Yifan Yue

Yifan Yue, Hongtao Chen, Shuo Zhang

In this work, we fully explore three refined convergence structures of the lowest-order rectangular Raviart-Thomas element in solving the Laplace eigenvalue problem. Firstly, the scheme possesses a property of supercloseness between the discrete eigenfunctions and the interpolated ones, so that post-processing can be easily constructed to improve the accuracy at most by one order. The essentially skillful method is the integral expansion for interpolation terms. Secondly, based on the supercloseness property, we derive the error expansions for not only simple eigenvalues but also multiple eigenvalues, and provide a rigorous proof for them, based on which Richardson extrapolation can be performed. As a byproduct, we prove that all eigenvalues converge from above. Moreover, by utilizing the supercloseness property and Rayleigh quotient analysis, we give a rigorous proof for the convergence behavior for multiple eigenvalues on uniform meshes for the problem on the square domain. Thirdly, the equivalence between the lowest-order rectangular Raviart-Thomas element and the enriched rotated bilinear element is also indicated. At the last of this work, several numerical experiments are designed to demonstrate our theory.

Shangyu Xing, Changhao Xiang, Yuteng Han et al.

Multimodal large language models (MLLMs) have made significant progress in integrating visual and linguistic understanding. Existing benchmarks typically focus on high-level semantic capabilities, such as scene understanding and visual reasoning, but often overlook a crucial, foundational ability: geometric perception. Geometric perception involves understanding geometric shapes, structures, and spatial relationships, which are essential for supporting higher-level semantic tasks. Despite its importance, this capability remains underexplored in current MLLM research. To address this gap, we introduce GePBench, a novel benchmark designed to assess the geometric perception abilities of MLLMs. Our extensive evaluations reveal that current state-of-the-art MLLMs exhibit significant deficiencies in geometric perception tasks. Furthermore, we show that models trained with GePBench data demonstrate substantial improvements on a wide range of benchmark tasks, highlighting the critical role of geometric perception in enabling advanced multimodal applications. Our code and datasets will be publicly available.