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.

Keywords: 35J25; 65N12; 65N30; Curvilinear Elements; High Orders; Lipschitz Continuous Boundaries or Interfaces; Second-Order PDEs; Traces of Polynomials; Weak Galerkin Method.

在这篇论文中，我们提出了弱伽辽金有限元方法在曲边或曲面单元（具有曲边或曲面的多边形或多面体）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构造而成。然后，我们在带有Lipschitz连续边缘或面的曲线多胞形网格上分析了弱伽辽金方法对椭圆方程和界面问题的修改版本。该方法旨在处理不那么平滑的复杂边界或界面。我们获得了$H^1$ 和 $L^2$ 误差的最佳收敛率，并且对于足够光滑的解可以达到任意高阶精度。讨论了数值算法并提供了测试结果以验证理论发现。

关键词: 35J25; 65N12; 65N30; 曲线单元；高阶方法；Lipschitz连续边界或界面；二阶偏微分方程；多项式迹；弱伽辽金方法。