How to prove the discrete reliability for nonconforming finite element methods

Abstract

Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity. One key ingredient is the discrete reliability of a residual-based a posteriori error estimator, which controls the error of two discrete finite element solutions based on two nested triangulations. In the error analysis of nonconforming finite element methods, like the Crouzeix-Raviart or Morley finite element schemes, the difference of the piecewise derivatives of discontinuous approximations to the distributional gradients of global Sobolev functions plays a dominant role and is the object of this paper. The nonconforming interpolation operator, which comes natural with the definition of the aforementioned nonconforming finite element in the sense of Ciarlet, allows for stability and approximation properties that enable direct proofs of the reliability for the residual that monitors the equilibrium condition. The novel approach of this paper is the suggestion of a right-inverse of this interpolation operator in conforming piecewise polynomials to design a nonconforming approximation of a given coarse-grid approximation on a refined triangulation. The results of this paper allow for simple proofs of the discrete reliability in any space dimension and multiply connected domains on general shape-regular triangulations beyond newest-vertex bisection of simplices. Particular attention is on optimal constants in some standard discrete estimates listed in the appendices.

Adaptive Morley FEM for the von Kármán equations with optimal convergence rates

“…Lemma 2.2 (Morley interpolation [13,20,25,28]). The Morley interpolation operator satisfies (a) the integral mean property D 2 pw I M = Π 0 D 2 pw of the Hessian, (b) the approximation and stability property…”

“…Given any triangle K ∈ T with its set of vertices N(K), its patch is Ω(K) := ∪ z ∈N(K) ω z and E(Ω(K)) denotes the set of edges E in T with dist(E, K) = 0. [8,20,25]). Given any T ∈ T there exists an enrichment or companion operator…”

“…The converse operation in [13,25] relies on a discrete Helmholtz decomposition. This paper follows [20] with a right-inverse I M E M . The key idea is first to compute the companion operator E M v M for some v M ∈ M(T ) and second to apply the interpolation operator I M of Lemma 2.2 on the finer triangulation T (rather than T ).…”

“…A modification of this idea is performed in [20, Def 6.9], [25] to define an operator J 2 : P 2 (K) → HCT (K) + P 5 (K) for each K ∈ T such that Ψ * M := I M (J 2 Ψ M ) ∈ V( T ) is well defined [20,Lem. 6.14] and satisfies [20,Thm. 6.19] that…”