In this paper we analyze a lowest order virtual element method for the load classic reaction-convection-diffusion problem and the convection-diffusion spectral problem, where the assumptions on the polygonal meshes allow to consider small edges for the polygons. Under well defined seminorms depending on a suitable stabilization for this geometrical approach, we derive the well posedness of the numerical scheme and error estimates for the load problem, whereas for the spectral problem we derive convergence and error estimates for the eigenvalues and eigenfunctions. We report numerical tests to assess the performance of the small edges on our numerical method for both problems under consideration.

翻译：本文分析了用于经典载荷反应-对流-扩散问题及对流-扩散谱问题的低阶虚拟元方法，其中多边形网格的假设允许考虑多边形的小边长。在依据该几何方法适当稳定化而定义的半范数下，我们推导了载荷问题数值格式的适定性及误差估计；对于谱问题，则推导了特征值与特征函数的收敛性及误差估计。我们报告了数值实验，以评估小边长对两种问题所采用数值方法性能的影响。