Reduced basis approximations of the solutions to spectral fractional diffusion problems

We consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.

Trace Gas Detection System Based on All-Optical Quartz-Enhanced Photoacoustic Spectroscopy

An all-optical quartz-enhanced photoacoustic spectroscopy system (QEPAS) with quadrature point stabilization for trace gas detection was reported. The extrinsic interferometry-based optical fiber Fabry–Perot sensor with quadrature point self-stabilization for detection of quartz prong vibration was used to replace the conventional one. The optimal coefficient of the modulation depth was ∼2.2 theoretically and experimentally, corresponding to the modulation depth of ∼0.1795 cm−1 at an acetylene (C2H2) absorption line of 6534.36 cm−1. Furthermore, the enhancement of QEPAS signal was obtained by using different microresonators. The minimum detectable limit of ∼580 parts per billion by volume (ppbv) was obtained. A normalized noise equivalent absorption coefficient for C2H2 of 2.95 × 10−7 cm−1·W·Hz–1/2 was obtained. The detection sensitivity was enhanced by a factor of ∼2.1 in comparison to the conventional QEPAS system. The linear correlation coefficient of the QEPAS signal response to the C2H2 concentration was 0.998 within the range from 10 parts per million by volume (ppmv) to 500 ppmv. Finally, the long-term stability of the QEPAS system was evaluated using Allan deviation analysis, and the ultimate detection limit of ∼130 ppbv was reached for an optimum averaging time of ∼108 s at atmospheric pressure and ambient temperature.

Improved Computational Efficiency of Nearest-Nodes Finite Element Method (NN-FEM)

The recently developed nearest-nodes finite element method (NN-FEM) has a number of advantages over the conventional finite element method (FEM). The most attractive one is that its performance is nearly not affected by element distortion. However, low computational efficiency of NN-FEM is a major concern, as in the original NN-FEM a local problem has to be solved at each quadrature point to construct shape functions there. In this paper, a new strategy is introduced in NN-FEM for constructing shape functions aiming at improving its computational efficiency. The strategy is, for regular-shape elements, shape functions are constructed at the element center and the obtained shape functions are used at all quadrature points in the element; only for severely distorted elements, shape functions are constructed separately at each quadrature point. Numerical results show that computational efficiency of the NN-FEM can be significantly improved with the above strategy, while other performance of the method is not affected.