This article presents a new primal-dual weak Galerkin finite element method for the div-curl system with tangential boundary conditions and low-regularity assumptions on the solution. The numerical scheme is based on a weak variational form involving no partial derivatives of the exact solution supplemented by a dual or ajoint problem in the general context of the weak Galerkin finite element method. Optimal order error estimates in $L^2$ are established for solution vector fields in $H^\theta(\Omega),\ \theta>\frac12$. The mathematical theory was derived on connected domains with general topological properties (namely, arbitrary first and second Betti numbers). Numerical results are reported to confirm the theoretical convergence.

翻译：本条为div-curl 系统提出了一种新的初等-两极弱的Galerkin 限制元素方法,该方法的边界条件相近,对解决方案的假设不固定。数字方法基于一种薄弱的变式形式,其中没有精确解决方案的部分衍生物,在弱的Galerkin 限制元素方法的总体背景下,没有以双重或共同问题作为补充。为解决方案矢量字段设定了最佳顺序错误估计值($H ⁇ theta(\Omega),\\\theta ⁇ zrac12美元)。数学理论来自与一般地貌特性(即任意的第一和第二Betti数字)相关的领域。报告的数字结果证实了理论趋同。