We design a virtual element method for the numerical treatment of the two-dimensional parabolic variational inequality problem on unstructured polygonal meshes. Due to the expected low regularity of the exact solution, the virtual element method is based on the lowest-order virtual element space that contains the subspace of the linear polynomials defined on each element. The connection between the nonnegativity of the virtual element functions and the nonnegativity of the degrees of freedom, i.e., the values at the mesh vertices, is established by applying the Maximum and Minimum Principle Theorem. The mass matrix is computed through an approximate $L^2$ polynomial projection, whose properties are carefully investigated in the paper. We prove the well-posedness of the resulting scheme in two different ways that reveal the contractive nature of the VEM and its connection with the minimization of quadratic functionals. The convergence analysis requires the existence of a nonnegative quasi-interpolation operator, whose construction is also discussed in the paper. The variational crime introduced by the virtual element setting produces five error terms that we control by estimating a suitable upper bound. Numerical experiments confirm the theoretical convergence rate for the refinement in space and time on three different mesh families including distorted squares, nonconvex elements, and Voronoi tesselations.

我们设计了一种虚拟元方法，用于对非结构多边形网格上的二维抛物线变分不等式问题进行数值处理。由于精确解的预期规则性较低，虚拟元素方法基于最低阶虚拟元素空间，该空间包含定义在每个元素上的线性多项式的子空间。虚元函数的非负性与自由度的非负性（即网格顶点处的值）之间的联系是通过应用最大最小原理定理建立的。质量矩阵通过近似计算${升}^2$多项式投影，其性质在论文中进行了仔细研究。我们以两种不同的方式证明了所得方案的适定性，揭示了 VEM 的收缩性质及其与二次泛函最小化的联系。收敛性分析需要存在一个非负的拟插值算子，论文中也讨论了其构造。虚拟元素设置引入的变分犯罪会产生五个误差项，我们通过估计合适的上限来控制这些误差项。数值实验证实了三种不同网格族（包括扭曲正方形、非凸单元和 Voronoi 镶嵌）的空间和时间细化的理论收敛率。