In this work, we primarily focus on the Q1 finite element spaces in any finite dimension, equipped with the discrete \ell_h2 inner product induced by the simple row-sum mass lumping. We establish the uniform (with respect to the mesh size h) equivalence between the discrete \ell_h2 norm and the L2 norm on these spaces, for both uniform and nonuniform meshes. Our main contribution is the derivation of sharp bounds for the equivalence between these two norms. Numerical examples demonstrate that these bounds are indeed sharp. Furthermore, we establish the equivalence between the discrete h_h^1 norm and the continuous H1 norm, along with corresponding numerical results. As an application, we also provide the equivalence between discrete and continuous norms in the context of error estimates for central FD solutions on polygonal domains.

冯新龙，二级教授，博士生导师；研究领域：计算数学、计算流体力学、不确定性量化、人工智能与机器学习等。曾在韩国首尔国立大学、香港浸会大学、巴西巴拉那联邦大学、加拿大阿尔伯塔大学从事博士后研究工作和短期访问。拥有中国准精算团队格，曾担任中国核学会计算物理学会理事、中国计算数学学会理事、中国数学会理事等。曾荣获教育部高等院校青年教师奖、自治区科学技术进步奖等。入选教育部重大人才计划、享受国务院政府特殊津贴专家等。主持完成20余项国家级和省部级自然科学基金项目。已在SIAM系列、IEEE系列等国际著名期刊合作发表学术论文100余篇。