A posteriori analysis of an iterative multi-discretization method for reaction–diffusion systems

• We develop a multidiscretization method for multiscale parabolic problems.
• We derive an accurate a posteriori error estimate.
• We account for the effects of finite iteration on the discrete solution.
• We apply the method to a number of test cases and an interesting application.

This paper is concerned with the accurate computational error estimation of numerical solutions of multi-scale, multi-physics systems of reaction–diffusion equations. Such systems can present significantly different temporal and spatial scales within the components of the model, indicating the use of independent discretizations for different components. However, multi-discretization can have significant effects on accuracy and stability. We perform an adjoint-based analysis to derive asymptotically accurate a posteriori error estimates for a user-defined quantity of interest. These estimates account for leading order contributions to the error arising from numerical solution of each component, an error due to incomplete iteration, an error due to linearization, and for errors arising due to the projection of solution components between different spatial meshes. Several numerical examples with various settings are given to demonstrate the performance of the error estimators.