Global $L^2$ Superconvergence of a Family of Quadratic Finite Volume Schemes on Tetrahedral Meshes

Source abstract

This paper investigates the global $L^2$ superconvergence of a family of quadratic finite volume schemes on tetrahedral meshes. By constraining the dual mesh with orthogonal conditions, we consider a family of quadratic finite volume schemes. We note that the orthogonal conditions are also fundamental for achieving optimal $L^2$ error estimates. First, we prove the weak estimate of the second type, which plays an important role in deriving the global $L^2$ superconvergence. We then prove the global superconvergence result $||u_h−I^2_h u||_0 = \mathcal{O}(h^4),$ where uh is the numerical solution of the quadratic finite volume scheme and $I^2_hu$ is the Lagrange quadratic finite element interpolant of the exact solution. Numerical experiments verify our theoretical results and also demonstrate that finite volume schemes formulated without the orthogonal conditions on the dual mesh fail to exhibit superconvergence and, in fact, achieve only a convergence order $\mathcal{O}(h^2)$ consistent with the $H^1$ error.