Trace gas sensors can be used to detect and identify very small quantities of gases for applications in such diverse fields as atmospheric chemistry, environmental and  industrial emissions monitoring, explosives detection, industrial  process control, and non-invasive medical diagnostics. The large-scale adoption of trace gas sensors requires sensor systems that are compact, portable, efficient, sensitive, cost-effective and highly reliable. Quartz Enhanced Photoacoustic Spectroscopy (QEPAS) sensors hold promise as a technology that may achieve many of these goals. In particular, QEPAS sensors can be as small as several cubic millimeters, whereas sensors based on other sensitive spectroscopic techniques require large cell volumes of tens to hundreds of cubic centimeters. QEPAS sensors use a quartz tuning fork to detect weak sound waves that are generated when a beam of light from a laser interacts with a trace gas. A major  engineering challenge to overcome before QEPAS  sensors can be widely deployed is to increase their sensitivity and lower their production cost. The overall goal of this project is to develop a computational model for QEPAS sensors that is a significant enhancement over existing models, and to then use this model to determine cost-effective designs that increase the sensitivity of QEPAS sensors. The major mathematical challenge of the project is to develop efficient computational methods to solve the multiphysics equations that form the basis of the model. The project will provide broad training in computational science for two mathematics graduate students from faculty mentors with complementary expertise in the  physics and engineering of the application, mathematical modeling, numerical analysis, and parallel computing.QEPAS sensors employ a resonantly vibrating quartz tuning fork to detect  weak acoustic pressure waves and thermal disturbances which are generated when optical radiation from a laser beam interacts with a trace gas. The project will involve the development and analysis of computational methods to solve a system of Helmholtz equations that describes the interaction between a thermo-visco-acoustic fluid and a resonantly vibrating mechanical structure (a quartz tuning fork). The model will be used to numerically  optimize the QEPAS signal as a function of the geometric parameters of the sensor. The cumulative effect of the damping of the tuning fork by the viscous fluid will be computed in terms of the geometric parameters of the system and physical constants. Consequently, the model will allow for realistic optimization of QEPAS sensors by varying the tuning fork geometry. Furthermore, in some situations, the thermal diffusion wave can dominate the acoustic pressure wave on the surface of the tuning fork, in a phenomenon known as Resonant Opto-Thermo-Acoustic DEtection (ROTADE). Current mathematical descriptions of these sensors cannot capture both QEPAS and ROTADE phenomena simultaneously, although experimental data indicates that depending on the position of the laser beam along the tuning fork axis, both types of trace gas sensing may occur. The new model will allow for simultaneous simulation of both types of sensor systems.  Preliminary analytical and computational results show that standard finite element methods for solving the equations in the model are ineffective due to small parameters in the equations and  the high wave number of the solution. The small parameters produce an ill-conditioned linear system resulting from the finite element discretizations of the equations, while the high wave numbers can cause large phase errors in the computed solution (pollution error). This project will advance knowledge in computational mathematics by developing and analyzing block preconditioners for the multiphysics Helmholtz system. In addition, methods for reducing the pollution error will be developed  by extending higher-order finite element and interior penalty stabilization methods originally proposed for scalar Helmholtz equations to the multiphysics Helmholtz system. The techniques developed will be relevant for more general coupled Helmholtz systems such as those which arise in the study of thermal phenomena near thin bodies, the design of hearing aid transducers and micro-electrical-mechanical devices.