动态扩散有限元方法的条件唯一性与最优能量范数收敛性
Optimal Energy-Norm Convergence for the Dynamic Diffusion Finite Element Method
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中文总结 AI 辅助
本文针对动态扩散有限元格式，推导其离散解的条件唯一性与最优一阶能量范数收敛估计，数值实验验证了理论结果，拓展了该格式的数学理解。
中文摘要 AI 辅助
本文对Santos等人（2021）针对稳态对流-扩散-反应问题进行数学分析的非线性两尺度动态扩散（Dynamic Diffusion, DD）有限元格式展开重新研究，在不修改离散格式的前提下建立两项额外理论结果。其一，定义非线性扩散算子$D_h$的人工扩散率被证明关于离散解是局部利普希茨连续的，该性质在依赖网格的小量条件下可导出离散解的条件唯一性。其二，通过将能量范数中的逼近误差与人工扩散的贡献相分离，推导得到连续分段线性有限元的最优一阶先验能量范数估计；该结果还阐明了此前$O(h^{1/2})$估计的适用场景——其针对更强的组合误差度量，因此无法单独确定能量误差的收敛速率。针对纯对流-扩散问题及对流-扩散-反应问题的数值实验，均验证了预测的一阶能量范数行为，以及平方根人工耗散度量的一阶衰减特性；具有尖锐出流层的强对流主导测试进一步展现了与$D_h$相关的稳定化效果。这些结果在保留原始DD格式底层离散结构的同时，拓展了对其的数学理解。
英文摘要
We revisit the nonlinear two-scale Dynamic Diffusion (DD) finite element formulation mathematically analyzed by Santos et al. (2021) for stationary advection--diffusion--reaction problems. We establish an explicit local Lipschitz estimate for the artificial diffusivity, with a constant of order $h_T$, and use it to prove uniqueness of the discrete solution for sufficiently fine meshes. By separating the approximation error in the energy norm from the contribution associated with artificial diffusion, we derive an optimal first-order a priori energy-norm estimate for continuous piecewise linear finite elements enriched with simplex bubble functions. We further prove that the square root of the nonlinear artificial dissipation is $O(h)$. Consequently, the combined energy-error and artificial-dissipation measure also converges with first order, sharpening the previously available $O(h^{1/2})$ estimate. A smooth manufactured problem, considered with and without reaction, corroborates the predicted convergence rates. A second, strongly advection-dominated problem with sharp outflow layers illustrates the stabilizing behavior of the DD formulation.
发表机构
- Federal University of Espírito Santo (UFES)（圣埃斯皮里图联邦大学）
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