Some observations on preconditioning for non-self-adjoint and time-dependent problems

Numerical Linear Algebra—specifically the computational solution of equations—forms a significant part of Computational Methods for Partial Differential Equations. Here we discuss the contrast between the solution of symmetric systems of equations that arise from self-adjoint problems and non-symmetric systems that arise from non-self-adjoint problems when iterative methods are employed; such methods are the only feasible methods for very large scale computation with PDEs. We then go on to consider non-symmetric all-at-once systems that arise in approximation of time-dependent problems, discuss causality and the parallel-in-time paradigm, suggesting an approach that involves preconditioning initial value problems with time-periodic problems.