We present and analyze a weak Galerkin finite element method for solving the transport-reaction equation in $d$ space dimensions. This method is highly flexible by allowing the use of discontinuous finite element on general meshes consisting of arbitrary polygon/polyhedra. We derive the \textcolor[rgb]{0.00,0.00,1.00}{$L_2$-error estimate} of $O(h^{k+\frac{1}{2}})$-order for the discrete solution when the $k$th-order polynomials are used for $k\geq 0$. Moreover, for a special class of meshes, we also obtain the \textcolor[rgb]{0.00,0.00,1.00}{optimal error} estimate of $O(h^{k+1})$-order in the $L_2$-norm. A derivative recovery formula is presented to approximate the convection \textcolor[rgb]{1.00,0.00,0.00}{directional derivative} and the corresponding superconvergence estimate is given. Numerical examples on compatible and non-compatible meshes are provided to show the effectiveness of this weak Galerkin method.
翻译：我们提出并分析一种用美元空间维度解决运输-反应方程式的微弱Galerkin 限制元素方法。 这种方法非常灵活, 允许在由任意多边形/ 波利赫德拉组成的普通模类上使用不连续的有限元素。 我们在$O( häk ⁇ frac{1 ⁇ 2 ⁇ 2 ⁇ 2 ⁇ ) 中得出 textcolor [rgb] 0.00, 0.00 $2$- error 估计值。 当$k$th- 单项多元成品用于 $k\geq 0美元时, 我们提出并分析一种较弱的离散溶方程式。 此外, 对于特殊类的meshes, 我们还获得了 extcolor [rgb] {0.00, 0.00, 1.00 ⁇ / pergometimal] 的估计值为$( häk+1} $O( häk+1}) 。 $L_ 2$- normum 。 的衍生物回收公式将接近对等量 / textcolgle [rgn- suffectal- complasses) 提供关于这种可比较和不相容和不相容和不相容不相容的方法的微的示例示例示例示例示例示例示例示例示例示例示例示例。
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