the references therein. Many numerical methods have been developed for solving such an important problem.…”

Section: Introductionmentioning

confidence: 99%

Accurate gradient computations at interfaces using finite element methods

Qin

Wang

et al. 2017

International Journal of Applied Mathematics and Computer Science

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New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is to get not only an accurate solution, but also an accurate first order derivative at the interface (from each side). The key in 1D is to use the idea of Wheeler (1974). For 2D interface problems, the point is to introduce a small tube near the interface and propose the gradient as part of unknowns, which is similar to a mixed finite element method, but only at the interface. Thus the computational cost is just slightly higher than in the standard finite element method. We present a rigorous one dimensional analysis, which shows a second order convergence order for both the solution and the gradient in 1D. For two dimensional problems, we present numerical results and observe second order convergence for the solution, and super-convergence for the gradient at the interface.

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“…However, these can have adverse effects on conditioning and require the determination of the stabilization parameters. Instead of using Lagrange multipliers or stabilization, the methods of [66,43,58,80,71,44,74] alter the basis functions to either satisfy the constraints directly, or simplify the process of doing so. In this regard, such methods represent the finite element analogues of the IIM.…”

Section: Introductionmentioning

confidence: 99%

A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions

Hellrung

Wang

Sifakis

et al. 2012

Journal of Computational Physics

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Keywords: Elliptic interface problems Embedded interface methods Virtual node methods Variational methods Multigrid methods a b s t r a c tWe present a numerical method for the variable coefficient Poisson equation in threedimensional irregular domains and with interfacial discontinuities. The discretization embeds the domain and interface into a uniform Cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions. The matrix associated with the discretization is symmetric positive definite and equal to the standard 7-point finite difference Poisson stencil away from embedded interfaces and boundaries. Numerical evidence suggests second order accuracy in the L 1 -norm. Our approach improves the treatment of Dirichlet and jump constraints in the recent work of Bedrossian et al. [1] and introduces innovations necessary in three dimensions. Specifically, we construct new constraint-based Lagrange multiplier spaces that significantly improve the conditioning of the associated linear system of equations; we provide a method for cell-local polyhedral approximation to the zero isocontour surface of a level set needed for three-dimensional embedding; and we show that the new Lagrange multiplier spaces naturally lead to a class of easy-to-implement multigrid methods that achieve near optimal efficiency, as shown by numerical examples. For the specific case of a continuous Poisson coefficient in interface problems, we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump constraints. As in [1], this is used in a discontinuity removal technique that yields the standard 7-point stencil across the interface and only requires a modification to the right-hand side of the linear system.

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An Analysis of a Broken $P_1$-Nonconforming Finite Element Method for Interface Problems

Cited by 80 publications

References 30 publications

A Modified $P_1$ -- Immersed Finite Element Method

A Modified $P_1$ -- Immersed Finite Element Method

Accurate gradient computations at interfaces using finite element methods

A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions

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