Numerical analysis

About: Numerical analysis is a research topic. Over the lifetime, 52236 publications have been published within this topic receiving 1224230 citations. The topic is also known as: numerical computation & numerical mathematics.

Convergence analysis of finite element methods for H(curl; Ω)-elliptic interface problems

Abstract: In this article we investigate the analysis of a finite element method for solving H(curl; Ω)-elliptic interface problems in general three-dimensional polyhedral domains with smooth interfaces. The continuous problems are discretized by means of the first family of lowest order Nedelec H(curl; Ω)-conforming finite elements on a family of tetrahedral meshes which resolve the smooth interface in the sense of sufficient approximation in terms of a parameter ? that quantifies the mismatch between the smooth interface and the triangulation. Optimal error estimates in the H(curl; Ω)-norm are obtained for the first time. The analysis is based on a ?-strip argument, a new extension theorem for H 1(curl; Ω)-functions across smooth interfaces, a novel non-standard interface-aware interpolation operator, and a perturbation argument for degrees of freedom for H(curl; Ω)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution.

The flow of thin liquid films over spinning disks: Hydrodynamics and mass transfer

Abstract: We study the hydrodynamics and mass transfer associated with gas absorption into a thin liquid film flowing over a spinning disk. We use the thin-layer approximation in conjunction with the Karman–Polhausen method to derive evolution equations for the film thickness and the volumetric flow rates in the radial and azimuthal directions. We also use the integral balance method to derive evolution equations for the thickness of the diffusion boundary layer as well as the concentration of solute at the disk surface. Numerical solutions of these partial differential equations, which govern the hydrodynamics and the associated mass transfer, reveal the formation of large finite-amplitude waves and elucidate their significant effect on the mass-transfer characteristics. We illustrate this dependence quantitatively by examining the effect of system parameters on the time-averaged and spatially averaged Sherwood numbers. The results are assessed by comparison with computations of the parabolized convective diffusio...

How far away is far enough for extracting numerical waveforms, and how much do they depend on the extraction method?

Abstract: We present a method for extracting gravitational waves from numerical spacetimes which generalizes and refines one of the standard methods based on the Regge-Wheeler-Zerilli perturbation formalism. At the analytical level, this generalization allows a much more general class of slicing conditions for the background geometry, and is thus not restricted to Schwarzschild-like coordinates. At the numerical level, our approach uses high-order multi-block methods, which improve both the accuracy of our simulations and of our extraction procedure. In particular, the latter is simplified since there is no need for interpolation, and we can afford to extract accurate waves at large radii with only little additional computational effort. We then present fully nonlinear three-dimensional numerical evolutions of a distorted Schwarzschild black hole in Kerr-Schild coordinates with an odd parity perturbation and analyse the improvement that we gain from our generalized wave extraction, comparing our new method to the standard one. In particular, we analyse in detail the quasinormal frequencies of the extracted waves, using both methods. We do so by comparing the extracted waves with one-dimensional high resolution solutions of the corresponding generalized Regge-Wheeler equation. We explicitly see that the errors in the waveforms extracted with the standard method at fixed, finite extraction radii do not converge to zero with increasing resolution. We find that even with observers as far out as R = 80M-which is larger than what is commonly used in state-of-the-art simulations-the assumption in the standard method that the background is close to having Schwarzschild-like coordinates increases the error in the extracted waves considerably. Furthermore, those errors are dominated by the extraction method itself and not by the accuracy of our simulations. For extraction radii between 20M and 80M and for the resolutions that we use in this paper, our new method decreases the errors in the extracted waves, compared to the standard method, by between one and three orders of magnitude. In a general scenario, for example a collision of compact objects, there is no precise definition of gravitational radiation at a finite distance, and gravitational wave extraction methods at such distances are thus inherently approximate. The results of this paper bring up the possibility that different choices in the wave extraction procedure at a fixed and finite distance may result in relative differences in the waveforms which are actually larger than the numerical errors in the solution.