三维非均匀笛卡尔划分上弱格架有限元方法数值梯度的超收敛性
Superconvergence of Numerical Gradient for Weak Galerkin Finite Element Methods on Nonuniform Cartesian Partitions in Three Dimensions
AI总结：
本文研究了在三维非均匀笛卡尔划分上使用弱格架有限元方法求解二阶椭圆问题的梯度近似超收敛性，分析了其误差估计，并通过数值实验验证了理论结果。
摘要：
本文通过弱格架有限元方案分析了在三维均匀和非均匀立方划分上二阶椭圆问题的梯度近似超收敛误差估计。由于从二维到三维时对称性丢失，三维中的超收敛结果并非最近二维中基于矩形划分到立方划分的超收敛结果的简单扩展。当使用最低阶弱格架有限元（元素内部为线性多项式，面上为常数）时，数值梯度在$ L^2 $-范数下的误差估计达到超收敛阶${\cal O}(h^r) $（$1.5 \leq r\leq 2$）。一系列数值实验被用来验证三维中的所建立的超收敛理论。
英文摘要：
A superconvergence error estimate for the gradient approximation of the second order elliptic problem in three dimensions is analyzed by using weak Galerkin finite element scheme on the uniform and non-uniform cubic partitions. Due to the loss of the symmetric property from two dimensions to three dimensions, this superconvergence result in three dimensions is not a trivial extension of the recent superconvergence result in two dimensions \cite{sup_LWW2018} from rectangular partitions to cubic partitions. The error estimate for the numerical gradient in the $L^{2}$-norm arrives at a superconvergence order of ${\cal O}(h^r) (1.5 \leq r\leq 2)$ when the lowest order weak Galerkin finite elements consisting of piecewise linear polynomials in the interior of the elements and piecewise constants on the faces of the elements are employed. A series of numerical experiments are illustrated to confirm the established superconvergence theory in three dimensions.
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