We propose an efficient method for the numerical solution of a general class of two dimensional semilinear parabolic problems on polygonal meshes. The proposed approach takes advantage of the properties of the serendipity version of the Virtual Element Method (VEM), which not only significantly reduces the number of degrees of freedom compared to the classical VEM but also, under certain conditions on the mesh allows to approximate the nonlinear term with an interpolant in the Serendipity VEM space; which substantially improves the efficiency of the method. An error analysis for the semi discrete formulation is carried out, and an optimal estimate for the error in the $L_2$-norm is obtained. The accuracy and efficiency of the proposed method when combined with a second order Strang operator splitting time discretization is illustrated in our numerical experiments, with high order approximations up to order $6$.

翻译：我们建议了一种有效的方法,用数字方法解决多边形模贝上两个维维半线性抛物线问题的一般类别,提议的方法利用了虚拟元素法(VEM)的精度版本的特性,该方法不仅大大降低了与传统VEM相比的自由度,而且在网格上的某些条件下,允许在Serendipity VEM空间内用一个内插器来近似非线性术语;这大大提高了该方法的效率。对半离散制剂进行了错误分析,并获得了对$L_2$-norm 错误的最佳估计。在与第二顺序分解时间操作器相结合时,拟议方法的准确性和效率在我们的数字实验中得到了说明,高排序近似值高达6美元。