带Robin边界条件的二阶线性椭圆问题的对称性继承与对称约化有限元分析
Symmetry Inheritance and Symmetry-Reduced Finite Element Analysis for Second-Order Linear Elliptic Problems with Robin Boundary Conditions
- School of Physics and Mechatronics Engineering, Guizhou Minzu University（贵州民族大学物理与机电工程学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对带Robin边界条件的二阶线性椭圆问题，建立对称性框架并推导变换规则，提出对称约化策略，给出有限元格式与数值结果验证其有效性。
AI中文摘要：
本文针对有界区域上带Robin边界条件的二阶线性椭圆方程，建立了一套对称性框架。区域的正交变换通过对标量场、矢量场和二阶张量场的左群作用来表示，详细推导了梯度、散度、扩散通量及法向导数边界项对应的变换规则。基于这些关系，针对主系数张量、一阶与零阶系数、微分算子、Robin系数及边界算子引入了对称群，随后将体积源项与边界源项的对称性纳入完整边值问题的公共对称群中。在唯一可解性假设下，证明该公共群的每个元素也是解的对称性；针对反射对称性，在人工对称边界上得到齐次广义Neumann条件，实现计算域的精确约化。本文提出了对应的有限元格式，并给出数值结果以验证理论对称性及所得区域约化策略的有效性，三个算例分别展示从多维到一维的径向约化、基于反射的区域约化，以及所有系数与源项均非零的变系数问题。
英文摘要：
A symmetry framework is developed for second-order linear elliptic equations subject to Robin boundary conditions on bounded domains. Orthogonal transformations of the domain are represented through left group actions on scalar, vector, and second-order tensor fields. The corresponding transformation rules for the gradient, divergence, diffusion flux, and conormal boundary term are derived in detail. Based on these relations, symmetry groups are introduced for the principal coefficient tensor, the first-order and zeroth-order coefficients, the differential operator, the Robin coefficient, and the boundary operator. The symmetry properties of the volume and boundary source terms are then incorporated into a common symmetry group for the complete boundary value problem. Under the assumption of unique solvability, it is proved that every element of this common group is also a symmetry of the solution. For reflection symmetries, homogeneous generalized Neumann conditions are obtained on artificial symmetry boundaries, leading to an exact reduction of the computational domain. A corresponding finite element formulation is presented, and numerical results are provided to verify the theoretical symmetry properties and the validity of the resulting domain-reduction strategy. Three examples illustrate radial reduction from multiple dimensions to one dimension, reflection-based domain reduction, and a variable-coefficient problem in which all coefficients and source terms are nonzero.
