发表机构
School of Mathematics, Jilin University; Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics（吉林大学数学学院; 应用物理与计算数学研究所计算物理实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出并分析了一种曲边四边形网格上的高阶有限体积元方法，通过基于高斯点的对偶网格克服几何变形问题，证明了稳定性和最优收敛性，数值实验验证了其在复杂边界和变形情况下的高效性与鲁棒性。
AI 中文摘要
本文针对椭圆方程，提出并分析了一种在曲边四边形网格上的高阶有限体积元方法。与现有主要基于直边网格的理论不同，本研究首次建立了曲边网格上有限体积方法的稳定性和最优收敛性分析。通过构造基于高斯点的对偶网格，我们克服了曲边网格几何变形和雅可比非均匀性引起的精度下降问题。我们在弱网格正则性条件下证明了离散双线性形式的强制性，从而获得了能量范数下的最优误差估计。此外，利用正交性和Aubin-Nitsche技巧，我们推导了最优的$L^2$误差估计。数值实验涵盖了常数和各向异性系数问题、不同的对偶剖分策略、复杂曲边边界域以及具有大变形的界面。数值结果表明，该方法在各种曲边网格上的$H^1$和$L^2$范数下均一致地达到最优收敛阶。与直边网格相比，曲边网格在逼近复杂曲边边界方面具有显著优势，并且在系数突变和界面大变形的情况下表现出更好的抗畸变能力。本文为曲边网格上的有限体积元方法提供了统一的理论框架，并验证了所提方法的效率和鲁棒性。
英文摘要
This paper proposes and analyzes a high-order finite volume element method on curved-edge quadrilateral meshes for elliptic equations. Unlike existing theories, which are primarily based on straight-edge meshes, this study is the first to establish an analysis of the stability and optimal convergence of the finite volume method on curved-edge meshes. By constructing a dual mesh based on Gaussian points, we overcome the accuracy degradation issues caused by geometric deformation and Jacobian non-uniformity of curved-edge meshes. We prove the coercivity of the discrete bilinear form under weak mesh regularity conditions, thereby obtaining an optimal error estimate in the energy norm. Furthermore, using orthogonality and the Aubin-Nitsche technique, we derive an optimal $L^2$ error estimate. Numerical experiments cover problems with constant and anisotropic coefficients, different dual partition strategies, complex curved boundary domains, and interfaces with large deformations. Numerical results indicate that this method consistently achieves the optimal convergence order in both the $H^1$ and $L^2$ norms on a variety of curved-edge meshes. Compared to straight-edge meshes, curved-edge meshes offer significant advantages in approximating complex curved boundaries and demonstrate better resistance to distortion in cases involving sudden changes in coefficients and large deformations at interfaces. This paper provides a unified theoretical framework for the finite volume element method on curved-edge meshes and verifies the efficiency and robustness of the proposed method.
