Error estimate of the Non Intrusive Reduced Basis method with finite volume schemes

TL;DR

This paper extends error estimation techniques for the Non Intrusive Reduced Basis method from finite element to finite volume schemes, demonstrating that FEM-based estimates also apply to FV methods used in industrial simulations.

Contribution

It generalizes the NIRB error estimates from FEM to finite volume schemes, including hybrid-mimetic and two-point flux approximation methods.

Findings

Error estimates for FEM also hold for FV schemes.

The method is applicable to hybrid-mimetic finite difference and TPFA schemes.

Provides theoretical foundation for using NIRB in industrial FV simulations.

Abstract

The context of this paper is the simulation of parameter-dependent partial differential equations (PDEs). When the aim is to solve such PDEs for a large number of parameter values, Reduced Basis Methods (RBM) are often used to reduce computational costs of a classical high fidelity code based on Finite Element Method (FEM), Finite Volume (FVM) or Spectral methods. The efficient implementation of most of these RBM requires to modify this high fidelity code, which cannot be done, for example in an industrial context if the high fidelity code is only accessible as a "black-box" solver. The Non Intrusive Reduced Basis method (NIRB) has been introduced in the context of finite elements as a good alternative to reduce the implementation costs of these parameter-dependent problems. The method is efficient in other contexts than the FEM one, like with finite volume schemes, which are more often used in an industrial environment. In this case, some adaptations need to be done as the degrees of freedom in FV methods have different meenings. At this time, error estimates have only been studied with FEM solvers. In this paper, we present a generalisation of the NIRB method to Finite Volume schemes and we show that estimates established for FEM solvers also hold in the FVM setting. We first prove our results for the hybrid-Mimetic Finite Difference method (hMFD), which is part the Hybrid Mixed Mimetic methods (HMM) family. Then, we explain how these results apply more generally to other FV schemes. Some of them are specified, such as the Two Point Flux Approximation (TPFA).