A posteriori analysis for iterative solvers for non-autonomous evolution problems

We derive, implement, and test a posteriori error estimates for numerical methods for a non-autonomous linear system that involve iterative solution of the discrete equations. We consider two iterations: the Picard iteration and the Jacobi iteration for solving the discrete matrix-vector equations. To carry out the analysis, we define an appropriate adjoint problem for the numerical approximations using the matricant. We present a number of examples with interesting characteristics to illustrate the effectiveness of the estimate. We also present a comparison between the a posteriori error estimate and a conceptually simpler estimate obtained with a “pseudoadjoint” problem.