A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients

Wang, Feng; Chen, JinRu; Huang, Peiqi

doi:10.1007/s11425-012-4406-y

Cited by 3 publications

(3 citation statements)

References 33 publications

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“…To this end, one needs to introduce a special projection of the flux solution to the element boundaries as the trace approximation. We refer to [12,6,13,1,29,36,28,40,48] for multigrid algorithms or preconditioning for the CR or CR-related nonconforming finite element methods. In particular, in [13], an optimal-order multigrid method was proposed and analyzed for the lowest-order Raviart-Thomas mixed element based on the equivalence between Raviart-Thomas mixed methods and certain nonconforming methods.…”

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confidence: 99%

BPX Preconditioner for Nonstandard Finite Element Methods for Diffusion Problems

Li¹,

Xie²

2016

SIAM J. Numer. Anal.

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This paper proposes and analyzes an optimal preconditioner for a general linear symmetric positive definite (SPD) system by following the basic idea of the well-known BPX framework.The SPD system arises from a large number of nonstandard finite element methods for diffusion problems, including the well-known hybridized Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM) mixed element methods, the hybridized discontinuous Galerkin (HDG) method, the Weak Galerkin (WG) method, and the nonconforming Crouzeix-Raviart (CR) element method. We prove that the presented preconditioner is optimal, in the sense that the condition number of the preconditioned system is independent of the mesh size. Numerical experiments are provided to confirm the theoretical results.

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confidence: 99%

BPX Preconditioner for Nonstandard Finite Element Methods for Diffusion Problems

Li¹,

Xie²

2016

SIAM J. Numer. Anal.

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“…[1][2][3][4][5] However, the BTBT mechanism inevitably causes the on-state current of TFETs to be lower than that of MOSFETs by approximately two to four orders of magnitude, limiting the applications of TFETs. [6][7][8][9][10][11] In addition, contrary to the theoretical prediction, it has been shown experimentally that the SS values of many fabricated TFET devices are much higher than 60 mV/dec. [12][13][14] Therefore, new channel materials and innovative device structures, such as narrow band-gap materials, [15][16][17] green FETs, [18] multi-gate TFETs, [19][20][21] and line TFETs (LTFETs) [22] have been developed to overcome the shortcomings of the TFETs.…”

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confidence: 70%

Heteromaterial-gate line tunnel field-effect transistor based on Si/Ge heterojunction

Zhang¹,

Liang²,

Wang³

et al. 2017

Chinese Phys. B

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A Si/Ge heterojunction line tunnel field-effect transistor (LTFET) with a symmetric heteromaterial gate is proposed. Compared to single-material-gate LTFETs, the heteromaterial gate LTFET shows an off-state leakage current that is three orders of magnitude lower, and steeper subthreshold characteristics, without degradation in the on-state current. We reveal that these improvements are due to the induced local potential barrier, which arises from the energy-band profile modulation effect. Based on this novel structure, the impacts of the physical parameters of the gap region between the pocket and the drain, including the work-function mismatch between the pocket gate and the gap gate, the type of dopant, and the doping concentration, on the device performance are investigated. Simulation and theoretical calculation results indicate that the gap gate material and n-type doping level in the gap region should be optimized simultaneously to make this region fully depleted for further suppression of the off-state leakage current.

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“…Because these nonconforming spaces are non‐nested, special intergrid transfer operators (restriction and prolongation) are needed in the analysis and implementation of the algorithms. Recently, the robustness of the BPX preconditioner using the non‐nested nonconforming coarse spaces was shown in for the jump coefficient problem .…”

Section: Introductionmentioning

confidence: 99%

Analysis of a multigrid preconditioner for Crouzeix-Raviart discretization of elliptic partial differential equation with jump coefficients

Zhu

2012

Numer. Linear Algebra Appl.

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In this paper, we present a multigrid V -cycle preconditioner for the linear system arising from piecewise linear nonconforming Crouzeix-Raviart discretization of second-order elliptic problems with jump coefficients. The preconditioner uses standard conforming subspaces as coarse spaces. We showed that the convergence rates of the (multiplicative) two-grid and multigrid V -cycle algorithms will deteriorate rapidly because of large jumps in coefficient. However, the preconditioned systems have only a fixed number of small eigenvalues depending on the large jump in coefficient, and the effective condition numbers are independent of the coefficient and bounded logarithmically with respect to the mesh size. As a result, the two-grid or multigrid preconditioned conjugate gradient algorithm converges nearly uniformly. We also comment on some major differences of the convergence theory between the nonconforming case and the standard conforming case. Numerical experiments support the theoretical results.Here,/ is an open polygonal domain and f 2 L 2 . /. We assume that the diffusion coefficientfor m ¤ n. Developing efficient solvers/preconditioners for the CR discretization is not only important for its own sake, but it has several important applications. In particular, it can be used to develop efficient solvers for other discretizations of (1.1). For example, by using the equivalence between CR discretization and mixed methods (cf. [1]), the preconditioners for CR discretization can be applied to solve mixed finite element discretizations for (1.1) (see [2,3] for the multigrid algorithms in the case of smooth coefficients). This relationship is also used in [4] to analyze multigrid algorithms for mixed finite element discretization on adaptively refined meshes. Another important application *Correspondence to: Yunrong Zhu, Physical Sciences 318 P. O. Box 8085, 25 is on the design of efficient multilevel solvers for interior penalty discontinuous Galerkin (IPDG) methods; we refer to [5] for the Poisson problem, and to [6] for (1.1) with jump coefficients.A lot of work has been done to develop multilevel solvers and preconditioners for the piecewise linear CR discretization of (1.1) in the context of smooth (or constant) coefficients. Multigrid algorithms were developed and analyzed in [7][8][9][10][11], and two-level additive Schwarz preconditioners are presented in [12]. Hierarchical and Bramble-Pasciak-Xu (BPX)-type multilevel preconditioners were proposed in [13], and their optimality was shown in [14]. Most of the aforementioned works define the multilevel structure using the natural sequences of nonconforming spaces. Because these nonconforming spaces are non-nested, special intergrid transfer operators (restriction and prolongation) are needed in the analysis and implementation of the algorithms. Recently, the robustness of the BPX preconditioner using the non-nested nonconforming coarse spaces was shown in [15] for the jump coefficient problem (1.1).On the other hand, we observe that the standard piecewise linea...

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A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients

Cited by 3 publications

References 33 publications

BPX Preconditioner for Nonstandard Finite Element Methods for Diffusion Problems

BPX Preconditioner for Nonstandard Finite Element Methods for Diffusion Problems

Heteromaterial-gate line tunnel field-effect transistor based on Si/Ge heterojunction

Analysis of a multigrid preconditioner for Crouzeix-Raviart discretization of elliptic partial differential equation with jump coefficients

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