Robust Finite Element Solution
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 212 (2008) 374–389 www.elsevier.com/locate/cam An elliptic singularly perturbed problem with two parameters II: Robust finite element solutionଁ Lj. Teofanova,∗, H.-G. Roosb aDepartment for Fundamental Disciplines, Faculty of Technical Sciences, University of Novi Sad, Trg D.Obradovi´ca 6, 21000 Novi Sad, Serbia bInstitute of Numerical Mathematics, Technical University of Dresden, Dresden D-01062, Germany Received 8 November 2005; received in revised form 8 December 2006 Abstract In this paper we consider a singularly perturbed elliptic model problem with two small parameters posed on the unit square. The problem is solved numerically by the finite element method using piecewise linear or bilinear elements on a layer-adapted Shishkin mesh. We prove that method with bilinear elements is uniformly convergent in an energy norm. Numerical results confirm our theoretical analysis. © 2007 Elsevier B.V. All rights reserved. MSC: 35J25; 65N30; 65N15; 76R99 Keywords: Singular perturbation; Convection–diffusion problems; Finite element method; Two small parameters; Shishkin mesh 1. Introduction In this paper, we consider a finite element method for the following singularly perturbed elliptic two-parameter problem posed on the unit square = (0, 1) × (0, 1) Lu := −1u + 2b(x)ux + c(x)u = f(x,y) in , u = 0onj, (1) b(x) > 0, c(x) > 0 for x ∈[0, 1], (2) where 0 < 1, 2>1 and and are constants. We assume that b, c and f are sufficiently smooth functions and that f satisfies the compatibility conditions f(0, 0) = f(0, 1) = f(1, 0) = f(1, 1) = 0.
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