AI 中文总结
本文结合群论与偏微分方程理论，推导二阶线性椭圆Dirichlet边值问题解的对称群性质，提出含镜像对称时的域约化方法，采用线性有限元法在子域求解以降低计算成本。
AI 中文摘要
本文结合经典群论与偏微分方程理论，研究n维有界域Ω上二阶线性椭圆边值问题唯一解u的对称群Sym(u)，该问题形式为：-∑_{i,j=1}^{n}a_{ij}(x)u_{x_ix_j} + ∑_{i=1}^{n}b_i(x)u_{x_i} + c(x)u = f(x)（x∈Ω），u(x)=h(x)（x∈∂Ω）。本文分别定义并刻画以下对称群：二阶系数矩阵函数A(x)=(a_{ij}(x))_{n×n}的对称群Sym(A)、一阶系数列向量函数b(x)=(b₁(x),b₂(x),…,bₙ(x))^T的对称群Sym(b)、零阶系数函数c(x)的对称群Sym(c)、内部源函数f(x)的对称群Sym(f)以及边界源函数h(x)的对称群Sym(h)。本文严格证明，上述对称群的交集Sym(A)∩Sym(b)∩Sym(c)∩Sym(f)∩Sym(h)是Sym(u)的子群。此外，若该公共对称群包含若干镜像对称元素，则可将整个域Ω上的原二阶线性椭圆边值问题约化为某一子域上对应的边值问题。本文严格证明，施加于子域边界的新边界条件为齐次广义Neumann边界条件，采用线性有限元法在该子域上数值求解二阶线性椭圆边值问题，从而实现域约化并显著降低计算成本。
英文摘要
Combining classical group theory and partial differential equation theory, this paper investigates the symmetry group $\operatorname{Sym}(u)$ of the unique solution $u$ to the second-order linear elliptic boundary value problem on an $n$-dimensional bounded domain $Ω$ $-\sum_{i,j=1}^{n} a_{ij}(x)u_{x_ix_j} + \sum_{i=1}^{n} b_i(x)u_{x_i} + c(x)u = f(x), x\in Ω$, $u(x) = h(x), x\in \partial Ω$, The following symmetry groups are defined and characterized respectively: the symmetry group $\operatorname{Sym}(A)$ of the second-order coefficient matrix function $A(x)=(a_{ij}(x))_{n\times n}$; the symmetry group $\operatorname{Sym}(b)$ of the first-order coefficient column vector function $b(x)=(b_{1}(x),b_{2}(x),\cdots,b_{n}(x))^{T}$; the symmetry group $\operatorname{Sym}(c)$ of the zero-order coefficient function $c(x)$; the symmetry group $\operatorname{Sym}(f)$ of the internal source function $f(x)$; and the symmetry group $\operatorname{Sym}(h)$ of the boundary source function $h(x)$. This paper rigorously proves that the common symmetry group $\operatorname{Sym}(A)\cap\operatorname{Sym}(b) \cap\operatorname{Sym}(c) \cap\operatorname{Sym}(f) \cap\operatorname{Sym}(h)$ is a subgroup of $\operatorname{Sym}(u)$. In addition, if the common symmetry group contains several mirror symmetry elements, the original second-order linear elliptic boundary value problem on the entire domain $Ω$ can be reduced to the corresponding boundary value problem on a certain subdomain. It is strictly proven in this paper that the new boundary condition imposed on the boundary of the subdomain is the homogeneous generalized Neumann boundary condition. The linear finite element method is used to numerically solve the second-order linear elliptic boundary value problem on the subdomain, thereby achieving domain reduction and significantly reducing the computational cost.
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