Virtual element approximation of two-dimensional parabolic variational inequalities

TL;DR

This paper develops a virtual element method for solving two-dimensional parabolic variational inequalities on complex meshes, ensuring stability and convergence despite low solution regularity.

Contribution

It introduces a novel virtual element scheme tailored for parabolic variational inequalities, with new analysis of stability, convergence, and error control on unstructured polygonal meshes.

Findings

The method is well-posed and contractive.

Convergence rates are confirmed numerically.

Effective on various complex mesh types.

Abstract

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.