A Reduced Basis Method for Darcy flow systems that ensures local mass conservation by using exact discrete complexes

TL;DR

This paper introduces a reduced basis method for Darcy flow in porous media that guarantees local mass conservation by leveraging exact discrete complexes and is applicable to various discretization techniques.

Contribution

It develops a novel reduced basis approach that ensures local mass conservation in Darcy flow simulations using exact co-chain complexes.

Findings

Mass conservation is guaranteed regardless of reduced basis quality.

The method is applicable to mixed finite and virtual element methods.

Extension to fractured porous media is demonstrated.

Abstract

A solution technique is proposed for flows in porous media that guarantees local conservation of mass. We first compute a flux field to balance the mass source and then exploit exact co-chain complexes to generate a solenoidal correction. A reduced basis method based on proper orthogonal decomposition is employed to construct the correction and we show that mass balance is ensured regardless of the quality of the reduced basis approximation. The method is directly applicable to mixed finite and virtual element methods, among other structure-preserving discretization techniques, and we present the extension to Darcy flow in fractured porous media.