An equation decomposition method for the numerical solution of a fourth‐order elliptic singular perturbation problem

Han, Houde; Huang, Zhongyi

doi:10.1002/num.20666

Cited by 11 publications

(13 citation statements)

References 14 publications

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“…Furthermore, the method was applied to solve the steady MHD duct fiow problems with boundary layers successfully [19]. TFPM also works well for time-dependent problem [21] and fourth-order singular perturbation problem [17]. More related work can be found in two review papers [5,37] and the references therein.…”

Section: Introductionmentioning

confidence: 98%

Tailored Finite Cell Method for Solving Helmholtz Equation in Layered Heterogeneous Medium

Huang

Yang

2012

JCM

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In this paper, we propose a tailored finite cell method for the computation of twodimensional Helmholtz equation in layered heterogeneous medium. The idea underlying the method is to construct a numerical scheme based on a local approximation of the solution to Helmholtz equation. This provides a computational tool of achieving high accuraicy with coarse mesh even for large wave number (high frequency). The stability analysis and error estimates of this method are also proved. We present several numerical results to show its efficiency and accuracy.Mathematics subject classification: 65N35, 35L10.

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Section: Introductionmentioning

confidence: 98%

Tailored Finite Cell Method for Solving Helmholtz Equation in Layered Heterogeneous Medium

Huang

Yang

2012

JCM

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“…To solve numerically singular perturbation problems of second-order elliptic equations, the TFPM was applied systematically by Han and Huang [15-17, 19, 31], Shih, Kellogg and Chang [55], Shih, Kellogg and Tsai [56]. Han and Huang proposed the TFPM scheme for the numerical solution of a singular perturbation problem of fourth-order elliptic equation [18], and an iterative TFPM scheme of which was given by Han, Huang and Zhang [23]. Hsieh, Shih and Yang [29] proposed a TFPM scheme for solving the steady magnetohydrodynamic (MHD) Duct ow problem with boundary layers, which is a singularly perturbed system of second-order elliptic equations.…”

Section: Introductionmentioning

confidence: 99%

The Tailored Finite Point Method

Han

Huang

2014

Computational Methods in Applied Mathematics

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Abstract:In this paper, a brief review of tailored nite point methods (TFPM) is given. The TFPM is a new approach to construct the numerical solutions of partial di erential equations. The TFPM has been tailored based on the local properties of the solution for each given problem. Especially, the TFPM is very e cient for solutions which are not smooth enough, e.g., for solutions possessing boundary/interior layers or solutions being highly oscillated. Recently, the TFPM has been applied to singular perturbation problems, the Helmholtz equation with high wave numbers, the rst-order wave equation in high frequency cases, transport equations with interface, second-order elliptic equations with rough or highly oscillatory coe cients, etc.

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“…The TFPM was successfully applied by Han, Huang ,and Kellogg to solve the Hemker problem [9,11]; they won the Hemker prize at the international conference BAIL 2008. Later the TFPM was developed to solve the second-order elliptic singular pertubation problem [6,21], the first order wave equation [13], the one-dimension Helmholtz equation with high wave number [5], second order elliptic equations with rough or highly oscillatory coefficients [10] and so on [7,8]. For the one dimensional singular perturbation problem the TFPM is close to the method of "exponential fitting" discussed in [1,3,16,20].…”

Section: Introductionmentioning

confidence: 99%

A Parameter-Uniform Tailored Finite Point Method for Singularly Perturbed Linear ODE Systems

Han

Miller

Tang

2013

JCM

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In scientific applications from plasma to chemical kinetics, a wide range of temporal scales can present in a system of differential equations. A major difficulty is encountered due to the stiffness of the system and it is required to develop fast numerical schemes that are able to access previously unattainable parameter regimes. In this work, we consider an initial-final value problem for a multi-scale singularly perturbed system of linear ordinary differential equations with discontinuous coefficients. We construct a tailored finite point method, which yields approximate solutions that converge in the maximum norm, uniformly with respect to the singular perturbation parameters, to the exact solution. A parameter-uniform error estimate in the maximum norm is also proved. The results of numerical experiments, that support the theoretical results, are reported.Mathematics subject classification: 37M05, 65G99.

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An equation decomposition method for the numerical solution of a fourth‐order elliptic singular perturbation problem

Cited by 11 publications

References 14 publications

Tailored Finite Cell Method for Solving Helmholtz Equation in Layered Heterogeneous Medium

Tailored Finite Cell Method for Solving Helmholtz Equation in Layered Heterogeneous Medium

The Tailored Finite Point Method

A Parameter-Uniform Tailored Finite Point Method for Singularly Perturbed Linear ODE Systems

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