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 $L_2$-error estimate of $O(h^{k+\frac{1}{2}})$-order for the discrete solution when the kth-order polynomials are used for $k\ge 0$. Moreover, for a special class of meshes, we also obtain the optimal error estimate of $O(h^{k+1})$-order in the $L_2$-norm. A derivative recovery formula is presented to approximate the convection 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.

我们提出并分析了弱Galerkin有限元方法来求解d空间维中的输运反应方程。通过允许在由任意多边形/多面体组成的普通网格上使用不连续有限元，此方法具有很高的灵活性。我们得出${\mathrm{大号}}_2$的误差估计 $Ø（H^{ķ+\frac{\mathrm{1个}}{2}}）$当使用k阶多项式来求解离散解时$ķ\ge 0$。此外，对于一类特殊的网格，我们还获得了$Ø（H^{ķ+\mathrm{1个}}）$中的顺序 ${\mathrm{大号}}_2$-规范。给出了一个导数恢复公式来近似对流方向导数，并给出了相应的超收敛估计。提供了有关兼容和不兼容网格的数值示例，以显示此弱Galerkin方法的有效性。