In this paper, we consider an interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell’s equations in cold plasma. In Huang and Li (J. Sci. Comput., 42 (2009), 321–340), for both semi and fully discrete DG schemes, we proved error estimates which are optimal in the energy norm, but sub-optimal in the L2-norm. Here by filling this gap, we show that these schemes are opt...

The paper examines relationships between the Shannon entropy and the `α-norm for n-ary probability vectors, n ≥ 2. More precisely, we investigate the tight bounds of the `α-norm with a fixed Shannon entropy, and vice versa. As applications of the results, we derive the tight bounds between the Shannon entropy and several information measures which are determined by the `α-norm. Moreover, we app...

Many axiomatic definitions of entropy, such as the Rényi entropy, of a random variable are closely related to the `α-norm of its probability distribution. This study considers probability distributions on finite sets, and examines the sharp bounds of the `β-norm with a fixed `α-norm, α 6= β, for n-dimensional probability vectors with an integer n ≥ 2. From the results, we derive the sharp bound...

This paper investigates string stability issues in homogeneous strings of strictly proper feedback control systems. We consider initial condition problems with unidirectional nearest neighbour communications, using only linear systems with two integrators in the loop. We define string stability as a type of stability that is uniform with respect to the string length. We show conditions under wh...

The symmetric interior penalty (SIP) method on graded meshes and its fast solution by multigrid methods are studied in this paper. We obtain quasi-optimal error estimates in both the energy norm and the L2 norm for the SIP method, and prove uniform convergence of the W -cycle multigrid algorithm for the resulting discrete problem. The performance of these methods is illustrated by numerical res...

In this paper, we provide uniform design and analysis framework for iterative learning control of a class of impulsive first-order distributed parameter systems in the time domain. In particular, P-type and D-type iterative learning controls with initial state learning are considered. We present convergence results for open-loop iterative learning schemes in the sense of the Lp-norm and λ-norm,...

In this paper we revisit and prove optimal order and mesh–independent convergence of an inexact Newton method where the linear Jacobian systems are solved with multigrid techniques. This convergence is shown using Banach spaces and the norm, max{‖ · ‖1, ‖ · ‖0,∞}, a stronger norm than is used in previous work. These results are valid for a class of second order, semi–linear, finite element, ell...

Abstract We develop a discrete counterpart of the De Giorgi–Nash–Moser theory, which provides uniform Hölder-norm bounds on continuous piecewise affine finite element approximations second-order linear elliptic problems form $-\nabla \cdot (A\nabla u)=f-\nabla F$ with $A\in L^\infty (\varOmega ; {{\mathbb{R}}}^{n\times n})$ uniformly matrix-valued function, $f\in L^{q}(\varOmega )$, $F\in L^p(\...

We prove a Bernstein type inequality for multivariate quasipolynomials and apply it to carry out the following results. (1) The evaluation of the uniform norm for a quasipolynomial on a convex body V/R by that on a measurable subset of V. (2) The estimate of the BMO-norm for a quasipolynomial in terms of its degree and exponential type. (3) The reverse Ho lder inequality with a dimensionless co...

Two quantitative notions of mixing are the decay correlations and a mix-norm -- negative Sobolev norm intensity can be measured by rates these quantities. From duality, uniformly dominated mix-norm; but they asymptotically faster than mix-norm? We answer this question constructing an observable with correlation that comes arbitrarily close to achieving rate mix-norm. Therefore is sharpest in bo...