Springer Science and Business Media LLC

The celebrated CFL condition for discretizations of hyperbolic PDEs is shown to be equivalent to some results of Jeltsch and Nevanlinna concerning regions of stability ofk-step,m-stage linear methods for the integration of ODEs. We characterize the methods for the numerical integration of the model equation,u t=u x which are weakly stable when the mesh-ratio takes the maximum value allowed by the CFL condition. We provide new equivalence theorems between stability and convergence, which improve on the classical results.

Optimal order error estimates in H 1, for the Q 1 isoparametric interpolation were obtained in Acosta and Durán (SIAM J Numer Anal37, 18–36, 1999) for a very general class of degenerate convex quadrilateral elements. In this work we show that the same conlusions are valid in W 1,p for 1≤ p < 3 and we give a counterexample for the case p  ≥  3, showing that the result cannot be generalized for more regular functions. Despite this fact, we show that optimal order error estimates are valid for any p  ≥  1, keeping the interior angles of the element bounded away from 0 and π, independently of the aspect ratio. We also show that the restriction on the maximum angle is sharp for p  ≥  3.

A unique correspondence between (m, n) rational approximations to exp (q) of order at leastm and a polynomial of degreen, theC-polynomial, is obtained. This polynomial is then used to find an effective result regarding theA-acceptability of these approximations.

In this paper we develop a sublinear-time compressive sensing algorithm for approximating functions of many variables which are compressible in a given Bounded Orthonormal Product Basis (BOPB). The resulting algorithm is shown to both have an associated best s-term recovery guarantee in the given BOPB, and also to work well numerically for solving sparse approximation problems involving functions contained in the span of fairly general sets of as many as $$\sim 10^{230}$$ orthonormal basis functions. All code is made publicly available. As part of the proof of the main recovery guarantee new variants of the well known CoSaMP algorithm are proposed which can utilize any sufficiently accurate support identification procedure satisfying a Support Identification Property (SIP) in order to obtain strong sparse approximation guarantees. These new CoSaMP variants are then proven to have both runtime and recovery error behavior which are largely determined by the associated runtime and error behavior of the chosen support identification method. The main theoretical results of the paper are then shown by developing a sublinear-time support identification algorithm for general BOPB sets which is robust to arbitrary additive errors. Using this new support identification method to create a new CoSaMP variant then results in a new robust sublinear-time compressive sensing algorithm for BOPB-compressible functions of many variables.

Some methods of accelerating the convergence of sequences are studied herein. It is shown that, in some special cases, the convergence of the ε-algorithm of Wynn is assured. A modification of this algorithm is then proposed which may still accelerate the convergence toward the limit. Finally, a new ϱ-algorithm is given in which the dependence with respect to a parameter is explicitely introduced.

Variational boundary integral equations for Maxwell's equations on Lipschitz surfaces in ${\mathbb R}^3$ are derived and their well-posedness in the appropriate trace spaces is established. An equivalent, stable mixed reformulation of the system of integral equations is obtained which admits discretization by Galerkin boundary elements based on standard spaces. On polyhedral surfaces, quasioptimal asymptotic convergence of these Galerkin boundary element methods is proved. A sharp regularity result for the surface multipliers on polyhedral boundaries with plane faces is established.

The proposed method is based on an additive decomposition of the differential operator and the subsequent fitted discretization of the resulting components. For standard situations, the derived stability and error estimates in the energy norm qualitatively coincide with well-known estimates. In the case of small diffusion, a uniform error estimate with reduced order is obtained.

Letf(z) be a function analytic in a neighbourhood of zero. For each pair of non-negative integers (m, n), form then byn Toeplitz determinantD(m/n) whose entries are the Maclaurin series coefficients off, namely, $$D(m/n): = det[f^{(m + j - k)} (0)/(m + j - k)!]_{j,k = 1'}^n $$ where we definef (s) (0)/s!≔0, ifs<0. A classical theorem of Kronecker asserts thatf(z) is a rational function if and only if there existm 0 andn 0 such thatD(m/n)=0 form≧m 0 andn≧n 0. In some important recent work, such as the solution of Meinardus's Conjecture, it has been found useful to form Padé approximants not at 0, but at different points near 0. In questions regarding normality of these Padé approximants with a shifting origin, one considers then byn determinantD(m/n; u) which is defined by (1), but with 0 replaced byu. In this spirit, we prove thatf(z) is a rational function if and only if there exists asingle pair of positive integers (m, n) such thatD(m/n; u) is identically zero foru in a neighbourhood of zero. Further, we deduce that except possibly for countably many values ofu, the Padé table of a non-rationalf(z) atz=u is normal, that isD(m/n; u)≠0, for allm, n=0, 1, 2,....

We shall consider an application of simple exponential splines to the numerical solution of singular perturbation problem. The computational effort involved in our collocation method is less than that required for the other methods of exponential type.