In this article, we present and analyze a stabilizer-free $C^0$ weak Galerkin (SF-C0WG) method for solving the biharmonic problem. The SF-C0WG method is formulated in terms of cell unknowns which are $C^0$ continuous piecewise polynomials of degree $k+2$ with $k\geq 0$ and in terms of face unknowns which are discontinuous piecewise polynomials of degree $k+1$. The formulation of this SF-C0WG method is without the stabilized or penalty term and is as simple as the $C^1$ conforming finite element scheme of the biharmonic problem. Optimal order error estimates in a discrete $H^2$-like norm and the $H^1$ norm for $k\geq 0$ are established for the corresponding WG finite element solutions. Error estimates in the $L^2$ norm are also derived with an optimal order of convergence for $k>0$ and sub-optimal order of convergence for $k=0$. Numerical experiments are shown to confirm the theoretical results.
翻译:在本篇文章中,我们提出并分析一种稳定无价的无价Galerkin(SF-C0WG)方法,以解决双调问题。SF-C0WG方法的制定方式是:细胞未知值为:C$0,连续的单件单向多角价2美元,每克元2美元,每克元0美元;面孔未知值为:不连续的单件多角价1美元,每克1美元,每克1美元。这一SF-C0WG方法的制定方式没有固定或处罚期,而且简单如:符合双调问题限定元素的$C1美元。一个离散标准中的最佳定序误差估计值为$H%2美元,对相应的工作组定值元素溶液为$H1美元,每克元0.00美元。根据美元标准得出的误差估计值为$k>0美元和美元=0美元次最佳趋同顺序。数字实验证实了理论结果。