Sandpiles and Superconductors: Nonconforming Linear Finite Element Approximations for Mixed Formulations of Quasi-Variational Inequalities

TL;DR

This paper introduces a simpler nonconforming linear finite element method for mixed formulations of variational inequalities modeling sandpiles and superconductors, proving convergence and demonstrating effectiveness through numerical experiments.

Contribution

It presents a novel, simpler finite element approximation for mixed formulations of quasi-variational inequalities, with proven convergence and practical numerical validation.

Findings

Proved subsequence convergence of the proposed approximation.

Demonstrated effectiveness through numerical experiments.

Simplified the numerical approach compared to previous Raviart--Thomas element methods.

Abstract

Similar evolutionary variational and quasi-variational inequalities with gradient constraints arise in the modeling of growing sandpiles and type-II superconductors. Recently, mixed formulations of these inequalities were used for establishing existence results in the quasi-variational inequality case. Such formulations, and this is an additional advantage, made it possible to determine numerically not only the primal variables, e.g. the evolving sand surface and the magnetic field for sandpiles and superconductors, respectively, but also the dual variables, the sand flux and the electric field. Numerical approximations of these mixed formulations in previous works employed the Raviart--Thomas element of the lowest order. Here we introduce simpler numerical approximations of these mixed formulations based on the nonconforming linear finite element. We prove (subsequence) convergence of these approximations, and illustrate their effectiveness by numerical experiments.