A Multiscale Hybrid-Mixed Method with Local Stabilization

Resumo

The Multiscale Hybrid-Mixed (MHM for short) method, first introduced in 2013 for elliptic problems, is based on a hybrid formulation defined at the continuous level over a coarse partition of the domain. The exact solution is decomposed into global and local components. Upon discretization, this structure leads to a natural decoupling between global and local problems: the global problem involves only the degrees of freedom on the skeleton of the coarse mesh, while the local problems are responsible for constructing the multiscale basis functions. Notably, these basis functions can be computed locally and independently, which leverages parallel computing and allows flexibility in solving the local problems. Different methods can be applied depending on the specific needs of each problem. Additionally, in 2015, the MHM method was extended to address diffusive-advective-reactive problems. However, accurately approximating these multiscale basis functions remains a challenge, especially in reaction-dominated or highly heterogeneous scenarios, such as the SPE-10 benchmark. These situations typically require fine local meshes, increasing computational effort. In this work, motivated by the need to improve local computations, we propose the MHM-UNUSUAL method. It incorporates the Unusual Stabilized Finite Element Method (UNUSUAL), originally developed in 2000 for reaction-advection-diffusion and Stokes problems in 2001, which adds terms depending on the residual of the Lagrange equation to the variational formulation, scaled by coefficients and mesh size. This stabilizing mechanism mitigates spurious oscillations in high-reaction regimes without excessive numerical diffusion, enabling the use of coarser local meshes and improving computational efficiency without sacrificing accuracy. The approach unfolds in two steps. First, we extend the UNUSUAL method to reaction-diffusion equation with general mixed boundary conditions and non-constant coefficients, establishing well-posedness and convergence, and proving its superiority over the standard Galerkin method in sharp gradients. Second, we use the UNUSUAL method to solve the local problems in the MHM framework, proving both theoretical robustness and numerical improvements over traditional Galerkin-based local solvers. Validated on Boundary Layer problems and SPE-10, the MHM-UNUSUAL method aims to contribute to the accurate and efficient simulation of complex phenomena in challenging multiscale scenarios.