Adaptive vertex-centered finite volume methods with convergence rates

TL;DR

This paper proves that an adaptive vertex-centered finite volume method achieves linear convergence with optimal algebraic rates, addressing challenges unique to finite volume methods without Galerkin orthogonality.

Contribution

It establishes the first convergence and optimal rate results for adaptive finite volume methods, overcoming the lack of Galerkin orthogonality.

Findings

Linear convergence of the adaptive algorithm

Optimal algebraic convergence rates achieved

Addresses challenges due to absence of Galerkin orthogonality

Abstract

We consider the vertex-centered finite volume method with first-order conforming ansatz functions. The adaptive mesh-refinement is driven by the local contributions of the weighted-residual error estimator. We prove that the adaptive algorithm leads to linear convergence with generically optimal algebraic rates for the error estimator and the sum of energy error plus data oscillations. While similar results have been derived for finite element methods and boundary element methods, the present work appears to be the first for adaptive finite volume methods, where the lack of the classical Galerkin orthogonality leads to new challenges.