  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.
