用于扭曲四边形网格的具有几何校正的拉维亚尔 - 托马斯元
Raviart--Thomas Elements with Geometric Correction for Distorted Quadrilateral Meshes
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中文总结 AI 辅助
针对四边形网格上拉维亚尔 - 托马斯元因双线性映射致几何扭曲问题，提出添加校正项的修改方法，构建通用框架并开发低阶元版本，理论分析与实验表明其能恢复兼容性且精度随扭曲增加而提升。
中文摘要 AI 辅助
基于拉维亚尔 - 托马斯空间的混合有限元方法广泛用于通量形式的二阶椭圆问题数值逼近。然而在四边形网格上，参考单元的双线性映射会引入空间变化的雅可比矩阵，可能违反标准拉维亚尔 - 托马斯空间的包含性质div V_h ⊂ W_h。本文提出对四边形网格上经典拉维亚尔 - 托马斯元进行简单修改，即给局部基函数添加几何校正项以补偿双线性映射引入的几何扭曲。所得空间与经典元有相同维度和自由度，同时恢复兼容性。给出构建此类修改空间的通用框架，并通过开发最低阶和次低阶拉维亚尔 - 托马斯元的修改版本进行说明。理论分析在四边形网格标准形状正则性假设下建立了最优逼近性质。在扭曲网格上的数值实验证实了预测的收敛速率，并表明随着几何扭曲增加，修改后的元比经典公式精度持续提高。
英文摘要
Mixed finite element methods based on Raviart--Thomas spaces are widely used for the numerical approximation of second--order elliptic problems in flux form. On quadrilateral meshes, however, the bilinear mapping from the reference element introduces a spatially varying Jacobian, which may violate the inclusion property $\mathrm{div}\,V_h \subset W_h$ for the standard Raviart--Thomas spaces. In this paper we propose a simple modification of the classical Raviart--Thomas elements on quadrilateral meshes. The modification consists of adding geometrically motivated correction terms to the local basis functions in order to compensate for the geometric distortion introduced by the bilinear mapping. The resulting spaces retain the same dimension and degrees of freedom as the classical Raviart--Thomas elements while restoring the compatibility property. We present a general framework for constructing such modified spaces and illustrate the approach by developing modified versions of the lowest order and next--to--lowest order Raviart--Thomas elements. Theoretical analysis establishes optimal approximation properties under the standard shape--regularity assumption for quadrilateral meshes. Numerical experiments on distorted meshes confirm the predicted convergence rates and show that the modified elements yield consistently improved accuracy over the classical Raviart--Thomas formulation as the geometric distortion increases.
