Fully Discrete Approximations to the Time-Dependent Navier–Stokes Equations with a Projection Method in Time and Grad-Div Stabilization

This paper studies fully discrete approximations to the evolutionary Navier{ Stokes equations by means of inf-sup stable H1-conforming mixed nite elements with a grad-div type stabilization and the Euler incremental projection method in time. We get error bounds where the constants do not depend on negative powers of the viscosity. We get the optimal rate of convergence in time of the projection method. For the spatial error we get a bound O(hk) for the L2 error of the velocity, k being the degree of the polynomials in the velocity approximation. We prove numerically that this bound is sharp for this method.MINECO grant MTM2016-78995-P (AEI)Junta de Castilla y León grant VA024P17Junta de Castilla y León grant VA105G18MINECO grant MTM2015-65608-

Grad-div stabilization for the time-dependent Boussinesq equations with inf-sup stable finite elements

In this paper we consider inf-sup stable nite element discretizations of the evolutionary Boussinesq equations with a grad-div type stabilization. We prove error bounds for the method with constants independent on the Rayleigh numbersMINECO grant MTM2016-78995-P (AEI)Junta de Castilla y León grant VA024P17Junta de Castilla y León grant VA105G18MINECO grant MTM2015-65608-

Postprocessing the Galerkin method: the finite-element case

A postprocessing technique, developed earlier for spectral methods, is extended here to Galerkin nite-element methods for dissipative evolution partial di erential equations. The postprocessing amounts to solving a linear elliptic problem on a ner grid (or higher-order space) once the time integration on the coarser mesh is completed. This technique increases the convergence rate of the nite-element method to which it is applied, and this is done at almost no additional computational cost. The numerical experiments presented here show that the resulting postprocessed method is computationally more e cient than the method to which it is applied (say, quadratic nite elements) as well as standard methods of similar order of convergence as the postprocessed one (say, cubic nite elements). The error analysis of the new method is performed in L2 and in L1 norms.DGICYT PB95-21

Second order error bounds for POD-ROM methods based on first order divided differences

This note proves for the heat equation that using BDF2 as time stepping scheme in POD-ROM methods with snapshots based on difference quotients gives both the optimal second order error bound in time and pointwise estimates

On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows

The kinetic energy of a flow is proportional to the square of the norm of the velocity. Given a sufficient regular velocity field and a velocity finite element space with polynomials of degree , then the best approximation error in is of order . In this survey, the available finite element error analysis for the velocity error in is reviewed, where is a final time. Since in practice the case of small viscosity coefficients or dominant convection is of particular interest, which may result in turbulent flows, robust error estimates are considered, i.e., estimates where the constant in the error bound does not depend on inverse powers of the viscosity coefficient. Methods for which robust estimates can be derived enable stable flow simulations for small viscosity coefficients on comparatively coarse grids, which is often the situation encountered in practice. To introduce stabilization techniques for the convection-dominated regime and tools used in the error analysis, evolutionary linear convection–diffusion equations are studied at the beginning. The main part of this survey considers robust finite element methods for the incompressible Navier–Stokes equations of order , , and for the velocity error in . All these methods are discussed in detail. In particular, a sketch of the proof for the error bound is given that explains the estimate of important terms which determine finally the order of convergence. Among them, there are methods for inf–sup stable pairs of finite element spaces as well as for pressure-stabilized discretizations. Numerical studies support the analytic results for several of these methods. In addition, methods are surveyed that behave in a robust way but for which only a non-robust error analysis is available. The conclusion of this survey is that the problem of whether or not there is a robust method with optimal convergence order for the kinetic energy is still open

Second order error bounds for POD-ROM methods based on first order divided differences

This note proves, for simplicity for the heat equation, that using BDF2 as time stepping scheme in POD-ROM methods with snapshots based on difference quotients gives both the optimal second order error bound in time and pointwise estimates.Comment: no comment

Error analysis of projection methods for non inf-sup stable mixed finite elements: The transient Stokes problem

A modified Chorin–Teman (Euler non-incremental) projection method and a modified Euler incremental projection method for non inf-sup stable mixed finite elements are analyzed. The analysis of the classical Euler non-incremental and Euler incremental methods are obtained as a particular case. We first prove that the modified Euler non-incremental scheme has an inherent stabilization that allows the use of non inf-sup stable mixed finite elements without any kind of extra added stabilization. We show that it is also true in the case of the classical Chorin–Temam method. For the second scheme, we study a stabilization that allows the use of equal-order pairs of finite elements. The relation of the methods with the so-called pressure stabilized Petrov Galerkin method (PSPG) is established. The influence of the chosen initial approximations in the computed approximations to the pressure is analyzed. Numerical tests confirm the theoretical resultsResearch sup-ported by Spanish MINECO under grants MTM2013-42538-P (MINECO, ES) and MTM2016-78995-P (AEI/FEDER UE