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In this paper, we consider the discretization of the two-dimensional stationary Stokes equation by Crouzeix-Raviart elements for the velocity of polynomial order $k\geq1$ on conforming triangulations and discontinuous pressure approximations of order $k-1$. We will bound the inf-sup constant from below independent of the mesh size and show that it depends only logarithmically on $k$. Our assumptions on the mesh are very mild: for odd $k$ we require that the triangulations contain at least one inner vertex while for even $k$ we assume that the triangulations consist of more than a single triangle., Comment: 46 pages, 6 figures

The Crouzeix-Raviart triangular finite elements are $\inf$-$\sup$ stable for the Stokes equations for any mesh with at least one interior vertex. This result affirms a {\em conjecture of Crouzeix-Falk} from 1989 for $p=3$. Our proof applies to {\em any odd degree} $p\ge 3$ and hence Crouzeix-Raviart triangular finite elements of degree $p$ in two dimensions and the piecewise polynomials of degree $p-1$ with vanishing integral form a stable Stokes pair {\em for all positive integers} $p$., Comment: 18 pages, 1 figure

In this paper, we prove that Crouzeix-Raviart finite elements of polynomial order $p\geq5$, $p$ odd, are inf-sup stable for the Stokes problem on triangulations. For $p\geq4$, $p$ even, the stability was proved by \'{A}. Baran and G. Stoyan in 2007 by using the \textit{macroelement technique,} a \textit{dimension formula}, the concept of \textit{critical points} in a triangulation and a representation of the corresponding \textit{critical functions}. Baran and Stoyan proved that these critical functions belong to the range of the divergence operator applied to Crouzeix-Raviart velocity functions and the macroelement technique implies the inf-sup stability. The generalization of this theory to cover odd polynomial orders $p\geq5$ is involved; one reason is that the macroelement classes, which have been used for even $p$, are unsuitable for odd $p$. In this paper, we introduce a new and simple representation of non-conforming Crouzeix-Raviart basis functions of odd degree. We employ only one type of macroelement and derive representations of all possible critical functions. Finally, we show that they are in the range of the divergence operator applied to Crouzeix-Raviart velocities from which the stability of the discretization follows., Comment: 27 pages

We discuss the stability theory and numerical analysis of the Helmholtz equation with variable and possibly non-smooth or oscillatory coefficients. Using the unique continuation principle and the Fredholm alternative, we first give an existence-uniqueness result for this problem, which holds under rather general conditions on the coefficients and on the domain. Under additional assumptions, we derive estimates for the stability constant (i.e., the norm of the solution operator) in terms of the data (i.e. PDE coefficients and frequency), and we apply these estimates to obtain a new finite element error analysis for the Helmholtz equation which is valid at high frequency and with variable wave speed. The central role played by the stability constant in this theory leads us to investigate its behaviour with respect to coefficient variation in detail. We give, via a 1D analysis, an a priori bound with stability constant growing exponentially in the variance of the coefficients (wave speed and/or diffusion coefficient). Then, by means a family of analytic examples (supplemented by numerical experiments), we show that this estimate is sharp

We consider elliptic problems with complicated, discontinuous diffusion tensor $A_{\scriptscriptstyle 0} $. One of the standard approaches to numerically treat such problems is to simplify the coefficient by some approximation, say $A_{\varepsilon}$, and to use standard finite elements. In \cite{Repin2012} a combined modelling-discretization strategy has been proposed which estimates the discretization and modelling errors by a posteriori estimates of functional type. This strategy allows to balance these two errors in a problem adapted way. However, the estimate of the modelling error is derived under the assumption that the difference $A_{\scriptscriptstyle 0} -A_{\varepsilon}$ is bounded in the $L^{\infty}$-norm, which requires that the approximation of the coefficient matches the discontinuities of the original coefficient. Therefore this theory is not appropriate for applications with discontinuous coefficients along \textit{complicated, curved} interfaces. Based on bounds for $A_{\scriptscriptstyle 0} -A_{\varepsilon}$ in an $L^{q}$-norm with $q<\infty$ we generalize the combined modelling-discretization strategy to a larger class of coefficients.