In this paper, we introduce and analyze a novel family of quadratic finite volume element (FVE) schemes defined over triangular meshes for solving elliptic equations. The schemes employ second-degree Gauss quadrature points on element edges and incorporate a two-parameter ( \(\alpha \) , \(\beta \) ) partition of the triangular elements. Utilizing an innovative mapping from the trial space to the test space, we establish that specific parameter choices \(\beta = 1 - \dfrac{2\alpha }{3} + \sqrt{ \left( 1-\dfrac{2\alpha }{3}\right) ^2 - \dfrac{2}{27\alpha } }\) endow the schemes with two distinct orthogonality conditions. Leveraging these orthogonality conditions, we derive a sufficient criterion ensuring the stability and \(L^2\) -optimal convergence of the FVE solution on triangular meshes. Significantly, for \(\alpha = \dfrac{5+2\sqrt{3}+\sqrt{1+2\sqrt{3}}}{12}\) and the corresponding \(\beta \) , the scheme’s stability and \(L^2\) -optimal convergence are proven to be independent of the minimum angle of the underlying triangulation. Numerical experiments confirm the theoretical convergence rates and validate the robustness of the proposed schemes.
