Novel superconvergence and ultraconvergence structures for the finite volume element method
Xiang Wang, Yuqing Zhang, Zhimin Zhang
TL;DR
The paper tackles achieving higher-than-expected convergence for bi-$k$-order finite volume element methods on rectangular meshes in diffusion–convection–reaction problems. It introduces tensorial $k$-$r$-order orthogonality and asymmetric-enabled M-decompositions (AMD-Super and AMD-Ultra) to construct superclose and ultraclose approximants, enabling a suite of convergence phenomena. The main results are $(k+1)$-order derivative superconvergence and $(k+2)$-order function-value superconvergence, along with $(k+2)$-order derivative ultraconvergence under diagonal diffusion and zero convection, with convergence points that can be tuned and may be asymmetric. Numerical experiments validate the theory, demonstrating the necessity of the orthogonality framework and the practical benefit of tunable ultraconvergence in FVE schemes on rectangles.
Abstract
This paper develops novel natural superconvergence and ultraconvergence structures for the bi-$k$-order finite volume element (FVE) method on rectangular meshes. These structures furnish tunable and possibly asymmetric superconvergence and ultraconvergence points. We achieve one-order-higher superconvergence for both derivatives and function values, and two-orders-higher ultraconvergence for derivatives--a phenomenon that standard bi-$k$-order finite elements do not exhibit. Derivative ultraconvergence requires three conditions: a diagonal diffusion tensor, zero convection coefficients, and the FVE scheme satisfying tensorial $k$-$k$-order orthogonality (imposed via dual mesh constraints). This two-dimensional derivative ultraconvergence is not a trivial tensor-product extension of the one-dimensional phenomena; its analysis is also considerably more complex due to directional coupling. Theoretically, we introduce the asymmetric-enabled M-decompositions (AMD-Super and AMD-Ultra) to rigorously prove these phenomena. Numerical experiments confirm the theory.