A new error analysis of Bergan’s energy-orthogonal element for a plate contact problem.pdfVIP

Applied Mathematics Letters 69 (2017) 67–74 Contents lists available at ScienceDirect Applied Mathematics Letters /locate/aml A new error analysis of Bergan’s energy-orthogonal element for a plate contact problem Lifang Pei, Dongyang Shi* School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China article info Article history: Received 3 December 2016 Received in revised form 25 January 2017 Accepted 25 January 2017 Available online 14 February 2017 Keywords: Plate contact problem Variational inequality Nonconforming FEM Bergan’s energy-orthogonal plate element Optimal error estimate abstract A nonconforming finite element method (FEM for short) is proposed and analyzed for a plate contact problem by employing the Bergan’s energy-orthogonal plate element. Because the shape function and its first derivatives of this element are discontinuous at the element’s vertices, which is quite different from the conventional finite elements used in the existing literature, some novel approaches, including interpolation operator splitting and energy orthogonality, are developed to present a new error analysis for deriving an optimal estimate of order O (h). At last, some numerical results are also provided to confirm the theoretical analysis. ? 2017 Elsevier Ltd. All rights reserved. 1. Introduction Let ? be a bounded convex polygon domain in R2, Γ = ?? be the boundary which is Lipschitz continuous and can be decomposed into three mutually disjoint parts: Γ = Γˉ1 ∪Γˉ2 ∪Γˉ3 such that Γ1, Γ2, Γ3 are relatively open, Γˉ1 ∩ Γˉ3 = ?, and meas(Γ1) 0. The unit outward normal vector is denoted by n = (n1, n2)T , and the tangential vector is s = (s1, s2)T with s1 = ?n2, s2 = n1. Both n and s exist a.e. on Γ . The outward normal derivative operator is denoted by ? ?n , and the tangential derivative operator is represented by ? ?s . Given f ∈ L2(? ), and g ∈ L2(Γ3) with g 0 a.e. on Γ3. Then in this paper we consider the following plate frictional contact problem (