A multiscale hybrid method for Darcy’s problems using mixed finite element local solvers

Abstract

Multiscale Hybrid Mixed (MHM) method refers to a numerical technique targeted to approximate systems of differential equations with strongly varying solutions. For fluid flow, normal fluxes (multiplier) over macro element boundaries, and coarse piecewise constant potential approximations in each macro element are computed (upscaling). Then, small details are resolved by local problems, using fine representations inside the macro elements, setting the multiplier as Neumann boundary conditions (downscaling). In this work a variant of the method is developed, denoted by MHM-H(div), adopting mixed finite elements at the dowscaling stage, instead of continuous finite elements used in all previous publications of the method. Thus, this alternative MHM method inherits improvements typical of mixed methods, as better flux accuracy, and local mass conservation at the micro scale level inside the macro elements which are important properties for multi-phase flows in rough heterogeneous media. Different two-scale stable space settings are considered. Vector face functions are supposed to have normal components restricted to a given finite dimensional trace space defined over the macro element boundaries. In each macro element, the internal flux components, with vanishing normal traces, and the potential approximations, may be enriched in different extents: with respect to internal mesh size, internal polynomial degree, or both, the choice being determined by the problem at hands. A unified general error analysis of the MHM-H(div) method is presented for all these two-scale space scenarios. Both MHM versions are compared for 2D test problems, with smooth solutions, for convergence rates verification, and for Darcy's flow in heterogeneous media. MHM-H(div) 3D simulations are presented for a known singular Darcy's solution, using adaptive macro partitions, and for an oscillatory permeability scenario.

The multiscale hybrid mixed method in general polygonal meshes

“…where the product on ∂K is the duality pairing between H − 1 2 (∂K) and H 1 2 (∂K). Following closely the arguments given in [2] we can prove that 14) and above and hereafter we lighten the notation and understand the supremum to be taken over sets excluding the zero function, even though this is not specifically indicated.…”

“…On the other hand, unless a mixed finite element method is used as a second order solver (see [14] for an example) in the second level mesh, this is not guaranteed. More precisely, for…”

“…In addition, it is worth mentioning that in the above case, σ h ∈ H(div, Ω). This idea was explored for simplicial elements in [14].…”

NewH(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes

“…• We can interpret the rows of the two-scale pair S γ × U γin as formed by two-scale Poisson-compatible pair V γ × P γin defined in [20], where V γ = V ∂ γ ⊕V γin is a constrained two-scale flux space. • One can derive stable two-scale FE spaces E γ for stress mixed formulations with reduced symmetry for other known single-scale FE settings.…”

“…Our method is based on a divide-and-conquer strategy combined with bubble enrichment techniques and static condensation, which are general-purpose tools widely adopted in multiscale simulations. It shall be denoted by the acronym MHM-WS, for its design is in the spirit of Multiscale Hybrid Mixed (MHM) methods (already applied for Darcy problems [20,27], for displacement-based elasticity formulations [26,35,36], and other contexts therein cited). In summary, this means that the MHM-WS scheme shares with these MHM methods the following characteristics:…”