SDFEM for singularly perturbed parabolic initial-boundary-value problems on equidistributed grids
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SDFEM for singularly perturbed parabolic initial‑boundary‑value problems on equidistributed grids D. Avijit1 · S. Natesan1 Received: 4 March 2020 / Revised: 13 June 2020 / Accepted: 27 July 2020 © Istituto di Informatica e Telematica (IIT) 2020
Abstract In this article, we study the convergence properties of the streamline-diffusion finite element method (SDFEM) for singularly perturbed 1D parabolic convection–diffusion initial-boundary-value problems. To discretize the spatial domain, we use a layer-adaptive nonuniform grids obtained through the equidistribution principle, whereas uniform grid is used in the time direction. Here, we use the backward-Euler method to discretize the temporal derivative and the SDFEM scheme for the spatial derivatives. The proposed method is uniformly convergent with first-order in time and second-order in space. Keywords Singularly perturbed 1D parabolic PDEs · Streamline-diffusion finite element method · Grid equidistribution · Uniform convergence Mathematics Subject Classification 34B27 · 65L05 · 65L11 · 65L20 · 65L50
1 Introduction In this article, we establish a uniformly stable numerical method for the following singularly perturbed 1D parabolic IBVP:
⎧ Lu ∶≡ 𝜕u − 𝜀 𝜕 2 u + b(x) 𝜕u + c(x)u = f (x, t), ⎪ 𝜕t 𝜕x 𝜕x2 ⎨ u(x, 0) = u (x), x ∈ Ω, 0 ⎪ ⎩ u(0, t) = u(1, t) = 0, t ∈ [0, T]
(x, t) ∈ G = Ω × (0, T], (1)
* S. Natesan [email protected] D. Avijit [email protected] 1
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, India
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where Ω = (0, 1) and 0 < 𝜀 ≪ 1 is a very small parameter, satisfies 0 < 𝜀 ≪ 𝛽 < b(x), ∀ x ∈ Ω and the coefficients b(x), c(x) and f(x, t) are assumed to be sufficiently smooth satisfying b(x) > 𝛽 > 0 and c(x) > 𝛾 > 0 for x ∈ Ω. Singularly perturbed 1D parabolic problems having variable coefficients can be seen in many application areas [5, 8, 13, 14], for example, in the convective-heat transport problems with high Peclét numbers, the drift diffusion equation of semiconductor device modeling, the mathematical modeling of unsteady viscous flow problems with high Reynolds number. The main characteristics of the IBVP (1) is, the solution of this problem possesses a regular boundary layer at the boundary x = 1 , when the parameter 𝜀 → 0 [14]. Due to the presence of boundary layer, the classical discretization methods like finite difference methods (FDMs) or finite element methods (FEMs) are unable to provide the required approximate solution unless the number of grid points become large [14]. To deal with such a situation we need to establish a uniformly accurate method using layer-adapted grids. There are several strategies to design the layer-adapted nonuniform grids, for example, one can construct layer-adapted grids by using the apriori information about the location and width of the boundary layers. In [5], Clavero et al. constructed a piecewise uniform grid to establish a uniform convergent FDM for convection–diffusion parabolic problems.
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