# Design and analysis of finite volume methods for elliptic equations with oblique derivatives; application to Earth gravity field modelling

## TL;DR

This paper develops and analyzes finite volume methods for elliptic equations with oblique boundary conditions, providing a generic framework, convergence proofs, and practical applications including Earth gravity field modeling.

## Contribution

It introduces a novel generic finite volume framework for oblique boundary conditions, with convergence analysis and specific flux schemes validated by numerical tests and real-world data.

## Key findings

- Optimal convergence rate of O(h) in energy norm.
- Validated methods through extensive 3D numerical tests.
- Successful application to Earth gravity potential data.

## Abstract

We develop and analyse finite volume methods for the Poisson problem with boundary conditions involving oblique derivatives. We design a generic framework, for finite volume discretisations of such models, in which internal fluxes are not assumed to have a specific form, but only to satisfy some (usual) coercivity and consistency properties. The oblique boundary conditions are split into a normal component, which directly appears in the flux balance on control volumes touching the domain boundary, and a tangential component which is managed as an advection term on the boundary. This advection term is discretised using a finite volume method based on a centred discretisation (to ensure optimal rates of convergence) and stabilised using a vanishing boundary viscosity. A convergence analysis, based on the 3rd Strang Lemma \cite{DPD18}, is conducted in this generic finite volume framework, and yields the expected $\mathcal O(h)$ optimal convergence rate in discrete energy norm. We then describe a specific choice of numerical fluxes, based on a generalised hexahedral meshing of the computational domain. These fluxes are a corrected version of fluxes originally introduced in \cite{Medla.et.al2018}. We identify mesh regularity parameters that ensure that these fluxes satisfy the required coercivity and consistency properties. The theoretical rates of convergence are illustrated by an extensive set of 3D numerical tests, including some conducted with two variants of the proposed scheme. A test involving real-world data measuring the disturbing potential in Earth gravity modelling over Slovakia is also presented.