Multiscale Hybrid-Mixed Method

Abstract

This work presents a priori and a posteriori error analyses of a new multiscale hybridmixed method (MHM) for an elliptic model. Specially designed to incorporate multiple scales into the construction of basis functions, this finite element method relaxes the continuity of the primal variable through the action of Lagrange multipliers, while assuring the strong continuity of the normal component of the flux (dual variable). As a result, the dual variable, which stems from a simple postprocessing of the primal variable, preserves local conservation. We prove existence and uniqueness of a solution for the MHM method as well as optimal convergence estimates of any order in the natural norms. Also, we propose a face-residual a posteriori error estimator, and prove that it controls the error of both variables in the natural norms. Several numerical tests assess the theoretical results.

A comprehensive performance evaluation of the BinLPT workload‐aware loop scheduler

“…The Elastodynamics Simulator relies on a family of finite element methods called Multiscale Hybrid‐Mixed (MHM) to carry out high‐order precision simulations of problems with highly heterogeneous coefficients. The simulator computes the numerical solution of each variable of interest (eg, displacement) as a direct sum of a solution to a global problem (defined over the skeleton of a “coarse” mesh) and the solutions to local problems (applied to element‐wise sub‐meshes).…”

“…The Elastodynamics Simulator relies on a family of finite element methods called Multiscale Hybrid-Mixed (MHM) 31,32 to carry out high-order…”

A local-global multiscale mortar mixed finite element method for multiphase transport in heterogeneous media

“…Parallel computed can be easily adopted for the local problems. We remark that the MMMFEM shares some similarities with the Multiscale Hybrid-Mixed finite element methods [4,20], both methods involve solving some local problems and a small dimension global problem. However, the local problems in the Multiscale Hybrid-Mixed finite element methods are Neumann boundary condition problems.…”

A Review of Hybrid High-Order Methods: Formulations, Computational Aspects, Comparison with Other Methods

“…Problem (55) is well-posed, since, according to (49) and (48),Ĥ T pt T , t T q is uniformly equivalent to }t T } 2 T and the inf-sup condition holds. The global (saddle point) problem resulting from the local elimination (55) has the same size and structure as that obtained with the Multiscale Hybrid-Mixed (MHM) method derived in [3,40] on simplicial meshes.…”