Finite element approximation and preconditioners for a coupled thermal–acoustic model

We consider a time-harmonic mathematical model for a coupled thermal–acoustic model. We demonstrate coercivity of the PDE system and hence well-posedness and optimal error estimates for standard Galerkin methods. The large scale and ill-conditioning of the discrete system motivates the study of preconditioners. Three block preconditioners are studied. The first two, applying block Jacobi and block Gauss–Seidel approaches to our problem, give provably mesh-independent spectral bounds. The third preconditioner, based on problem-specific physical reasoning, does not have this property. However, on moderate-sized meshes, it considerably outperforms the existing mesh techniques. Numerical experiments in two and three dimensions confirm the theoretical results.