Steady filtration problems with seawater intrusion: macro‐hybrid penalized finite element approximations

Abstract

SUMMARYMacro-hybrid penalized ÿnite element approximations are studied for steady ÿltration problems with seawater intrusion. On the basis of nonoverlapping domain decompositions with vertical interfaces, sections of coastal aquifers are decomposed into subsystems with simpler geometries and small scales, interconnected via transmission conditions of pressure and ux continuity. Corresponding local penalized formulations are derived from the global penalized variational formulation of the two-free boundary ow problem, with continuity transmission conditions modelled variationally in a dual sense. Then, macrohybrid ÿnite element approximations are derived for the system, deÿned on independent subdomain grids. Parallel relaxation penalty-duality algorithms are proposed from ÿxed-point problem characterizations. Numerical experiments exemplify the macro-hybrid penalized theory, showing a good agreement with previous primal conforming penalized ÿnite element approximations (Comput. Methods Appl. Mech. Engng. 2000; 190:609-624).

Analysis of augmented three-field macro-hybrid mixed finite element schemes

“…With respect to the series of Darcy flow examples, we should mention that parallel penalty-duality algorithms have been proved to be very efficient in [6], as well as for related filtration problems with seawater intrusion in [5].…”

Steady filtration problems with seawater intrusion: macro-hybrid primal mixed variational analysis

“…Following our previous study [15], we take into account the fundamental macro-hybrid compatibility property of the Dirichlet internal boundary trace operators [ À e ],…”

“…There, on the basis of nonoverlapping domain decompositions, with dualized transmission conditions, across vertical interfaces, finite element schemes, defined on non-matching grids, as well as parallel relaxation penalty-duality algorithms, were considered. As demonstrated in [15], the verticality of the subdomain interfaces turns out to be an essential constraint for spatial decompositions, in order to guarantee well-posedness of corresponding local flow subproblems.…”

“…Then, through duality principles and methods [16,17], that establish relationship between primal and dual solution components, we are able to apply the pressure-primal analysis developed in [1,2,15] to corresponding penalized macro-hybrid formulations, based on non-overlapping domain decompositions, with vertical interfaces. Importantly, all of these decomposed mixed models may be implemented on independent subdomain finite element grids, to which parallel relaxation penalty-duality algorithms are derivable, for numerical resolution.…”

“…Recently in [15], we proceeded to study macro-hybrid penalized formulations of the two-free boundary problem. There, on the basis of nonoverlapping domain decompositions, with dualized transmission conditions, across vertical interfaces, finite element schemes, defined on non-matching grids, as well as parallel relaxation penalty-duality algorithms, were considered.…”