In this paper, the adaptive numerical solution of a 2D Poisson model problem by Crouzeix-Raviart elements ($\operatorname*{CR}_{k}$ $\operatorname*{FEM}$) of arbitrary odd degree $k\geq1$ is investigated. The analysis is based on an established, abstract theoretical framework: the \textit{axioms of adaptivity} imply optimal convergence rates for the adaptive algorithm induced by a residual-type a posteriori error estimator. Here, we introduce the error estimator for the $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ discretization and our main theoretical result is the proof ot Axiom 3: \textit{discrete reliability}. This generalizes results for adaptive lowest order $\operatorname*{CR}_{1}$ $\operatorname*{FEM}$ in the literature. For this analysis, we introduce and analyze new local quasi-interpolation operators for $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ which are key for our proof of discrete reliability. We present the results of numerical experiments for the adaptive version of $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ for some low and higher (odd) degrees $k\geq1$ which illustrate the optimal convergence rates for all considered values of $k$.
Let Ω ⊆ R 2 be a bounded polygonal Lipschitz domain. We consider the Galerkin discretization of the Poisson equation
satisfying homogeneous Dirichlet boundary conditions, by the non-conforming Crouzeix-Raviart finite element (CR k FEM) of arbitrary odd degree k ≥ 1. The Crouzeix-Raviart element was first introduced in [19] as a finite dimensional function space W h over a triangulation T by imposing orthogonality conditions of the jumps across interelement boundaries. This provides substantial freedom in designing concrete Crouzeix-Raviart type elements. Some examples of such elements in 2D include, but are not limited to, the lowest order Crouzeix-Raviart element [19] for degree k = 1 (see, e.g, [9] for a survey), the Fortin-Soulie element [24] for k = 2, the Crouzeix-Falk element [18] for k = 3, the Gauss-Legendre elements [3] for k ≥ 4 and of course the standard conforming hp-finite elements. Furthermore in [16] the cases k = 4, 6 are discussed for the Stokes equation in 2D. Some examples of 3D Crouzeix-Raviart elements include the linear CR 1 FEM [19] and [23] provides a local basis construction for quadratic Crouzeix-Raviart elements. In [17] a family of maximal Crouzeix-Raviart elements which allow a local basis is presented and most recently, the work [6] extends the 2D Crouzeix-Raviart elements to any space dimension d ≥ 2 for any degree k ≥ 1 and provides a family of local interpolations operators for odd degrees k ≥ 1.
In this paper, the adaptive version of the Crouzeix-Raviart finite element method (CR k AFEM) generated by loops of the form solve → estimate → mark → refine (1.2) driven by an residual a posteriori error estimator (cf., (3.3)) is investigated. We employ Dörfler marking [21] as the marking strategy and newest vertex bisection [27, Sec. 2.1] as a refinement strategy. This work strives towards a proof of optimal convergence of CR k AFEM of arbitrary odd degree k ≥ 1 driven by our residual error estimator.
The axioms of adaptivity [12]: stability (A1), reduction (A2), discrete reliability (A3) and general quasi orthogonality (A4) in combination with Dörfler marking and newest vertex bisection, provide a general theoretical framework that is sufficient for a proof of optimal convergence rates of an adaptive process as in (1.2). For standart h-version conforming AFEM, optimal convergence with respect to its respective residual error estimator is known, see e.g., [15,33]. Section 5.1 of [12] provides a proof of optimal convergence of conforming AFEM within the framework of the axioms of adaptivity. For non-conforming methods there are two popular methodical approaches: first, fully discontinuous methods within discontinuous Galerkin (dG) discretizations (see, e.g., [2] for a unified analysis) and hybrid high order methods (HHO) (see e.g, [20]). Both of these methods need stabilization of the bilinear form. Second, methods such as CR k FEM which do not need stabilization of the bilinear form. For the lowest order CR 1 AFEM optimal convergence is known (cf., e.g, [4,31]) and [12,Sec. 5.2] provides a proof of optimal convergence within the axioms of adaptivity framework. Optimal convergence of high-order CR k AFEM remains an open question. In [8], optimal convergence of an adaptive symmetric interior penalty dG method is shown. For HHO the picture is less complete. In [20,Sec. 4] a reliable and efficient error estimator which contains a stabilization term is presented for HHO; however the reduction property (A2) is still open for HHO. Recent research provides an efficient and reliable stabilization free residual estimator for HHO [5] but discrete reliability as in (A3) remains open.
In this paper we follow the theoretical framework of the axioms of adaptivity. We prove that the CR k FEM of arbitrary odd degree k ≥ 1 is discretely reliable (A3) (cf. Theorem 3.2) and satisfies the stability and reduction properties (A1) and (A2) with respect to our residual error estimator. The main difficulty for non-conforming finite element methods, like CR k FEM, in the context of the axioms of adaptivity is that the finite element spaces are generally not nested under mesh refinement. For example, if T is a refinement of the triangulation T , then the CR k spaces are generally non-nested, i.e., CR k (T ) ⊈ CR k ( T ). One way of dealing with this non-nestedness is the introduction of conforming companion/enrichment operators J, which were already used in [10]. The paper [14] provides a general dimension-independent approach for the proof of discrete reliability for non-conforming finite element methods via the use of an operator J, which is a conforming companion operator to a non-conforming quasi interpolation operator I NC : V + V (T ) → V (T ), where V is the space of weak solutions and V (T ) ⊈ V is a non-conforming finite element space.
The approach in [14] relies on two key assumptions. It requires the non-conforming quasi-interpolation I NC to (a) be the orthogonal projection with respect to t
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