Finite volumes and mixed Petrov-Galerkin finite elements : the unidimensional problem

TL;DR

This paper introduces a new finite volume method for the 1D Laplace operator using mixed Petrov-Galerkin finite elements, establishing stability and convergence with optimal accuracy under certain conditions.

Contribution

It formulates a novel finite volume scheme with a universal weighting function satisfying a compatibility condition, linking it to the inf-sup stability criterion.

Findings

Proves the equivalence of the compatibility condition to the inf-sup property.

Demonstrates convergence of the scheme under regularity assumptions.

Achieves optimal order of accuracy in Hilbert space norms.

Abstract

For Laplace operator in one space dimension, we propose to formulate the heuristic finite volume method with the help of mixed Petrov-Galerkin finite elements. Weighting functions for gradient discretization are parameterized by some universal function. We propose for this function a compatibility interpolation condition and we prove that such a condition is equivalent to the inf-sup property when studying stability of the numerical scheme. In the case of stable scheme and under two distinct hypotheses concerning the regularity of the solution, we demonstrate convergence of the finite volume method in appropriate Hilbert spaces and with optimal order of accuracy.