Guan Qingguang, Queisser Gillian, Zhao Wenju
School of Mathematics and Natural Sciences, University of Southern Mississippi, Hattiesburg, MS 39406.
Department of Mathematics, Temple University, Philadelphia, PA 19122, USA.
Comput Math Appl. 2023 Oct 15;148:282-292. doi: 10.1016/j.camwa.2023.08.017. Epub 2023 Sep 5.
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 and 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.
在本文中，我们为弱伽辽金有限元方法提出了定义在曲线单元（具有弯曲边或面的多边形或多面体）的弯曲边或面上的新基函数。这些基函数是通过收集多项式在弯曲边/面上的线性无关迹线来构造的。然后，我们分析了在具有利普希茨连续边或面的曲线多面体网格上求解椭圆方程和界面问题的修正弱伽辽金方法。该方法旨在处理不太光滑的复杂边界或界面。得到了关于(H^1)和(L^2)误差的最优收敛率，并且对于足够光滑的解可以达到任意高阶。文中讨论了数值算法并给出了测试以验证理论结果。
