A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes

Most classical finite element schemes for the (Navier--)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior forcing term, using H(div)-conforming velocity reconstruction operators. This approach has so far only been analyzed on shape-regular triangulations. The novelty of the present contribution is that the reconstruction approach for the Crouzeix--Raviart method, which has a stable Fortin operator on arbitrary meshes, is combined with results on the interpolation error on anisotropic elements for reconstruction operators of Raviart--Thomas and Brezzi--Douglas--Marini type, generalizing the method to a large class of anisotropic triangulations. Numerical examples confirm the theoretical results in a 2D and a 3D test case

A posteriori error estimation for the Stokes problem: Anisotropic and isotropic discretizations

The paper presents a posteriori error estimators for the stationary Stokes problem. We consider anisotropic finite element discretizations (i.e. elements with very large aspect ratio) where conventional, isotropic error estimators fail. Our analysis covers two- and three-dimensional domains, conforming and nonconforming discretizations as well as different elements. This large variety of settings requires different approaches and results in different estimators. Furthermore many examples of finite element pairs that are covered by the analysis are presented. Lower and upper error bounds form the main result with minimal assumptions on the elements. The lower error bound is uniform with respect to the mesh anisotropy with the exception of nonconforming 3D discretizations made of pentahedra or hexahedra. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. Some of the corresponding results seem to be novel (in particular for 3D domains), and cover element pairs of practical importance. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimators

A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes

Most classical finite element schemes for the (Navier-)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior forcing term, using $\mathbf{H}(\operatorname{div})$-conforming velocity reconstruction operators. This approach has so far only been analyzed on shape-regular triangulations. The novelty of the present contribution is that the reconstruction approach for the Crouzeix-Raviart method, which has a stable Fortin operator on arbitrary meshes, is combined with results on the interpolation error on anisotropic elements for reconstruction operators of Raviart-Thomas and Brezzi-Douglas-Marini type, generalizing the method to a large class of anisotropic triangulations. Numerical examples confirm the theoretical results in a 2D and a 3D test case

Anisotropic weakly over-penalised symmetric interior penalty method for the Stokes equation

In this study, we investigate an anisotropic weakly over-penalised symmetric interior penalty method for the Stokes equation. Our approach is a simple discontinuous Galerkin method similar to the Crouzeix--Raviart finite element method. As our primary contribution, we show a new proof for the consistency term, which allows us to obtain an estimate of the anisotropic consistency error. The key idea of the proof is to apply the relation between the Raviart--Thomas finite element space and a discontinuous space. While inf-sup stable schemes of the discontinuous Galerkin method on shape-regular mesh partitions have been widely discussed, our results show that the Stokes element satisfies the inf-sup condition on anisotropic meshes. Furthermore, we also provide an error estimate in an energy norm on anisotropic meshes. In numerical experiments, we compare calculation results for standard and anisotropic mesh partitions, and the results show the effectiveness of using anisotropic meshes for problems with boundary layers.Comment: 41 pages, 9 figures, 11 table

Recovery Techniques For Finite Element Methods And Their Applications

Recovery techniques are important post-processing methods to obtain improved approximate solutions from primary data with reasonable cost. The practical us- age of recovery techniques is not only to improve the quality of approximation, but also to provide an asymptotically exact posteriori error estimators for adaptive meth- ods. This dissertation presents recovery techniques for nonconforming finite element methods and high order derivative as well as applications of gradient recovery. Our first target is to develop a systematic gradient recovery technique for Crouzeix- Raviart element. The proposed method uses finite element solution to build a better approximation of the exact gradient based on local least square fittings. Due to poly- nomial preserving property of least square fitting, it is easy to show that the new proposed method preserves quadratic polynomials. In addition, the proposed gra- dient recovery is linearly bounded. Numerical tests indicate the recovered gradient is superconvergent to the exact gradient for both second order elliptic equation and Stokes equation. The gradient recovery technique can be used in a posteriori error estimates for Crouzeix-Raviart element, which is relatively simple to implement and problem independent. Our second target is to propose and analyze a new effective Hessian recovery for continuous finite element of arbitrary order. The proposed Hessian recovery is based on polynomial preserving recovery. The proposed method preserves polynomials of degree (k + 1) on general unstructured meshes and polynomials of degree (k + 2) on translation invariant meshes. Based on it polynomial preserving property, we can able to prove superconvergence of the proposed method on mildly structured meshes. In addition, we establish the ultraconvergence result for the new Hessian recovery technique on translation invariant finite element space of arbitrary order. Our third target is to demonstrate application of gradient recovery in eigenvalue computation. We propose two superconvergent two-grid methods for elliptic eigen- value problems by taking advantage of two-gird method, two-space method, shifted- inverse power method, and gradient recovery enhancement. Theoretical and numer- ical results reveal that the proposed methods provide superconvergent eigenfunction approximation and ultraconvergent eigenvalue approximation. In addition, two mul- tilevel adaptive methods based recovery type a posterior error estimate are proposed

Anisotropic finite elements for the Stokes problem: a posteriori error estimator and adaptive mesh

AbstractWe propose an a posteriori error estimator for the Stokes problem using the Crouzeix–Raviart/P0 pair. Its efficiency and reliability on highly stretched meshes are investigated. The analysis is based on hierarchical space splitting whose main ingredients are the strengthened Cauchy–Schwarz inequality and the saturation assumption. We give a theoretical proof of a method to enrich the Crouzeix–Raviart element so that the strengthened Cauchy constant is always bounded away from unity independently of the aspect ratio. An anisotropic self-adaptive mesh refinement approach for which the saturation assumption is valid will be described. Our theory is confirmed by corroborative numerical tests which include an internal layer, a boundary layer, a re-entrant corner and a crack simulation. A comparison of the exact error and the a posteriori one with respect to the aspect ratio will be demonstrated