基于L型域上泊松问题的虚拟元方法

Virtual Element Method Based on Poisson Problem in L-Type Domain

摘要: 本文应用虚拟元方法研究L型域上泊松问题。首先构造虚拟元函数空间，并给出空间中函数的自由度。其次对空间进行虚拟元离散，构造与问题相关的投影算子。然后借助自由度来计算投影算子，得到连续问题的虚拟元离散形式，而后对离散形式进行误差分析。最后给出泊松方程的数值计算，通过不同范数意义下的相对误差与绝对误差，可以看出随着网格剖分的细化，数值解的收敛效果变得更好。当网格剖分最细时，数值解的收敛效果最好，验证了虚拟元方法的有效性和准确性。

Abstract: In this paper, the virtual element method is applied to study the Poisson problem on the L-type field. Firstly, the virtual element space is constructed, and the degrees of freedom of the functions in the space are given. Secondly, the virtual element discrete of the space is constructed, and the pro-jection operator related to the problem is constructed. Then, with the help of degrees of freedom, the projection operator is calculated to obtain the virtual element discrete form of the continuous problem, and then the error analysis of the discrete form is carried out. Finally, the numerical cal-culation of Poisson equation is given, and through the relative error and absolute error in the meaning of different norms, it can be seen that with the refinement of meshing, the convergence ef-fect of the numerical solution becomes better. When the mesh is the finest, the convergence effect of the numerical solution is the best, which verifies the effectiveness and accuracy of the virtual ele-ment method.