六阶问题的混合高阶方法：完全非协调与$C^0$协调离散
Mixed Hybrid High-Order Methods for Sixth-Order Problems: Fully Non-conforming and $C^0$-Conforming Discretizations
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- IIT Roorkee（印度理工学院罗尔基拉分校）
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中文总结 AI 辅助
针对六阶椭圆问题，基于Ciarlet-Raviart重构提出混合系统，并设计完全非协调HHO与$C^0$协调HHO-有限元两种离散方法，证明稳定性与最优误差估计，数值实验验证收敛性。
中文摘要 AI 辅助
我们考虑一类在二维和三维空间中满足简支和Cahn-Hilliard型边界条件的六阶椭圆偏微分方程。基于Ciarlet-Raviart重构，我们将六阶问题改写为涉及二阶和四阶方程的等价混合系统，并在适当假设下建立其适定性。对于所得混合公式，我们提出并分析两种离散框架：一种是在一般多面体网格上的完全非协调混合高阶（HHO）方法，另一种是在单纯形网格上的$C^0$协调HHO-有限元方法。我们证明了稳定性，并在适当的正则性假设下推导了主变量的最优阶先验误差估计。对于完全非协调HHO方法，我们进一步推导了可靠的基于残差的后验误差估计器。数值实验证实了理论收敛速率，并展示了所提方法在一系列迁移率参数下的鲁棒性。
英文摘要
We consider a class of sixth-order elliptic partial differential equations in two and three dimensions subject to simply supported and Cahn--Hilliard-type boundary conditions. Based on the Ciarlet--Raviart reformulation, we recast the sixth-order problem as an equivalent mixed system involving second- and fourth-order equations and establish its well-posedness under suitable assumptions. For the resulting mixed formulation, we propose and analyse two discretization frameworks: a fully non-conforming hybrid high-order (HHO) method on general polytopal meshes and a $C^0$-conforming HHO--finite element method on simplicial meshes. We prove stability and derive optimal-order a priori error estimates for the primary variables under appropriate regularity assumptions. For the fully non-conforming HHO method, we further derive a reliable residual-based a posteriori error estimator. Numerical experiments confirm the theoretical convergence rates and illustrate the robustness of the proposed methods for a range of mobility parameters.
