A family of non-conforming elements and the analysis of Nitsche’s method for a singularly perturbed fourth order problem

Abstract

In this paper we address several issues arising from a singularly perturbed fourth order problem with small parameter ε. First, we introduce a new family of nonconforming elements. We then prove that the corresponding finite element method is robust with respect to the parameter ε and uniformly convergent to order h 1/2 . In addition, we analyze the effect of treating the Neumann boundary condition weakly by Nitsche's method. We show that such treatment is superior when the parameter ε is smaller than the mesh size h and obtain sharper error estimates. Such error analysis is not restricted to the proposed elements and can easily be carried out to other elements as long as the Neumann boundary condition is imposed weakly. Finally, we discuss the local error estimates and the pollution effect of the boundary layers in the interior of the domain.

A C0 interior penalty method for a singularly‐perturbed fourth‐order elliptic problem on a layer‐adapted mesh

“…Results similar to the one above have been proven through recent years and for other elements, see e.g. .…”

“…Thus, for k = 2 this bound is not uniform in ε. However, uniform bounds can also be derived, e.g. where for some solves the reduced problem: We leave out the proof of these estimates, as it follows (, Theorem 1), only being shorter because of the Galerkin orthogonality. The concept of proof requires the H 1 ‐stability of a projector onto the finite element space.…”

“…In both and , the case was analyzed. Recently, with Theorem 1 in , a result as was proven for some class of nonconforming elements, and mentioned without proof to hold also for the CIP method, see the final remark after Theorem in this article.…”

Uniformly Convergent Nonconforming Element for 3-D Fourth Order Elliptic Singular Perturbation Problem

“…A nonconforming non-C 0 tetrahedral element was constructed and analyzed in [14] by the similar way used in [9], and a nonconforming C 0 tetrahedral element was constructed in [15]. Recently, a nonconforming C 0 tetrahedral element was constructed in [16] by Nitsche's method. In this paper, we introduce an C 0 cuboid element, which was constructed in [17] by us, but the error estimate was valid only for ε = 1.…”

A solution decomposition for a singularly perturbed fourth-order problem

“…In contrast to our problem, this kind of problems is well understood and numerical analysis can be found, see [2,3,5,7,8], just to name a few. The main difference, however, is that the equations treated in the references cited do not contain third-order terms.…”