In this paper, we propose new basis functions defined on curved sides or faces of curvilinear elements (polygons or polyhedrons with curved sides or faces) for the weak Galerkin finite element method. Those basis functions are constructed by collecting linearly independent traces of polynomials on the curved sides/faces. We then analyze the modified weak Galerkin method for the elliptic equation and the interface problem on curvilinear polytopal meshes with Lipschitz continuous edges or faces. The method is designed to deal with less smooth complex boundaries or interfaces. Optimal convergence rates for $H^1$ and $L^2$ errors are obtained, and arbitrary high orders can be achieved for sufficiently smooth solutions. The numerical algorithm is discussed and tests are provided to verify theoretical findings.

翻译：本文针对弱Galerkin有限元方法，提出了定义在曲线单元（具有曲线边或面的多边形或多面体）弯曲边或面上的新基函数。这些基函数通过收集曲边/面上多项式的线性无关迹来构造。随后，我们分析了在具有Lipschitz连续边或面的曲线多胞网格上，改进后的弱Galerkin方法对椭圆方程和界面问题的适用性。该方法旨在处理不够光滑的复杂边界或界面。我们获得了$H^1$和$L^2$误差的最优收敛阶，且对于足够光滑的解可实现任意高阶精度。文中讨论了数值算法，并通过算例验证了理论结果。