Convergence of Chandrashekar’s Second-Derivative Finite-Volume Approximation

Abstract

We consider a slightly modified local finite-volume approximation of the Laplacian operator originally proposed by Chandrashekar (Int J Adv Eng Sci Appl Math 8(3):174–193, 2016, https://doi.org/10.1007/s12572-015-0160-z). The goal is to prove consistency and convergence of the approximation on unstructured grids. Consequently, we propose a semi-discrete scheme for the heat equation augmented with Dirichlet, Neumann and Robin boundary conditions. By deriving a priori estimates for the numerical solution, we prove that it converges weakly, and subsequently strongly, to a weak solution of the original problem. A numerical simulation demonstrates that the scheme converges with a second-order rate.