Numerical analysis of a multi-physics model for trace gas sensors.

Trace gas sensors are currently used in many applications from leak detection to national security and may some day help with disease diagnosis. These sensors are modelled by a coupled system of complex elliptic partial differential equations for pressure and temperature. Solutions are approximated using the finite element method which we will show admits a continuous and coercive variational problem with optimal H¹ and L² error estimates. Numerically, the finite element discretization yields a skew-Hermitian dominant matrix for which classical algebraic preconditioners quickly degrade. We develop a block preconditioner that requires scalar Helmholtz solutions to apply but gives a very low outer iteration count. To handle this, we explore three preconditioners for the resulting linear system. First we analyze the classical block Jacobi and block Gauss-Seidel preconditions before presenting a custom, physics based preconditioner. We also present analysis showing eigenvalues of the preconditioned system are mesh-dependent but with a small coefficient. Numerical experiments confirm our theoretical discussion