发表机构
School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文首次在稀疏网格上构造高阶有限元 de Rham 复形，结合 Alpert 多小波与积分对应物，建立可交换插值、误差界、正合性及稳定离散势，并通过 curl-curl 和 Maxwell 特征值实验验证其有效性。
AI 中文摘要
我们首次在张量积稀疏网格上构造了一族高阶有限元微分形式。该构造始于相容的一维连续分段多项式空间和低一次的不连续分段多项式空间，并通过各坐标方向上的微分将它们联系起来。其分层分解将 Alpert 多小波与其积分对应物相结合。我们建立了可交换的典范插值算子，并在混合 Sobolev 正则性下给出了相应的逼近误差界。对于任意维单位立方体上任意多项式次的稀疏网格 de Rham 复形，我们还通过发展一种新的稳定同伦算子，证明了其正合性、多项式次鲁棒的稳定离散势，以及一个 H(d) 有界可交换投影。立方体上 curl-curl 源问题和方环上 Maxwell 特征值问题的数值实验，说明了所提出的高阶稀疏网格方法的有效性。
英文摘要
We construct, for the first time, a family of higher-order finite element differential forms on tensor-product sparse grids. The construction starts from compatible one-dimensional spaces of continuous piecewise polynomials and discontinuous piecewise polynomials of one degree lower, linked by differentiation in each coordinate direction. Their hierarchical decompositions combine Alpert multiwavelets with their integrated counterparts. We establish commuting canonical interpolation operators and corresponding approximation error bounds under mixed Sobolev regularity. For the sparse-grid de Rham complex of arbitrary polynomial degree on the unit cube in arbitrary dimension, we also prove exactness, polynomial-degree-robust stable discrete potentials, and an H(d)-bounded commuting projection by developing a novel stable homotopy operator. Numerical experiments for curl-curl source problems on a cube and a Maxwell eigenproblem on a square-annulus illustrate the effectiveness of the proposed higher-order sparse-grid method.
Comments29 pages
