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

Abstract

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 divergencefree 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 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 pressuredominant situations. This is also confirmed by some numerical examples with the novel pressurerobust modifications of the Taylor-Hood and mini finite element methods.incompressible Navier-Stokes equations and mixed finite elements and pressure robustness and a posteriori error estimators and adaptive mesh refinement

A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation

“…Space discretizations that remain accurate in the presence of dominant gradient fields in the momentum balance -leading to strong pressure gradients -have recently triggered a notable research activity [29,48,40,38,24,53,16,34,35,2,39,49,50,41,21,22] and have been called pressure-robust [42,3]. This concept explains how these equivalence classes of forces and the special role of gradient-type forces affect the notion of dominant advection in Navier-Stokes flows.…”

A Pressure-Robust Discretization of Oseen's Equation Using Stabilization in the Vorticity Equation

“…A similar starting point was used in the meteorology community [53] where a residual SUPG-like method built from (1.8) for the two-dimensional case (although different from the one proposed in this work, and no analysis was presented). The same principle has also been applied in recent work on pressure-robust residual-based a posteriori error control [47,44].…”

“…Space discretizations that remain accurate in the presence of dominant gradient fields in the momentum balance -leading to strong pressure gradients -have recently triggered a notable research activity [35,56,47,45,28,62,20,40,41,2,46,57,58,48,25,26] and have been called pressure-robust [49,3]. We remark that the classical grad-div stabilization is a means to enhance the pressure-robustness of popular methods like the Taylor-Hood element [18].…”

Fortin operator for the Taylor–Hood element

“…A common tool to verify infsup stability are Fortin operators, which are bounded interpolation operators preserving the discrete divergence of a function. Apart from stability results, Fortin operators are important in the design of a posteriori error estimators [21], their quasi-local approximation properties are needed when discretizing nonlinear incompressible fluid equations [17], they are used in the investigation of pre-conditioners [22], and they allow for stability results in different norms such as W 1,∞ [16,19]. Hence, there are numerous contributions on the design of Fortin operators for various finite element pairs, including several papers [2,9,14,17,22] on the Taylor-Hood element in dimension d = 2.…”