Functional Error Estimates for Mimetic Difference Approximations to the Poisson Problem

Functional a posteriori error estimates provide guaranteed upper bounds on the deviation between the exact solution and any approximation in the appropriate functional space, making them agnostic to the discretization method used to obtain the approximation. However, numerical solutions often do not belong to the correct functional space, requiring postprocessing techniques to ensure compatibility with the error estimation framework. In this article, we consider the Poisson equation as a model problem and demonstrate how to postprocess solutions obtained using mimetic differences of the Corbino-Castillo type to enable the application of functional error estimates. Numerical experiments in two dimensions confirm that the proposed postprocessing techniques yield fully computable error estimates that recover ideal convergence rates in the energy norm.