The equation y = / + Ky is considered in a separable Hubert space H, with K assumed compact and linear. It is shown that every approximation to y of the form yXn = £nanl«. (where {u-} is a given complete set in H, and the an¡, 1 < / < n, are arbitrary numbers) is less accurate than the best approximation of the form y2n = f + ZnbnjKUj, if ii is sufficiently large. Specifically it is shown that ...

In a recent paper of Arnold et al. [D.N. Arnold, F. Brezzi, L.D. Marini, A family of discontinuous Galerkin finite elements for the Reissner–Mindlin plate, J. Sci. Comput. 22 (2005) 25–45], the ideas of discontinuous Galerkin methods were used to obtain and analyze two new families of locking free finite element methods for the approximation of the Reissner–Mindlin plate problem. By following t...

We introduce generalized Galerkin variational integrators, which are a natural generalization of discrete variational mechanics, whereby the discrete action, as opposed to the discrete Lagrangian, is the fundamental object. This is achieved by approximating the action integral with appropriate choices of a finite-dimensional function space that approximate sections of the configuration bundle a...

In the last five years, the scientific computing community has taken great interest in the so-called stochastic Galerkin schemes (SGS). These schemes are typically used to address the following type of problem: Suppose that I have a deterministic differential equation model for which I can find an approximate solution using standard numerical discretization techniques. Now suppose that, in an a...

Runge-Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws employing useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes, TVD Runge-Kutta time discretizations and limiters. In most of the RKDG papers in the literature, the LaxFriedrichs numeri...

Adaptive DNS/LES is used to compute the drag coefficient cD for the flow past a sphere at Reynolds number Re = 10. Using less than 10 mesh points, cD is computed to an accuracy of a few percent corresponding to experimental precision, which is at least an order of magnitude cheaper than standard non-adaptive Large Eddy Simulation LES computations in the literature. Adaptive DNS/LES is a General...

We introduce a general, analytical framework to express and approximate partial differential equations (PDEs) numerically on graphs networks of surfaces – generalized by the term hypergraphs. To this end, we consider PDEs hypergraphs as singular limits in thin domains (such fault planes, pipes, etc.), observe that (mixed) hybrid formulations offer useful tools formulate such PDEs. Thus, our num...

This paper presents a unified analysis of Discontinuous Galerkin methods to approximate Friedrichs' symmetric systems. An abstract set of conditions is identified at the continuous level to guarantee existence and uniqueness of the solution in a subspace of the graph of the differential operator. Then a general Discontinuous Galerkin method that weakly enforces boundary conditions and mildly pe...

In this paper, central discontinuous Galerkin methods are developed for solving ideal magnetohydrodynamic (MHD) equations. The methods are based on the original central discontinuous Galerkin methods designed for hyperbolic conservation laws on overlapping meshes, and use different discretization for magnetic induction equations. The resulting schemes carry many features of standard central dis...