Discontinuous Galerkin Methods for an Elliptic Variational Inequality of Fourth-Order

Abstract

Discontinuous Galerkin (DG) methods are studied for solving an elliptic variational inequality of fourth-order. Numerous discontinuous Galerkin schemes for the Kirchhoff plate bending problem are extended to the variational inequality. Numerical results are presented to illustrate convergence orders of the different methods.

C⁰ Discontinuous Galerkin Methods for a Plate Frictional Contact Problem

“…We will extend these three methods and additionally propose two more CDG methods to solve the 4th-order elliptic variational inequality of second kind. For 4th-order elliptic variational inequalities of first kind, some DG methods were developed in [26]; however, no error estimates were derived. In [8], a quadratic C 0 IP method for Kirchhoff plates problem with the displacement obstacle was studied, and errors in the energy norm and the L ∞ norm are given by O(h α ), where 0.5 < α ≤ 1.…”

“…In [26], a general primal formulation of CDG methods was presented for a 4th-order elliptic variational inequality of first kind. The process of deriving CDG schemes for 4th-order elliptic equations can also be found in [17].…”

“…The process of deriving CDG schemes for 4th-order elliptic equations can also be found in [17]. Based on the discussions in [26] and [17], we introduce five CDG methods for the problem (1.6) as follows: Find…”

A C0 Nonconforming Virtual Element Method for the Kirchhoff Plate Obstacle Problem

“…Our focus is on the plate obstacle problem, which is expressed as a first-kind fourthorder elliptic variational inequality [40,41]. Given a downward force f in the center of an elastic thin plate with a fixed and non-rotatable boundary, there exists an obstacle ψ beneath the plate.…”

Discontinuous Galerkin methods for solving a hyperbolic inequality

“…Moreover, locality of the discretization makes the DG methods ideally suited for parallel computing (see and the references therein). Recently, DG methods have been applied for solving VIs, such as gradient plasticity problem , obstacle problems , Signorini problem , quasistatic contact problems , plate contact problem , two membranes problem and Stokes or Navier–Stokes flows with slip boundary condition . A posteriori error analysis of DG methods for VIs was also considered in .…”