Optimal Order of Uniform Convergence for Finite Element Method on Bakhvalov-Type Meshes
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Optimal Order of Uniform Convergence for Finite Element Method on Bakhvalov-Type Meshes Jin Zhang1
· Xiaowei Liu2
Received: 6 March 2020 / Revised: 6 September 2020 / Accepted: 10 September 2020 / Published online: 19 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We propose a new analysis of convergence for a kth order (k ≥ 1) finite element method, which is applied on Bakhvalov-type meshes to a singularly perturbed two-point boundary value problem. A novel interpolant is introduced, which has a simple structure and is easy to generalize. By means of this interpolant, we prove an optimal order of uniform convergence with respect to the perturbation parameter. Numerical experiments illustrate these theoretical results. Keywords Singular perturbation · Convection–diffusion equation · Finite element method · Bakhvalov-type mesh · Uniform convergence Mathematics Subject Classification 65N30 · 65N50
1 Introduction We consider the two-point boundary value problem Lu := −εu − b(x)u + c(x)u = f (x) in Ω := (0, 1), u(0) = u(1) = 0,
(1)
where ε is a positive parameter, b, c and f are sufficiently smooth functions such that b(x) ≥ β > 1 on Ω¯ and 1 c(x) + b (x) ≥ γ > 0 2
on Ω¯
(2)
This research is supported by National Natural Science Foundation of China (11771257, 11601251), Shandong Provincial Natural Science Foundation, China (ZR2017MA003).
B
Jin Zhang [email protected] Xiaowei Liu [email protected]
1
School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
2
School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China
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2
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Journal of Scientific Computing (2020) 85:2
with some constants β and γ . The condition (2) ensures that the boundary value problem has a unique solution. In the cases of interest the diffusion parameter ε can be arbitrarily small and satisfies 0 < ε 1. Thus this problem is singularly perturbed and its solution typically features a boundary layer of width O(ε ln(1/ε)) at x = 0 (see [13]). Solutions to singularly perturbed problems are characterized by the presence of boundary or interior layers, where solutions change rapidly. Numerical solutions of these problems are of significant mathematical interest. Classical numerical methods are often inappropriate, because in practice it is very unlikely that layers are fully resolved by common meshes. Hence specialised numerical methods are designed to compute accurate approximate solutions in an efficient way. For example, standard numerical methods on layer-adapted meshes, which are fine in layer regions and standard outside, are commonly used; see [11,13] and many references therein. On these meshes, classical numerical methods are uniformly convergent with respect to the singular perturbation parameter; see [9]. Among them, there are two kinds of popular grids: Bakhvalov-type meshes (B-meshes) and Shishkin-type meshes (S-meshes); see [9]. The accuracy of finite differenc
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