We prove new optimality results for adaptive mesh refinement algorithms non-symmetric, indefinite, and time-dependent problems by proposing a generalization of quasi-orthogonality which follows directly from the inf-sup stability underlying problem. This completely removes central technical difficulty in modern proofs optimal convergence leads to simple Taylor-Hood discretization stationary Sto...

In this lecture, we develop the theory of supermodular games; key references are the papers of Topkis [7], Vives [8], and Milgrom and Roberts [3]. Our development closely follows that of Milgrom and Roberts, though we will also note other references where necessary. We start with some basic definitions and facts about lattices. Given a set X, a binary relation is a partial ordering on X if it i...

The discretisation of the Oseen problem by finite element methods may suffer in general from two shortcomings. First, the discrete inf-sup (Babuška-Brezzi) condition can be violated. Second, spurious oscillations occur due to the dominating convection. One way to overcome both difficulties is the use of local projection techniques. Studying the local projection method in an abstract setting, we...

We present a theoretical analysis of the CORSING (COmpRessed SolvING) method for the numerical approximation of partial differential equations based on compressed sensing. In particular, we show that the best s-term approximation of the weak solution of a PDE with respect to an orthonormal system of N trial functions, can be recovered via a Petrov-Galerkin approach using m N orthonormal test fu...

The problem of simultaneous Chebyshev approximation of a set F of uniformly bounded, real-valued functions on a compact interval / by a set P of continuous functions is equivalent to the problem of simultaneous approximation of two real-valued functions F+ (x), F~ (x), with F~ (x) ¿ F+ (x), for all x in I, where F~ is lower semicontinuous and F+ is upper semicontinuous. 1. Formulation of the Ap...

The inf-sup constant for the divergence, or LBB constant, is explicitly known for only few domains. For other domains, upper and lower estimates are known. If more precise values are required, one can try to compute a numerical approximation. This involves, in general, approximation of the domain and then the computation of a discrete LBB constant that can be obtained from the numerical solutio...

We consider space-time methods for time-periodic problems and discuss well-posedness of the variational formulation as well as the use of Reduced Basis Methods (RBM) in a parameterized setting. We propose a possible discretization enforcing periodic boundary conditions in time and provide lower bounds for the space-time inf-sup constant. Rigorous RBM space-time a-posteriori error bounds for sta...

Let ( , σ ) be the one-sided shift space onm symbols. For any x = (xi)i 1 ∈ and positive integer n, define Rn(x) = inf{j n : x1x2 · · · xn = xj+1xj+2 · · · xj+n}. We prove that for each pair of numbersα, β ∈ [0,∞] withα β, the following recurrent set Eα,β = { x ∈ : lim inf n→∞ logRn(x) n = α, lim sup n→∞ logRn(x) n = β } has Hausdorff dimension one. AMS classification scheme number: 28A80