High-order extended finite element methods for solving interface problems

Xiao, Yuanming; Xu, Jinchao; Wang, Fei

doi:10.1016/j.cma.2020.112964

Cited by 44 publications

(24 citation statements)

References 35 publications

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“…The XFE space has optimal approximation quality for piecewise smooth functions in H p ( 1 ∪ 2 ). The following theorem is proved in [28] as an analogue of Cea's lemma.…”

Section: Lemma 2 Letmentioning

confidence: 99%

“…The analysis reveals optimal order error bounds with respect to h for the geometry approximation and for the finite element approximation. In [14,28], various issues related to unfitted methods was addressed, including the dependence of error estimates on the diffusion coefficients, the condition number of the discrete system, and the choice of stabilization parameters.…”

Section: Introductionmentioning

confidence: 99%

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High-Order Finite Element Methods for Interface Problems: Theory and Implementations

Xiao

Zhai

Zhang

et al. 2020

Lecture Notes in Computational Science and Engineering

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We propose arbitrary order extended finite element (XFE) methods for solving both two and three dimensional elliptic interface problems. The meshes in our methods do not need to fit the interface and discontinuous Galerkin (DG) schemes are used in the vicinity of the interface to incorporate the transmission conditions between subdomains. Optimal error estimates in the piecewise H1-norm and in the L2-norm are rigorously proved for the scheme. Two implementation aspects are addressed in this paper: (1) An optimal multigrid solver is introduced for the generated linear system, which converges uniformly with respect to the mesh size. (2) A high order algorithm is presented to compute the integral over elements intersecting the interface. Numerical experiments are reported to support the theoretical results.

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“…The XFE space has optimal approximation quality for piecewise smooth functions in H p ( 1 ∪ 2 ). The following theorem is proved in [28] as an analogue of Cea's lemma.…”

Section: Lemma 2 Letmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High-Order Finite Element Methods for Interface Problems: Theory and Implementations

Xiao

Zhai

Zhang

et al. 2020

Lecture Notes in Computational Science and Engineering

Self Cite

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show abstract

“…For finite element methods, roughly speaking, there are two approaches to recover the optimal convergence. The first one is to enrich the standard finite element space by augmenting extra degrees of freedom on interface elements and if necessary add some integral terms into the variational form to weakly enforce interface conditions, for example, the extended finite element method [9], the unfitted Nitsche's method [17,37,38,5] and the enriched finite element method [36]. The other approach is to modify the traditional finite element space on interface elements according to interface conditions to achieve the optimal approximation, while keeping the degrees of freedom and the structure unchanged, for example, the multiscale finite method [7] and the IFE method [29,32,31,19,18,14,15] that we utilize in this paper.…”

Section: Introductionmentioning

confidence: 99%

A new parameter free partially penalized immersed finite element and the optimal convergence analysis

Ji¹,

Wang²,

Chen³

et al. 2021

Preprint

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This paper presents a new parameter free partially penalized immersed finite element method and convergence analysis for solving second order elliptic interface problems. The optimal approximation capabilities of the immersed finite element space is proved via a novel new approach that is much simpler than that in the literature. A new trace inequality which is necessary to prove the optimal convergence of immersed finite element methods is established on interface elements. Optimal error estimates are derived rigorously even though the curved interface is approximated by line segments. The new method and analysis have also been extended to problems with variable coefficients. Numerical examples are also provided to confirm the theoretical analysis and efficiency of the new method.Keywords interface problem • partially penalized immersed finite element • unfitted mesh • trace inequality • optimal error estimates of IFEM

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“…The design and analysis of finite element methods on unfitted meshes with optimal convergence rates was started in [2,3]. Since then, many unfitted mesh finite element methods have been developed, for example, the unfitted Nitsche's method [18,34,35,8], the extended finite element method [12], the enriched finite element method [33], the multiscale finite method [10], the finite element method for high-contrast problems [17] and the immersed finite element (IFE) methods [27,29,28,20,19,16,15,21], to name only a few. In this paper, we focus on the IFE method.…”

Section: Introductionmentioning

confidence: 99%

An immersed Raviart-Thomas mixed finite element method for elliptic interface problems on unfitted meshes

Ji¹

2021

Preprint

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This paper presents a lowest-order immersed Raviart-Thomas mixed triangular finite element method for solving elliptic interface problems on unfitted meshes independent of the interface. In order to achieve the optimal convergence rates on unfitted meshes, an immersed finite element finite (IFE) is constructed by modifying the traditional Raviart-Thomas element. Some important properties are derived including the unisolvence of IFE basis functions, the optimal approximation capabilities of the IFE space and the corresponding commuting digram. Optimal error estimates are rigorously proved for the mixed IFE method and some numerical examples are also provided to validate the theoretical analysis.

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High-order extended finite element methods for solving interface problems

Cited by 44 publications

References 35 publications

High-Order Finite Element Methods for Interface Problems: Theory and Implementations

High-Order Finite Element Methods for Interface Problems: Theory and Implementations

A new parameter free partially penalized immersed finite element and the optimal convergence analysis

An immersed Raviart-Thomas mixed finite element method for elliptic interface problems on unfitted meshes

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