2025/10/12 by Wang, Xiang, Zhang, Yuqing, Zhang, Zhimin
#65N08 #65N12 #65N30 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2510.10668
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.