$L^2$ Error Estimates for High Order Finite Volume Methods on Triangular Meshes

Abstract

We establish a unified framework for $L^2$ error analysis for high order Lagrange finite volume methods on triangular meshes. Orthogonal conditions are originally proposed to construct dual partitions on triangular meshes, such that the corresponding finite volume method (FVM) schemes hold optimal $L^2$ norm convergence order. Moreover, with the Aubin--Nitsche technique, we prove the optimal $L^2$ error estimate for high order FVM schemes on triangular meshes. Some numerical experiments are presented to demonstrate the proved result.