Preprint 2023-29
Rodolfo Araya, Fabrice Jaillet, Diego Paredes, Frederic Valentin:
Generalizing the Multiscale Hybrid-Mixed Method for Reactive-Advective-Diffusive Equations
Abstract:
We propose a new family of multiscale hybrid mixed methods (MHM) for the reactive-advective-diffusive (RAD) equation in complex domains. It generalizes the MHM methods originally proposed in Harder, Paredes and Valentin (2013 and 2015) to polytopal meshes and covers all asymptotic regimes of the model within a single mathematical framework. As a result, the skeletal MHM method changes its structure automatically, from primal to mixed forms, depending on the asymptotic of local RAD solutions, which respond to multiscale basis functions at the element level. We establish the existence, uniqueness, and optimality of the MHM solution with respect to two-scale mesh parameters, relating it to the solution of a discrete primal hybrid version of the RAD model. Furthermore, we estimate the conditioning of the matrices associated with the local problems responsible for upscaling, from which we establish upper limits for the condition number of the algebraic system associated with the MHM method. Numerical experiments validate theoretical results.
