AI 中文总结
针对光滑闭曲面双调和方程，提出基于重构表面拉普拉斯算子的CDG方法，证明其误差收敛速率，经基准计算验证，并可扩展至非线性模型。
AI 中文摘要
我们针对光滑闭曲面上的双调和方程，开发并分析了一种基于重构表面拉普拉斯算子的连续/间断伽辽金（CDG）方法。在适配的r≥1阶参数网格上使用k≥2阶的连续映射有限元，同时采用间断的(k-2)阶提升项修正断裂的拉普拉斯-贝尔特拉米算子，以处理两侧法向通量的跳跃。所得的完全平方形式对任意固定的β>0，在零均值空间上是强制性的，无需足够大的罚参数。在标准几何假设下，我们证明能量和重构拉普拉斯误差为O(h^{k-1}+h^r)，L^2误差为O(h^{q_k}+h^{r+1})，其中q_2=2，k≥3时q_k=k+1。基准计算验证了这些收敛速率，而表面Swift-Hohenberg实验展示了该方法向以拉普拉斯为主的非线性模型的扩展。
英文摘要
We develop and analyze a continuous/discontinuous Galerkin (CDG) method based on reconstructed surface Laplacians for the biharmonic equation on a smooth closed surface. Continuous mapped finite elements of degree $k\ge2$ are used on fitted parametric meshes of degree $r\ge1$, while a discontinuous degree-$(k-2)$ lifting corrects the broken Laplace-Beltrami operator for two-sided conormal-flux jumps. The resulting completed-square form is coercive on the mean-zero space for every fixed $β>0$, without requiring a sufficiently large penalty parameter. Under the standard geometric assumptions, we prove that the energy and reconstructed-Laplacian errors are $\mathcal O(h^{k-1}+h^r)$, and the $L^2$-error is $\mathcal O(h^{q_k}+h^{r+1})$, where $q_2=2$ and $q_k=k+1$ for $k\ge3$. Benchmark computations support these rates, while a surface Swift-Hohenberg experiment illustrates the extension of the method to nonlinear Laplacian-dominated models.
Comments30 pages
