二阶椭圆型方程有限元逼近的一致Hölder范数界
Uniform Hölder-norm bounds for finite element approximations of second-order elliptic equations
摘要：
我们开发了De Giorgi-Nash-Moser理论的离散对应物，该理论提供了二阶线性椭圆问题的连续分段仿射有限元近似的均匀Hölder范数界，其形式为$-\nabla \cdot(A\nabla u)=f-\nabla\cdot F$与$A\in L^\infty(Ω;\mathbb{R}^{n\times n})$一致椭圆矩阵值函数，$f\in L^{q}(Ω)$，$F\in L^p(Ω;\mathbb{R}^n)$，与$p > n$和$q > n/2$，$A$ -非钝角形状规则三角剖分，不需要是准均匀的，关于有界多面体Lipschitz域$Ω\subset \mathbb{R}^n$ .
英文摘要：
We develop a discrete counterpart of the De Giorgi-Nash-Moser theory, which provides uniform Hölder-norm bounds on continuous piecewise affine finite element approximations of second-order linear elliptic problems of the form $-\nabla \cdot(A\nabla u)=f-\nabla\cdot F$ with $A\in L^\infty(Ω;\mathbb{R}^{n\times n})$ a uniformly elliptic matrix-valued function, $f\in L^{q}(Ω)$, $F\in L^p(Ω;\mathbb{R}^n)$, with $p > n$ and $q > n/2$, on $A$-nonobtuse shape-regular triangulations, which are not required to be quasi-uniform, of a bounded polyhedral Lipschitz domain $Ω\subset \mathbb{R}^n$.
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