On the extrapolation for a singularly perturbed boundary value problem

Vulanović, Relja; Herceg, Dejana; Petrović, Nenad

doi:10.1007/bf02238193

Cited by 30 publications

(31 citation statements)

References 4 publications

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“…This is the two-dimensional analog of (4) - (5) Once again, this problem is singularly perturbed since (15) reduces to the analog of (6) (20) by taking M large can only be done at the expense of adding more terms to (19). The situation is analogous to that discussed i n R emark 3.…”

Section: The Heat Transfer Problemmentioning

confidence: 94%

The hp finite element method for problems in mechanics with boundary layers

Schwab

Suri

Xenophontos

1998

Computer Methods in Applied Mechanics and Engineering

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We consider the numerical approximation of boundary layer phenomena occuring in many singularly perturbed problems in mechanics, such as plate and shell problems. We present guidelines for the e ective resolution of such l a yers in the context of exisiting, commercial p and hp nite element (FE) version codes. We show that if high order, \spectral" elements are available, then just two elements are su cient to approximate these layers at a near-exponential rate, independently of the problem parameters thickness or Reynolds number. We present hp mesh design principles for situations where both corner singularities and boundary layers are present.

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Section: The Heat Transfer Problemmentioning

confidence: 94%

The hp finite element method for problems in mechanics with boundary layers

Schwab

Suri

Xenophontos

1998

Computer Methods in Applied Mechanics and Engineering

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“…Most available references analyze the convergence of finite difference or finite element schemes of fixed (usually low) polynomial degree in conjunction with various mesh refinements (the h version); see, e.g., [4,6,13,18,19], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…In practical terms, the amount of discretization required with such schemes for satisfactory resolution of the boundary layers may be infeasible when d is very small. On the other hand, strongly graded d-dependent mesh refinement, like the one from [20] presented in §6, does lead to robust convergence, at an optimal rate that is algebraic (see [4,16,18,19,20], where this and other graded meshes are discussed).…”

Section: Introductionmentioning

confidence: 99%

The $p$ and $hp$ versions of the finite element method for problems with boundary layers

1996

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Abstract. We study the uniform approximation of boundary layer functions exp(−x/d) for x ∈ (0, 1), d ∈ (0, 1], by the p and hp versions of the finite element method. For the p version (with fixed mesh), we prove super-exponential convergence in the range p + 1/2 > e/(2d). We also establish, for this version, an overall convergence rate of O(p −1 √ ln p) in the energy norm error which is uniform in d, and show that this rate is sharp (up to the √ ln p term) when robust estimates uniform in d ∈ (0, 1] are considered. For the p version with variable mesh (i.e., the hp version), we show that exponential convergence, uniform in d ∈ (0, 1], is achieved by taking the first element at the boundary layer to be of size O(pd).Numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates, even when few degrees of freedom are used and when d is as small as, e.g., 10 −8 . They also illustrate the superiority of the hp approach over other methods, including a low-order h version with optimal "exponential" mesh refinement.The estimates established in this paper are also applicable in the context of corresponding spectral element methods.

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“…Vulanovi6 et al [30] and Vulanovi6 [29]. Here we shall talk about classical finite difference schemes on special nonequidistant meshes.…”

Section: Ig'(x)imentioning

confidence: 99%

“…In the nonlinear case this problem was considered in Vulanovi6 [27], * This research was partly supported by NSF and SIZ for Science of SAP Vojvodina through funds made available to the U.S.-Yugoslav Joint Board on Scientific and Technological Cooperation (grants JF 544, JF 799) Boglaev [-3], Herceg [15], Vulanovi6 et al [30] and Vulanovi6 [29]. We note that the problem (1.1) with conditions (1.2) and (1.6) O<~2<__cu (x,u), (x,u)elxR occurs frequently in the literature.…”

Section: Ig'(x)imentioning

confidence: 99%