On the error bounds of nonconforming finite elements

Abstract

We prove that the error estimates of a large class of nonconforming finite elements are dominated by their approximation errors, which means that the well-known Cea's lemma is still valid for these nonconforming finite element methods. Furthermore, we derive the error estimates in both energy and L 2 norms under the regularity assumption u ∈ H 1+s (Ω) with any s > 0. The extensions to other related problems are possible.

Adaptive Nonconforming Finite Element Approximation of Eigenvalue Clusters

“…The idea behind these operators is to design for a nonconforming nite element function ℓ some conforming companion +1 ℓ ∈ with certain conservation properties. For = 2, operators of this kind have been constructed in [14] and independently in [41]. The following result extends [14] to any dimension ≥ 2.…”

“…An important methodological tool is the higher-order 2 control for the eigenfunction approximations which is proven by means of conforming companion operators. Operators of this kind were introduced in [14,41] in the twodimensional case and are generalised in this paper to higher space dimensions ≥ 2. The resulting 2 error estimates compare the 2 error directly with the energy error and therefore do not employ any a priori results of the eigenfunction approximation.…”

A new family of nonconforming finite elements on quadrilaterals

“…(4.4) is usually referred to as the approximation error and the second term is referred to as the consistency error. To analyze the consistency error, we need to modify the technique from [11,13,14].…”

“…Note that the second term cannot be coped with by the technique from [11,13,14]. Herein we follow a similar technique from [12].…”

“…As the second purpose of this paper, we follow the idea of [11][12][13][14] to present a medium a priori error analysis for this family of nonconforming elements. However the method therein cannot be directly used herein since there is an extra average term, see Theorem 4.2 below.…”