A stopping criterion for the iterative solution of partial differential equations

Highlights: • Mesh independent volume norms for both vectors and matrices. • Inexpensive methods for estimating the minimum singular value of a PDE matrix. • A robust stopping criteria for complex, nonlinear, non-monotonic PDE problems. A stopping criterion for iterative solution methods is presented that accurately estimates the solution error using low computational overhead. The proposed criterion uses information from prior solution changes to estimate the error. When the solution changes are noisy or stagnating it reverts to a less accurate but more robust, low-cost singular value estimate to approximate the error given the residual. This estimator can also be applied to iterative linear matrix solvers such as Krylov subspace or multigrid methods. Examples of the stopping criterion's ability to accurately estimate the non-linear and linear solution error are provided for a number of different test cases in incompressible fluid dynamics.