Mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media

TL;DR

This paper introduces hybrid finite volume element schemes for optimal control of immiscible flow in porous media, providing error estimates and demonstrating optimal convergence rates.

Contribution

It presents a novel combination of mixed and discontinuous finite volume methods with an optimise-then-discretise approach for better numerical approximation.

Findings

Derived error estimates with optimal convergence rates

Developed schemes with minimum regularity requirements

Validated the effectiveness of the proposed methods

Abstract

We introduce a family of hybrid discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media. The proposed schemes are based on mixed and discontinuous finite volume element methods in combination with the optimise-then-discretise approach for the approximation of the optimal control problem, leading to nonsymmetric algebraic systems, and employing minimum regularity requirements. Estimates for the error (between a local reference solution of the infinite dimensional optimal control problem and its hybrid approximation) measured in suitable norms are derived, showing optimal orders of convergence.