发表机构
Monash University; Australian National University; Bureau of Meteorology（莫纳什大学; 澳大利亚国立大学; 气象局）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出内蕴有限元框架，在参数空间离散流形上的标量与向量PDE，通过拓扑粘合构造de Rham复形，数值验证最优收敛、无几何误差且高效。
AI 中文摘要
我们提出一个内蕴有限元框架，用于在由图册描述的普遍流形上数值逼近标量与向量值偏微分方程。弱形式及其离散化完全在流形的平坦参数空间中表达，精确几何仅通过度量张量在求积点处的求值进入。通过纯拓扑的图间粘合，在多图册上获得整个de Rham复形的任意阶协调有限元空间。我们为多图册上的向量值空间发展了一种节点基变换构造，其中节点自由度通过传输映射耦合具有不兼容坐标系的图。图间连续性在节点处成立，跳跃量级为逼近误差阶，所得离散场精确切于流形。对于曲面Stokes问题，数值上观察到该方法的最优收敛阶；未提供先验误差分析。在二维和三维立方球流形上对Hodge拉普拉斯问题的数值实验表明，对于足够的求积阶，内蕴框架无几何一致性误差，且内蕴组装比外蕴对应方法显著更便宜。进一步考虑包含地形作为几何扰动的旋转浅水方程，表明质量精确守恒，能量守恒至时间离散误差阶。
英文摘要
We present an intrinsic finite element framework for the numerical approximation of scalar- and vector-valued partial differential equations on general manifolds described by atlases of charts. Weak formulations and their discretisation are expressed exclusively in the flat parametric space of the manifold, where the exact geometry enters only through the metric tensor that is evaluated at quadrature points. Conforming finite element spaces of arbitrary order for the whole de Rham complex are obtained on multi-chart atlases through purely topological inter-chart gluing. We develop a nodal change of basis construction for vector-valued spaces on multi-chart atlases, where nodal degrees of freedom couple charts with incompatible coordinate systems through transmission maps. Inter-chart continuity holds at the nodes with jumps of the order of the approximation error, and the resulting discrete field is exactly tangent to the manifold. Optimal convergence rates for this method are observed numerically for the surface Stokes problem; no a priori error analysis is provided. Numerical experiments on the cubed sphere manifold in two and three dimensions for the Hodge Laplacian problems demonstrate the intrinsic framework is free of the geometric consistency errors for a sufficient quadrature degree, and that intrinsic assembly is substantially cheaper than its extrinsic counterpart. Further considering the rotating shallow water equations that include orography as a perturbation of the geometry demonstrates that mass is conserved exactly, and energy is conserved up to the time discretisation error.
