Guaranteed energy error estimators for a modified robust Crouzeix-Raviart Stokes element

This paper provides guaranteed upper energy error bounds for a modified lowest-order nonconforming Crouzeix-Raviart finite element method for the Stokes equations. The modification from [A. Linke 2014, On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime] is based on the observation that only the divergence-free part of the right-hand side should balance the vector Laplacian. The new method has optimal energy error estimates and can lead to errors that are smaller by several magnitudes, since the estimates are pressure-independent. An efficient a posteriori velocity error estimator for the modified method also should involve only the divergence-free part of the right-hand side. Some designs to approximate the Helmholtz projector are compared and verified by numerical benchmark examples. They show that guaranteed error control for the modified method is possible and almost as sharp as for the unmodified method

Guaranteed energy error estimators for a modified robust Crouzeix--Raviart Stokes element

This paper provides guaranteed upper energy error bounds for a modified lowest-order nonconforming Crouzeix--Raviart finite element method for the Stokes equations. The modification from [A. Linke 2014, On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime] is based on the observation that only the divergence-free part of the right-hand side should balance the vector Laplacian. The new method has optimal energy error estimates and can lead to errors that are smaller by several magnitudes, since the estimates are pressure-independent. An efficient a posteriori velocity error estimator for the modified method also should involve only the divergence-free part of the right-hand side. Some designs to approximate the Helmholtz projector are compared and verified by numerical benchmark examples. They show that guaranteed error control for the modified method is possible and almost as sharp as for the unmodified method

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

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

Optimal L2 velocity error estimate for a modified pressure-robust Crouzeix-Raviart Stokes element

Recently, a novel approach for the robust discretization of the incompressible Stokes equations was proposed that slightly modifies the nonconforming Crouzeix-Raviart element such that its velocity error becomes pressure-independent. The modification results in an O(h) consistency error that allows straightforward proofs for the optimal convergence of the discrete energy norm of the velocity and of the L2 norm of the pressure. However, though the optimal convergence of the velocity in the L2 norm was observed numerically, it appeared to be nontrivial to prove. In this contribution, this gap is closed. Moreover, the dependence of the energy error estimates on the discrete inf-sup constant is traced in detail, which shows that classical error estimates are extremely pessimistic on domains with large aspect ratios. Numer-ical experiments in 2D and 3D illustrate the theoretical findings

Optimal L2 velocity error estimate for a modified pressure-robust Crouzeix--Raviart Stokes element

Recently, a novel approach for the robust discretization of the incompressible Stokes equations was proposed that slightly modifies the nonconforming Crouzeix--Raviart element such that its velocity error becomes pressure-independent. The modification results in an O(h) consistency error that allows straightforward proofs for the optimal convergence of the discrete energy norm of the velocity and of the L2 norm of the pressure. However, though the optimal convergence of the velocity in the L2 norm was observed numerically, it appeared to be nontrivial to prove. In this contribution, this gap is closed. Moreover, the dependence of the error estimates on the discrete inf-sup constant is traced in detail, which shows that classical error estimates are extremely pessimistic on domains with large aspect ratios. Numerical experiments in 2D and 3D illustrate the theoretical findings

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 L2-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 NavierStokes equations. The novel error estimators only take the curl of the righthand 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 TaylorHood and mini finite element methods

Adaptive Algorithms

Overwhelming empirical evidence in computational science and engineering proved that self-adaptive mesh-generation is a must-do in real-life problem computational partial differential equations. The mathematical understanding of corresponding algorithms concerns the overlap of two traditional mathematical disciplines, numerical analysis and approximation theory, with computational sciences. The half workshop was devoted to the mathematics of optimal convergence rates and instance optimality of the Dörfler marking or the maximum strategy in various versions of space discretisations and time-evolution problems with all kind of applications in the efficient numerical treatment of partial differential equations

Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity

A stress equilibration procedure for linear elasticity is proposed and analyzed in this paper with emphasis on the behavior for (nearly) incompressible materials. Based on the displacement-pressure approximation computed with a stable finite element pair, it constructs an $H (\text{div})$-conforming, weakly symmetric stress reconstruction. Our focus is on the Taylor-Hood combination of continuous finite element spaces of polynomial degrees $k+1$ and $k$ for the displacement and the pressure, respectively. Our construction leads then to reconstructed stresses by Raviart-Thomas elements of degree $k$ which are weakly symmetric in the sense that its anti-symmetric part is zero tested against continuous piecewise polynomial functions of degree $k$. The computation is performed locally on a set of vertex patches covering the computational domain in the spirit of equilibration \cite{BraSch:08}. Due to the weak symmetry constraint, the local problems need to satisfy consistency conditions associated with all rigid body modes, in contrast to the case of Poisson's equation where only the constant modes are involved. The resulting error estimator is shown to constitute a guaranteed upper bound for the error with a constant that depends only on the shape regularity of the triangulation. Local efficiency, uniformly in the incompressible limit, is deduced from the upper bound by the residual error estimator