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：本文针对具有切向边界条件及解的低正则性假设的div-curl系统，提出了一种新的原始-对偶弱Galerkin有限元方法。该数值格式基于不涉及精确解偏导数的弱变分形式，并在弱Galerkin有限元方法的广义框架中补充了一个对偶或伴随问题。对于解向量场属于$H^\theta(\Omega),\ \theta>\frac12$的情形，建立了$L^2$范数下的最优阶误差估计。该数学理论在具有一般拓扑性质（即任意第一和第二贝蒂数）的连通区域上推导得出。数值实验结果验证了理论收敛性。