An unconditionally stable and L² optimal quadratic finite volume scheme over triangular meshes for anisotropic elliptic equations

In this paper, we propose an unconditionally stable and L² optimal quadratic finite volume (FV) scheme for solving the two-dimensional anisotropic elliptic equation on triangular meshes. In quadratic FV schemes, the construction of the dual partition is closely related to the L² error estimate. While many dual partitions over triangular meshes have been investigated in the literature, only the one proposed by Wang and Li (SIAM J. Numer. Anal. 54:2729–2749, 2016) has been proven to achieve optimal L² norm convergence rate. This paper introduces a novel approach for constructing the dual partition using multiblock control volumes, which is also shown to optimally converge in the L² norm (O (h³)). Furthermore, we present a new mapping from the trial space to the test space, which enables us to demonstrate that the inf-sup condition of the scheme holds independently of the minimal angle of the underlying mesh. To the best of our knowledge, this is the first unconditionally stable quadratic FV scheme over triangular meshes that achieves optimal L² norm convergence rate. We provide numerical experiments to validate our findings. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.