Abstract Various computational problems as, e.g., equations with fractional diffusion operators, evaluation of high-dimensional integrals, the Møller–Plesset approach in quantum chemistry, etc., are easily solved by using approximations by rational functions or by exponential sums. In the case of Cauchy–Stieltjes or, respectively, Lebesgue–Stieltjes functions we provide a uniform proof of upper bounds of the convergence rates of their best approximations by rational functions or exponential sums. It turns out that the convergence rate by rational approximation is better than for exponential sums. We extend the analysis also to the approximation on infinite intervals and to the best approximation of the relative error. Instead of looking for the best approximation one can use the computationally cheaper quadrature method, in particular the sinc quadrature. The corresponding sharp error estimates are determined. The theoretical results are supported by numerical results.

Abstract In this paper, we propose a Barrett–Garcke–Nürnberg (BGN) method for evolving curves under a prescribed background velocity field in $${{\mathbb {R}}}^2$$ R 2 and present the corresponding convergence analysis. Unlike mean curvature flow and surface diffusion, where the evolution velocities inherently exhibit parabolicity, this case is dominated by transport which poses a significant difficulty in establishing convergence proofs. To address the challenges imposed by this transport-dominant nature, we derive several discrete energy estimates of the transport type on discretized polynomial curves within the framework of the projection error. The use of the projection error is indispensable as it provides crucial additional stability through its orthogonality structure. We prove that the proposed method converges sub-optimally in the $$L^2$$ L 2 norm, and this is the first convergence proof for a fully discrete numerical method solving the evolution of curves driven by general flows.

Abstract We present a simple universal algorithm for high-dimensional integration which has the optimal error rate (independent of the dimension) in all weighted Korobov classes both in the randomized and the deterministic setting. Our theoretical findings are complemented by numerical tests.

Abstract Superlinear convergence occurs frequently as a desirable second stage in a Krylov iteration, and its understanding has undergone a deep development. This paper aims to contribute to this long history of analysis for a class of nonlinear systems of partial differential equations (PDEs) covering reaction–diffusion–convection processes, discretized with finite elements (FE), linearized with a Newton–Krylov method and applying suitable operator preconditioning. We give practical estimates of the rate of superlinear convergence of the arising Krylov iterations in terms of the accessible data of the PDEs. We are in particular interested in the effect of integrability properties of possibly unbounded sources. We obtain robust superlinear rates both in the sense of mesh independence and of uniform behaviour w.r.t. the outer Newton iterations. The numerical examples reinforce the theoretical results.

Abstract In this work, we investigate the boundary controllability of time-discrete parabolic systems, uniformly with respect to the discretization parameter. To establish our main results, we adapt and extend the moment method first introduced by Fattorini and Russell, to the time-discrete setting. While this method has proven effective in the continuous framework and in the space-discrete case, its adaptation to time-discrete systems is challenging due to the fact that the more accurate results in the field are based on several complex analysis tools. To overcome this, we introduce a new alternative proof for constructing biorthogonals to (generalized) exponential functions in the continuous setting, based on the original proof by Fattorini and Russell which avoids the use of these tools, and taking ideas from the block moment method introduced by Benabdallah, Morancey and the first author of this work. We then manage to adapt this strategy to the discrete setting, enabling the construction and estimation of biorthogonal families for some time-discrete functions that play the same role as the exponentials at the discrete level. Our results show that these biorthogonals can be uniformly estimated for a finite portion of the spectrum, determined by the discretization parameter $$\tau$$ , and that converges to the whole spectrum as $$\tau$$ goes to zero. Using this tool, we prove a relaxed null-controllability result for discrete parabolic systems. It says that there exists a bounded sequence of time discrete controls that makes the solution reach a target at final time tending to zero exponentially fast as the discretization parameter $$\tau$$ goes to zero. We will also study the effects of the existence of a minimal null-control time and how this phenomenon translates into the discrete world.