# A second-order face-centred finite volume method for elliptic problems

## TL;DR

This paper introduces a second-order face-centred finite volume method (FCFV) that defines solutions on mesh faces, offering robustness against mesh distortion and eliminating the need for gradient reconstruction, applicable to complex 3D geometries.

## Contribution

It presents a novel second-order FCFV method based on a mixed formulation that computes solutions on faces, improving robustness and efficiency over traditional cell-centred and vertex-centred FV methods.

## Key findings

- Method achieves second-order accuracy.
- Insensitive to mesh distortion and stretching.
- Effective in complex 3D geometries.

## Abstract

A second-order face-centred finite volume method (FCFV) is proposed. Contrary to the more popular cell-centred and vertex-centred finite volume (FV) techniques, the proposed method defines the solution on the faces of the mesh (edges in two dimensions). The method is based on a mixed formulation and therefore considers the solution and its gradient as independent unknowns. They are computed solving an element-by-element problem after the solution at the faces is determined. The proposed approach avoids the need of reconstructing the solution gradient, as required by cell-centred and vertex-centred FV methods. This strategy leads to a method that is insensitive to mesh distortion and stretching. The current method is second-order and requires the solution of a global system of equations of identical size and identical number of non-zero elements when compared to the recently proposed first-order FCFV. The formulation is presented for Poisson and Stokes problems. Numerical examples are used to illustrate the approximation properties of the method as well as to demonstrate its potential in three dimensional problems with complex geometries. The integration of a mesh adaptive procedure in the FCFV solution algorithm is also presented.