Notes on lattice points of zonotopes and lattice-face polytopes

Minkowski's second theorem on successive minima gives an upper bound on the volume of a convex body in terms of its successive minima. We study the problem to generalize Minkowski's bound by replacing the volume by the lattice point enumerator of a convex body. In this context we are interested in bounds on the coefficients of Ehrhart polynomials of lattice polytopes via the successive minima. Our results for lattice zonotopes and lattice-face polytopes imply, in particular, that for 0-symmetric lattice-face polytopes and lattice parallelepipeds the volume can be replaced by the lattice point enumerator.Comment: 16 pages, incorporated referee remarks, corrected proof of Theorem 1.2, added new co-autho

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

Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier-Stokes equations

Recently, it was understood how to repair a certain L2-orthogonality of discretely-divergence-free vector fields and gradient fields such that the velocity error of inf-sup stable discretizations for the incompressible Stokes equations becomes pressure-independent. These new pressure-robust Stokes discretizations deliver a small velocity error, whenever the continuous velocity field can be well approximated on a given grid. On the contrary, classical inf-sup stable Stokes discretizations can guarantee a small velocity error only, when both the velocity and the pressure field can be approximated well, simultaneously. In this contribution, pressure-robustness is extended to the time-dependent Navier-Stokes equations. In particular, steady and time-dependent potential flows are shown to build an entire class of benchmarks, where pressure-robust discretizations can outperform classical approaches significantly. Speedups will be explained by a new theoretical concept, the discrete Helmholtz projector of an inf-sup stable discretization. Moreover, different discrete nonlinear convection terms are discussed, and skew-symmetric pressure-robust discretizations are proposed

Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier--Stokes equations

Recently, it was understood how to repair a certain L2-orthogonality of discretely-divergence-free vector fields and gradient fields such that the velocity error of inf-sup stable discretizations for the incompressible Stokes equations becomes pressure-independent. These new 'pressure-robust' Stokes discretizations deliver a small velocity error, whenever the continuous velocity field can be well approximated on a given grid. On the contrary, classical inf-sup stable Stokes discretizations can guarantee a small velocity error only, when both the velocity and the pressure field can be approximated well, simultaneously. In this contribution, 'pressure-robustness' is extended to the time-dependent Navier--Stokes equations. In particular, steady and time-dependent potential flows are shown to build an entire class of benchmarks, where pressure-robust discretizations can outperform classical approaches significantly. Speedups will be explained by a new theoretical concept, the 'discrete Helmholtz projector' of an inf-sup stable discretization. Moreover, different discrete nonlinear convection terms are discussed, and skew-symmetric pressure-robust discretizations are proposed

On spurious oscillations due to irrotational forces in the Navier--Stokes momentum balance

This contribution studies the influence of the pressure on the velocity error in finite element discretisations of the Navier--Stokes equations. Three simple benchmark problems that are all close to real-world applications convey that the pressure can be comparably large and is not to be underestimated. For widely used finite element methods like the Taylor--Hood finite element method, such relatively large pressures can lead to spurious oscillations and arbitrarily large errors in the velocity, even if the exact velocity is in the ansatz space. Only mixed finite element methods, whose velocity error is pressure-independent, like the Scott--Vogelius finite element method can avoid this influence

In situ benthic fluxes from an intermittently active mud volcano at the Costa Rica convergent margin

Along the erosive convergent margin off Costa Rica a large number of mound-shaped structures exist built by mud diapirism or mud volcanism. One of these, Mound 12, an intermittently active mud volcano, currently emits large amounts of aqueous dissolved species and water. Chemosynthetic vent communities, authigenic carbonates, and methane plumes in the water column are manifestations of that activity. Benthic flux measurements were obtained by a video-guided Benthic Chamber Lander (BCL) deployed at a vent site located in the most active part of Mound 12. The lander was equipped with 4 independent chambers covering adjacent areas of the seafloor. Benthic fluxes were recorded by repeated sampling of the enclosed bottom waters while the underlying surface sediments were recovered with the lander after a deployment time of one day. One of the chambers was placed directly in the centre of an active vent marked by the occurrence of a bacterial mat while the other chambers were located at the fringe of the same vent system at a lateral distance of only 40 cm. A transport-reaction model was developed and applied to describe the concentration profiles in the pore water of the recovered surface sediments and the temporal evolution of the enclosed bottom water. Repeated model runs revealed that the best fit to the pore water and benthic chamber data is obtained with a flow velocity of 10 cm yr− 1 at the centre of the vent. The flux rates to the bottom water are strongly modified by the benthic turnover (benthic filter). The methane flux from below at the bacterial mat site is as high as 1032 μmol cm− 2 yr− 1, out of which 588 μmol cm− 2 yr− 1 is oxidised in the surface sediments by microbial consortia using sulphate as terminal electron acceptor and 440 μmol cm− 2 yr− 1 are seeping into the overlaying bottom water. Sulphide is transported to the surface by ascending fluids (238 μmol cm− 2 yr− 1) and is formed within the surface sediment by the anaerobic oxidation of methane (AOM, 588 μmol cm− 2 yr− 1). However, sulphide is not released into the bottom water but completely oxidized by oxygen and nitrate at the sediment/water interface. The oxygen and nitrate fluxes into the sediment are high (781 and 700 μmol cm− 2 yr− 1, respectively) and are mainly driven by the microbial oxidation of sulphide. Benthic fluxes were much lower in the other chambers placed in the fringe of the vent system. Thus, methane and oxygen fluxes of only 28 and 89 μmol cm− 2 yr− 1, respectively were recorded in one of these chambers. Our study shows that the aerobic oxidation of methane is much less efficient than the anaerobic oxidation of methane so that methane which is not oxidized within the sediment by AOM is almost completely released into the bottom water. Hence, anaerobic rather than aerobic methane oxidation plays the major role in the regulation of benthic methane fluxes. Moreover, we demonstrate that methane and oxygen fluxes at cold vent sites may vary up to 3 orders of magnitude over a lateral distance of only 40 cm indicating an extreme focussing of fluid flow and methane release at the seafloor

Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations

In this contribution, classical mixed methods for the incompressible Navier-Stokes equations that relax the divergence constraint and are discretely inf-sup stable, are reviewed. Though the relaxation of the divergence constraint was claimed to be harmless since the beginning of the 1970ies, Poisson locking is just replaced by another more subtle kind of locking phenomenon, which is sometimes called poor mass conservation. Indeed, divergence-free mixed methods and classical mixed methods behave qualitatively in a different way: divergence-free mixed methods are pressure-robust, which means that, e.g., their velocity error is independent of the continuous pressure. The lack of pressure-robustness in classical mixed methods can be traced back to a consistency error of an appropriately defined discrete Helmholtz projector. Numerical analysis and numerical examples reveal that really locking-free mixed methods must be discretely inf-sup stable and pressure-robust, simultaneously. Further, a recent discovery shows that locking-free, pressure-robust mixed methods do not have to be divergence-free. Indeed, relaxing the divergence constraint in the velocity trial functions is harmless, if the relaxation of the divergence constraint in some velocity test functions is repaired, accordingly

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