Novel superconvergence and ultraconvergence structures for the finite volume element method
Abstract
This paper develops novel natural superconvergence and ultraconvergence structures for the bi--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--order finite elements do not exhibit. Derivative ultraconvergence requires three conditions: a diagonal diffusion tensor, zero convection coefficients, and the FVE scheme satisfying tensorial --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.
Cite
@article{arxiv.2510.10668,
title = {Novel superconvergence and ultraconvergence structures for the finite volume element method},
author = {Xiang Wang and Yuqing Zhang and Zhimin Zhang},
journal= {arXiv preprint arXiv:2510.10668},
year = {2025}
}