An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons

In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included.

New-on-line method for water isotope analysis of speleothem fluid inclusions using laser absorption spectroscopy (WS-CRDS)

A new online method to analyse water isotopes of speleothem fluid inclusions using a wavelength scanned cavity ring down spectroscopy (WS-CRDS) instrument is presented. This novel technique allows us simultaneously to measure hydrogen and oxygen isotopes for a released aliquot of water. To do so, we designed a new simple line that allows the online water extraction and isotope analysis of speleothem samples. The specificity of the method lies in the fact that fluid inclusions release is made on a standard water background, which mainly improves the δ D robustness. To saturate the line, a peristaltic pump continuously injects standard water into the line that is permanently heated to 140 °C and flushed with dry nitrogen gas. This permits instantaneous and complete vaporisation of the standard water, resulting in an artificial water background with well-known δ D and δ18O values. The speleothem sample is placed in a copper tube, attached to the line, and after system stabilisation it is crushed using a simple hydraulic device to liberate speleothem fluid inclusions water. The released water is carried by the nitrogen/standard water gas stream directly to a Picarro L1102-i for isotope determination. To test the accuracy and reproducibility of the line and to measure standard water during speleothem measurements, a syringe injection unit was added to the line. Peak evaluation is done similarly as in gas chromatography to obtain &delta D; and δ18O isotopic compositions of measured water aliquots. Precision is better than 1.5 ‰ for δ D and 0.4 ‰ for δ18O for water measurements for an extended range (−210 to 0 ‰ for δ D and −27 to 0 ‰ for δ18O) primarily dependent on the amount of water released from speleothem fluid inclusions and secondarily on the isotopic composition of the sample. The results show that WS-CRDS technology is suitable for speleothem fluid inclusion measurements and gives results that are comparable to the isotope ratio mass spectrometry (IRMS) technique.