Zhimin Zhang

This paper develops a unified theoretical framework for constructing B-spline basis function spaces with structural equivalence to finite element spaces. The theory rigorously establishes that these bases emerge as explicit linear combinations of B-spline element bases. For any prescribed smoothness requirements, this element-wise formulation enables the Hermite interpolation at nodes, which directly utilizes function values and derivatives without solving global linear systems. By focusing on explicit interpolation properties, element-wise analysis establishes optimal approximation errors, even when the space smoothness attains its theoretical maximum for the space degree. In isogeometric analysis (IgA), the construction naturally decomposes geometric mappings into element-level representations, allowing efficient computations across elements regardless of node distribution. Notably, the same Hermite interpolation framework simultaneously handles domain parameterization and IgA solutions, allowing direct imposition of boundary conditions through function and derivative matching. Numerical tests demonstrate optimal convergence rates and superconvergence properties in 2D IgA under uniform knot configurations, and improved computational efficiency in 3D IgA with non-uniform knot distributions.

This paper presents a study focused on solving hyperbolic conservation laws using arbitrary-order spectral difference (SD) methods. The study is structured around several crucial aspects. Firstly, in order to ensure the maximum principle for scalar equations and positivity preservation for Euler systems, we adopt the concept of flux limiters. This adoption leads to the development of the structure-preserving SD scheme with a flux limiter (SDFL), which is proven to preserve the original high-order accuracy. However, the SDFL scheme with lower order might lack conservational properties, despite its strong performance in short-term simulations. Consequently, we have developed a specific variant of the SDFL scheme with conservational properties, referred to as the CSDFL scheme. Secondly, we introduce a modified WENO-ZQ (MWENO-ZQ) reconstruction to suppress spurious oscillations when simulating problems with strong discontinuities. Finally, we conduct extensive numerical experiments to validate the effectiveness of the proposed high-order SD (SDFL,CSDFL) methods with MWENO-ZQ reconstruction. The results demonstrate the robustness and efficiency of these techniques in solving problems involving strong discontinuities, low pressure, and low density.

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.