Instance optimal Crouzeix-Raviart adaptive finite element methods for the Poisson and Stokes problems

We extend the ideas of Diening, Kreuzer, and Stevenson [Instance optimality of the adaptive maximum strategy, Found. Comput. Math. (2015)], from conforming approximations of the Poisson problem to nonconforming Crouzeix-Raviart approximations of the Poisson and the Stokes problem in 2D. As a consequence, we obtain instance optimality of an AFEM with a modified maximum marking strategy

Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem

In this paper, we analyze the convergence and optimality of a standard adaptive nonconforming linear element method for the Stokes problem. After establishing a special quasi--orthogonality property for both the velocity and the pressure in this saddle point problem, we introduce a new prolongation operator to carry through the discrete reliability analysis for the error estimator. We then use a specially defined interpolation operator to prove that, up to oscillation, the error can be bounded by the approximation error within a properly defined nonlinear approximate class. Finally, by introducing a new parameter-dependent error estimator, we prove the convergence and optimality estimates

A new generalization of the P1P_1 non-conforming FEM to higher polynomial degrees

This paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions

Optimal adaptivity for a standard finite element method for the Stokes problem

We prove that the a standard adaptive algorithm for the Taylor-Hood discretization of the stationary Stokes problem converges with optimal rate. This is done by developing an abstract framework for indefinite problems which allows us to prove general quasi-orthogonality proposed in [Carstensen et al., 2014]. This property is the main obstacle towards the optimality proof and therefore is the main focus of this work. The key ingredient is a new connection between the mentioned quasi-orthogonality and $LU$-factorizations of infinite matrices.Comment: Updated version after Paper has been accepted. arXiv admin note: text overlap with arXiv:1710.0608

Nonconforming Finite Element Discretisation for Semilinear Problems with Trilinear Nonlinearity

The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of 2D Navier-Stokes problem and in the von K\'{a}rm\'{a}n equations. This paper establishes a best-approximation a~priori error analysis and an a~posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity. The analysis avoids any smallness assumptions on the data and so has to provide discrete stability by a perturbation analysis before the Newton-Kantorovic theorem can provide the existence of discrete solutions. An abstract framework for the stability analysis in terms of discrete operators from the medius analysis leads to new results on the nonconforming Crouzeix-Raviart FEM for second-order linear non-selfadjoint and indefinite elliptic problems with $L^\infty$ coefficients. The paper identifies six parameters and sufficient conditions for the local a~priori and a~posteriori error control of conforming and nonconforming discretisations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications. This leads to new best-approximation error estimates and to a~posteriori error estimates in terms of explicit residual-based error control for the conforming and Morley FEM

Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods

Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the $L^2$-norm of the right-hand side. However, the velocity is only steered by the divergence-free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier--Stokes equations. The novel error estimators only take the $\mathrm{curl}$ of the right-hand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressure-dominant situations. This is also confirmed by some numerical examples with the novel pressure-robust modifications of the Taylor--Hood and mini finite element methods

Morley Finite Element Method for the Eigenvalues of the Biharmonic Operator

This paper studies the nonconforming Morley finite element approximation of the eigenvalues of the biharmonic operator. A new $C^1$ conforming companion operator leads to an $L^2$ error estimate for the Morley finite element method which directly compares the $L^2$ error with the error in the energy norm and, hence, can dispense with any additional regularity assumptions. Furthermore, the paper presents new eigenvalue error estimates for nonconforming finite elements that bound the error of (possibly multiple or clustered) eigenvalues by the approximation error of the computed invariant subspace. An application is the proof of optimal convergence rates for the adaptive Morley finite element method for eigenvalue clusters.Comment: to appear in IMA Journal of Numerical Analysi

A Nonconforming Finite Element Method for Fourth Order Curl Equations in R^3

In this paper we present a nonconforming finite element method for solving fourth order curl equations in three dimensions arising from magnetohydrodynamics models. We show that the method has an optimal error estimate for a model problem involving both curl^2 and curl^4 operators. The element has a very small number of degrees of freedom and it imposes the inter-element continuity along the tangential direction which is appropriate for the approximation of magnetic fields. We also provide explicit formulae of basis functions for this element.Comment: 16 pages, submitte

Variational formulation and numerical analysis of linear elliptic equations in nondivergence form with Cordes coefficients

This paper studies formulations of second-order elliptic partial differential equations in nondivergence form on convex domains as equivalent variational problems. The first formulation is that of Smears \& S\"uli [SIAM J.\ Numer.\ Anal.\ 51(2013), pp.\ 2088--2106.], and the second one is a new symmetric formulation based on a least-squares functional. These formulations enable the use of standard finite element techniques for variational problems in subspaces of $H^2$ as well as mixed finite element methods from the context of fluid computations. Besides the immediate quasi-optimal a~priori error bounds, the variational setting allows for a~posteriori error control with explicit constants and adaptive mesh-refinement. The convergence of an adaptive algorithm is proved. Numerical results on uniform and adaptive meshes are included

Adaptive Nonconforming Finite Element Approximation of Eigenvalue Clusters

This paper analyses an adaptive nonconforming finite element method for eigenvalue clusters of self-adjoint operators and proves optimal convergence rates (with respect to the concept of nonlinear approximation classes) for the approximation of the invariant subspace spanned by the eigenfunctions of the eigenvalue cluster. Applications include eigenvalues of the Laplacian and of the Stokes system