In this paper, we propose an unconditionally stable and \(L^2\) 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 \(L^2\) 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 \(L^2\) 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 \(L^2\) norm (\(O(h^3)\)). 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 \(L^2\) norm convergence rate. We provide numerical experiments to validate our findings.

在本文中，我们提出了一种无条件稳定且\(L^2\)最优二次有限体积（FV）方案来求解三角形网格上的二维各向异性椭圆方程。在二次 FV 方案中，对偶划分的构造与\(L^2\)误差估计密切相关。虽然文献中已经研究了许多三角形网格上的对偶划分，但只有 Wang 和 Li 提出的一种（SIAM J. Numer. Anal. 54:2729–2749, 2016）被证明可以实现最优\(L^2\ ）范数收敛速度。本文介绍了一种使用多块控制卷构建双分区的新方法，该方法也被证明可以在\(L^2\)范数 ( \(O(h^3)\) ）中最佳收敛。此外，我们提出了从试验空间到测试空间的新映射，这使我们能够证明该方案的 inf-sup 条件独立于底层网格的最小角度而成立。据我们所知，这是第一个在三角形网格上无条件稳定的二次 FV 方案，可实现最佳\(L^2\)范数收敛速度。我们提供数值实验来验证我们的发现。