Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?

The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important problems in these fields are discussed

Higher-order finite element methods for elliptic problems with interfaces

We present higher-order piecewise continuous finite element methods for solving a class of interface problems in two dimensions. The method is based on correction terms added to the right-hand side in the standard variational formulation of the problem. We prove optimal error estimates of the methods on general quasi-uniform and shape regular meshes in maximum norms. In addition, we apply the method to a Stokes interface problem, adding correction terms for the velocity and the pressure, obtaining optimal convergence results.Comment: 26 pages, 6 figures. An earlier version of this paper appeared on November 13, 2014 in http://www.brown.edu/research/projects/scientific-computing/reports/201

Weighted Sobolev spaces and regularity for polyhedral domains

We prove a regularity result for the Poisson problem $-\Delta u = f$, u |\_{\pa \PP} = g on a polyhedral domain \PP \subset \RR^3 using the \BK\ spaces \Kond{m}{a}(\PP). These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges \cite{Babu70, Kondratiev67}. In particular, we show that there is no loss of \Kond{m}{a}--regularity for solutions of strongly elliptic systems with smooth coefficients. We also establish a "trace theorem" for the restriction to the boundary of the functions in \Kond{m}{a}(\PP)

Maximum–norm a posteriori error estimates for an optimal control problem

We analyze a reliable and efficient max-norm a posteriori error estimator for a control-constrained, linear–quadratic optimal control problem. The estimator yields optimal experimental rates of convergence within an adaptive loop. © 2019, Springer Science+Business Media, LLC, part of Springer Nature

Longer time accuracy for incompressible Navier-Stokes simulations with the EMAC formulation

In this paper, we consider the recently introduced EMAC formulation for the incompressible Navier-Stokes (NS) equations, which is the only known NS formulation that conserves energy, momentum and angular momentum when the divergence constraint is only weakly enforced. Since its introduction, the EMAC formulation has been successfully used for a wide variety of fluid dynamics problems. We prove that discretizations using the EMAC formulation are potentially better than those built on the commonly used skew-symmetric formulation, by deriving a better longer time error estimate for EMAC: while the classical results for schemes using the skew-symmetric formulation have Gronwall constants dependent on $\exp(C\cdot Re\cdot T)$ with $Re$ the Reynolds number, it turns out that the EMAC error estimate is free from this explicit exponential dependence on the Reynolds number. Additionally, it is demonstrated how EMAC admits smaller lower bounds on its velocity error, since {incorrect treatment of linear momentum, angular momentum and energy induces} lower bounds for $L^2$ velocity error, and EMAC treats these quantities more accurately. Results of numerical tests for channel flow past a cylinder and 2D Kelvin-Helmholtz instability are also given, both of which show that the advantages of EMAC over the skew-symmetric formulation increase as the Reynolds number gets larger and for longer simulation times.Comment: 21 pages, 5 figure