13th International Workshop on Variational Multiscale and Stabilized Finite Elemements (VMS 18) - Abstract

Barrenechea, Gabriel

The multiscale hybrid-mixed finite element method in polygonal meshes

In this talk the recent extension of the Multiscale Hybrid-Mixed (MHM) method, originally proposed in [1], to the case of general polygonal meshes (that can be non-convex and non-conforming as well) will be presented. We present new stable multiscale finite elements such that they preserve the well-posedness, super-convergence and local conservation properties of the original M HM method under mild regularity conditions on the polygons. More precisely, we show that piecewise polynomial of degree k-1 and k, k ≥ 1, for the Lagrange mu ltipliers (flux) along with continuous piecewise polynomial interpolations of degree $k$ posed on second-level sub-meshes are stable if the latter is refined enough. Suc h one- and two-level discretization impact the error in a way that the discrete prima l (pressure) and dual (velocity) variables achieve super-convergence in the natural norms under extra local regularity only. Numerical tests illustrate theoretical resu lts and the flexibility of the approach.
This work has been carried out in collaboration with Fabrice Jaillet (Lyon 1, France), Diego Paredes (UCV, Valparaiso, Chile), and Frederic Valentin (LNCC, Brazil).

References
[1] Araya, R., Harder, C. , Paredes, D. and Valentin, F.: Multiscale hybrid-mixed method, Journal on Numerical Analysis, 51(6), 3505--3531, (2013).