Sandpiles and superconductors: nonconforming linear finite element approximations for mixed formulations of quasi-variational inequalities

Abstract

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 RaviartThomas 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.

Analysis of an iterative scheme of fractional steps type associated to the reaction–diffusion equation endowed with a general nonlinearity and Cauchy–Neumann boundary conditions

“…Furthermore, following the same reasoning as in work [10], we conclude that problem (1.1) has a solution ϕ ∈ W 1,2 p (Q) (see [32, p. 195]). Finally, estimate (2.19) and the embeddings W The uniqueness of the solution.…”

“…In [33] the reader can found more details relative to a more extensive class of problems on the type those treated in this paper (reaction-diffusion equation), as well as different types for the nonlinear term F (ϕ). For numerical methods we direct the readers to consult the works [2,3,[6][7][8][9]11,15,16,18,21,23,28,29], and references therein.…”

Hyperbolic Maxwell Variational Inequalities for Bean's Critical-State Model in Type-II Superconductivity

“…Barrett and Prigozhin [4] analyzed the associated parabolic quasi-variational inequality problem in a scalar 2D setting and its dual formulation. Recently, they [6] introduced a nonconforming finite element method. They proved the convergence of their nonconforming method and illustrated its efficiency numerically.…”

Elliptic quasi-variational inequalities under a smallness assumption: uniqueness, differential stability and optimal control

“…We will not cover the general situation of a set-valued mapping Q : V ⇒ V , but we restrict the treatment of (1.1) to the case in which Q(y) is a moving set, i.e., Q(y) = K + Φ(y) (1.2) for some non-empty, closed and convex subset K ⊂ V and Φ : V → V . It is well-known that QVIs have many important real-world applications, we refer exemplarily to Bensoussan, Lions, 1987;Prigozhin, 1996;Barrett, Prigozhin, 2013;Alphonse, Hintermüller, Rautenberg, 2019 and the references therein.…”