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

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

Discretization of Navier--Stokes equations using pressure-robust finite element methods is considered for the high Reynolds number regime. To counter oscillations due to dominating convection we add a stabilization based on a bulk term in the form of a residual-based least squares stabilization of the vorticity equation supplemented by a penalty term on (certain components of) the gradient jump over the elements faces. Since the stabilization is based on the vorticity equation, it is independent of the pressure gradients, which makes it pressure-robust. Thus, we prove pressure-independent error estimates in the linearized case, known as Oseen's problem. In fact, we prove an $O(h^{k+\frac12})$ error estimate in the $L^2$-norm that is known to be the best that can be expected for this type of problem. Numerical examples are provided that, in addition to confirming the theoretical results, show that the present method compares favorably to the classical residual-based streamline upwind Petrov--Galerkin stabilization

Optimal L² Velocity Error Estimates 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. This modified robust Crouzeix–Raviart element employs lowest-order Raviart–Thomas elements in a variational crime in the right-hand side of the Stokes discretization, reestablishing the $L^2$-orthogonality of divergence-free and irrotational forces in the momentum balance. The modification results in an $\mathcal{O}(h)$ consistency error that allows straightforward proofs for the optimal convergence of the discrete energy norm of the velocity and of the $L^2$ norm of the pressure. However, though the optimal convergence of the velocity in the $L^2$ norm was observed numerically, it has not yet been proven for the lowest-order Raviart–Thomas elements as the proof available for higher-order elements is not applicable in this case. In this contribution, this gap is closed and the observed $L^2$ convergence rate for the velocities is shown for the lowest-order Raviart–Thomas elements. 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. Numerical experiments in two and three dimensions illustrate the theoretical findings

A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem

A novel notion for constructing a well-balanced scheme - a gradient-robust scheme - is introduced and a showcase application for the steady compressible, isothermal Stokes equations in a nearly-hydrostatic situation is presented. Gradient-robustness means that gradient fields in the momentum balance are well-balanced by the discrete pressure gradient - which is possible on arbitrary, unstructured grids. The scheme is asymptotic-preserving in the sense that it reduces for low Mach numbers to a recent inf-sup stable and pressure-robust discretization for the incompressible Stokes equations. The convergence of the coupled FEM-FVM scheme for the nonlinear, isothermal Stokes equations is proved by compactness arguments. Numerical examples illustrate the numerical analysis, and show that the novel approach can lead to a dramatically increased accuracy in nearly-hydrostatic low Mach number flows. Numerical examples also suggest that a straightforward extension to barotropic situations with nonlinear equations of state is feasible. (C) 2020 Elsevier B.V. All rights reserved.DAAD (German Academic Exchange Service) scholarshipDeutscher Akademischer Austausch Dienst (DAAD)This work has been partially funded by a DAAD (German Academic Exchange Service) scholarship, which supported a research stay of Mine Akbas at the Weierstrass Institute in Berlin in the period January to September 2018.WOS:0005620239000092-s2.0-8508467247

Inverse modeling of thin layer flow cells for detection of solubility, transport and reaction coefficients from experimental data

Thin layer flow cells are used in electrochemical research as experimental devices which allow to perform investigations of electrocatalytic surface reactions under controlled conditions using reasonably small electrolyte volumes. The paper introduces a general approach to simulate the complete cell using accurate numerical simulation of the coupled flow, transport and reaction processes in a flow cell. The approach is based on a mass conservative coupling of a divergence-free finite element method for fluid flow and a stable finite volume method for mass transport. It allows to perform stable and efficient forward simulations that comply with the physical bounds namely mass conservation and maximum principles for the involved species. In this context, several recent approaches to obtain divergence-free velocities from finite element simulations are discussed. In order to perform parameter identification, the forward simulation method is coupled to standard optimization tools. After an assessment of the inverse modeling approach using known realistic data, first results of the identification of solubility and transport data for O2 dissolved in organic electrolytes are presented. A plausibility study for a more complex situation with surface reactions concludes the paper and shows possible extensions of the scope of the presented numerical tools