Optimal-order preconditioners for the Morse–Ingard equations

Abstract

The Morse-Ingard equations of thermoacoustics [1] are a system of coupled time-harmonic equations for the temperature and pressure of an excited gas. They form a critical aspect of modeling trace gas sensors. In this paper, we analyze a reformulation of the system that has a weaker coupling between the equations than the original form. We give a Gårding-type inequality for the system that leads to optimal-order asymptotic finite element error estimates. We also develop preconditioners for the coupled system. These are derived by writing the system as a 2 × 2 block system with pressure and temperature unknowns segregated into separate blocks and then using either the block diagonal or block lower triangular part of this matrix as a preconditioner. Consequently, the preconditioner requires inverting smaller, Helmholtz-like systems individually for the pressure and temperature. Rigorous eigenvalue bounds are given for the preconditioned system, and these are supported by numerical experiments.

A Two-Way Coupled Model of Visco-Thermo-Acoustic Effects in Photoacoustic Trace Gas Sensors

“…We reformulate the pressure-temperature subsystem by using (2.1) to eliminate \Delta \tau F from (2.2), which we expect will simplify the theoretical analysis of the finite element preconditioners for the system [19]. This gives \xi \Delta P + a 1 P + a 2 \tau F = ia 3 S. (A.1)…”