Morley-Wang-Xu element methods with penalty for a fourth order elliptic singular perturbation problem
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Morley-Wang-Xu element methods with penalty for a fourth order elliptic singular perturbation problem Wenqing Wang1 · Xuehai Huang2 Ruiyue Zhou2
· Kai Tang2 ·
Received: 23 December 2016 / Accepted: 13 November 2017 © Springer Science+Business Media, LLC, part of Springer Nature 2017
Abstract Two Morley-Wang-Xu element methods with penalty for the fourth order elliptic singular perturbation problem are proposed in this paper, including the interior penalty Morley-Wang-Xu element method and the super penalty Morley-WangXu element method. The key idea in designing these two methods is combining the Morley-Wang-Xu element and penalty formulation for the Laplace operator. Robust a priori error estimates are derived under minimal regularity assumptions on the exact solution by means of some established a posteriori error estimates. Finally, we present some numerical results to demonstrate the theoretical estimates. Keywords Fourth order singular perturbation problem · Morley-Wang-Xu element · Interior penalty method · Super penalty method · Error analysis Mathematics Subject Classification (2010) 65N30 · 65N15 · 35J40 · 35J35
Communicated by: Long Chen Xuehai Huang
[email protected] Wenqing Wang [email protected] Kai Tang [email protected] Ruiyue Zhou [email protected] 1
Department of Basic Teaching, Wenzhou Business College, Wenzhou 325035, China
2
College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China
W. Wang et al.
1 Introduction Assume that ⊂ Rd with d = 2, 3 is a bounded polytope. Let f ∈ L2 (), then the fourth order elliptic singular perturbation problem is to find u ∈ H02 () satisfying 2 2 ε u − u = f in , (1.1) u = ∂n u = 0 on ∂, where n is the unit outward normal to ∂, and ε is a real small and positive parameter. Problem (1.1) models the thin buckling plates with u being the displacement in two dimensions [24]. It can also be viewed as a simplification of the stationary Cahn-Hilliard equation in three dimensions [39] and the linearization of the vanishing moment method for the fully nonlinear Monge-Amp`ere equation [6]. The variational formulation of problem (1.1) is to find u ∈ H02 () such that ε2 a(u, v) + b(u, v) = (f, v) ∀v ∈ H02 (), (1.2) where a(u, v) = (∇ 2 u, ∇ 2 v),
b(u, v) = (∇u, ∇v)
with (·, ·) being the L2 inner product over . The H 2 -conforming finite elements are suitable to discretize the fourth order operator and the second order operator in problem (1.1) simultaneously [41], such as the Argyris element [2] and its three-dimensional counterpart [55], Hsieh-CloughToucher element [19]. However the H 2 -conforming finite elements require higher degree polynomials or macroelement techniques which are completely not necessary since the boundary layers of problem (1.1) (cf. [38, section 5]). To this end, nonconforming finite element methods are more popular for the fourth order elliptic singular perturbation problem. To be specific, many H 2 -nonconforming elements which are H 1 -conforming have been constructed for problem (
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