A Priori Error Estimates for Some Discontinuous Galerkin Immersed Finite Element Methods

Lin, Tao; Yang, Qing; Zhang, Xu

doi:10.1007/s10915-015-9989-3

Cited by 35 publications

(21 citation statements)

References 39 publications

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“…Later in [1,2], Adjerid and Lin extended the IFE approximation to arbitrary polynomial degree, and proved the optimal error estimates in the energy and the L 2 -norms. In the past decade, IFE methods have also been extensively studied for a variety of interface problems in two dimension [19,26,27,31,32,33] and three dimension [23,37].…”

Section: Introductionmentioning

confidence: 99%

Superconvergence of immersed finite element methods for interface problems

Cao

Zhang

2017

Adv Comput Math

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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.

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Section: Introductionmentioning

confidence: 99%

Superconvergence of immersed finite element methods for interface problems

Cao

Zhang

2017

Adv Comput Math

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“…For the first three examples, we consider a diffusion interface problem with a smooth elliptical interface curve which has been reported in [25,27]. Let Ω = [−1, 1] 2 , and the interface Γ is an ellipse centered at (x 0 , y 0 ) = (0, 0) with horizontal semiaxis a = π 6.28 and the vertical semi-axis b = 3 2 a.…”

Section: )mentioning

confidence: 99%

Residual-Based a Posteriori Error Estimation for Immersed Finite Element Methods

Zhang

2019

J Sci Comput

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In this paper we introduce and analyze the residual-based a posteriori error estimation of the partially penalized immersed finite element method for solving elliptic interface problems. The immersed finite element method can be naturally utilized on interface-unfitted meshes. Our a posteriori error estimate is proved to be both reliable and efficient with reliability constant independent of the location of the interface. Numerical results indicate that the efficiency constant is independent of the interface location and that the error estimation is robust with respect to the coefficient contrast.Key words. immersed finite element methods, a posteriori error estimation, interface problems, residual-based. AMS subject classifications. 35R05, 65N15, 65N301. Introduction. Interface problems arise widely in the multi-physics and multimaterial applications in the fluid mechanics and material science. The governing partial differential equations (PDEs) for interface problems are usually characterized with discontinuous coefficients that represent different material properties. The solutions to the interface problems often involve kinks, singularities, discontinuity, and other non-smooth behaviors. It is therefore challenging to obtain accurate numerical approximations for interface problems. Moreover, the complexity of the interface geometry may add an extra layer of difficulty to the numerical approximation.In general, there are two classes of numerical methods for solving interface problems. The first class of methods use interface-fitted meshes, i.e., the meshes are tailored to fit the interface, see the left plot in Figure 1.1. Methods of this type include classical finite element methods [14], discontinuous Galerkin methods [4] and recently developed weak Galerkin methods [30], to name a few. The second class of methods use unfitted meshes which are independent of the interface, as illustrated in the right plot in Figure 1.1. In the past few decades, many numerical methods based on unfitted meshes have been developed. In the finite difference framework, since the pioneering work of immersed boundary method [31] by Peskin, many numerical methods of finite difference type have been developed such as immersed interface method [20,23], matched interface and boundary method [35]. In finite element framework, there are quite a few numerical methods developed, for instance, the general finite element method [5], unfitted finite element method [17], multi-scale finite element method [19], extended finite element method [29], and immersed finite element method (IFEM) [22,24]. A great advantage for unfitted numerical methods is that they can circumvent (re)meshing procedure which can be very expensive especially for time dependent problems with complex interface geometry or for shape optimization processes that require repeated updates of the mesh.The IFEM was first developed in [22] for a one-dimensional elliptic interface problem and then extended to higher-order approximations [1,2,10,11] and to higher- *

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“…Although many works on the 2-D IFE methods have appeared in the literature, such as [22,23,25,27,28,32,43,44] for theoretical analysis and [2,3,7,26,51] for applications, to name just a few, the study on the 3-D case is relatively sparse, see [36] for a linear IFE method and [49] for a trilinear IFE method. Even though the research reported here is within the direction of that in [49], but our work has three distinct new contributions.…”

Section: Introductionmentioning

confidence: 99%