A Boundary-Layer Preconditioner for Singularly Perturbed Convection Diffusion
Motivated by a wide range of real-world problems whose solutions exhibit boundary and interior layers, the numerical analysis of discretizations of singularly perturbed differential equations is an established sub-discipline within the study of the numerical approximation of solutions to differential equations. Consequently, much is known about how to accurately and stably discretize such equations on \textit{a priori} adapted meshes, in order to properly resolve the layer structure present in their continuum solutions. However, despite being a key step in the numerical simulation process, much less is known about the efficient and accurate solution of the linear systems of equations corresponding to these discretizations. In this paper, we discuss problems associated with the application of direct solvers to these discretizations, and we propose a preconditioning strategy that is tuned to the matrix structure induced by using layer-adapted meshes for convection-diffusion equations, proving a strong condition-number bound on the preconditioned system in one spatial dimension, and a weaker bound in two spatial dimensions. Numerical results confirm the efficiency of the resulting preconditioners in one and two dimensions, with time-to-solution of less than one second for representative problems on $1024\times 1024$ meshes and up to $40\times$ speedup over standard sparse direct solvers.