An A Posteriori Analysis of $C^0$ Interior Penalty Methods for the Obstacle Problem of Clamped Kirchhoff Plates

Abstract

We develop an a posteriori analysis of C 0 interior penalty methods for the displacement obstacle problem of clamped Kirchhoff plates. We show that a residual based error estimator originally designed for C 0 interior penalty methods for the boundary value problem of clamped Kirchhoff plates can also be used for the obstacle problem. We obtain reliability and efficiency estimates for the error estimator and introduce an adaptive algorithm based on this error estimator. Numerical results indicate that the performance of the adaptive algorithm is optimal for both quadratic and cubic C 0 interior penalty methods.2010 Mathematics Subject Classification. 65N30, 65N15, 65K15, 74K20, 74S05. Key words and phrases. Kirchhoff plates, obstacle problem, a posteriori analysis, adaptive, C 0 interior penalty methods, discontinuous Galerkin methods, fourth order variational inequalities.

Finite Element Methods for Elliptic Distributed Optimal Control Problems with Pointwise State Constraints (Survey)

“…2. An interesting phenomenon concerning fourth order variational inequalities is that a posteriori error estimators originally designed for fourth order boundary value problems can be directly applied to fourth order variational inequalities [61,68]. This is different from the second order case where a posteriori error estimators for boundary value problems are not directly applicable to variational inequalities.…”

Virtual enriching operators

“…Our goal is to construct a linear operator E h : (1.5) where Π h : C(Ω) −→ V h is the Lagrange nodal interpolation operator, and the positive constants C and C only depend on the shape regularity of T h and k. Moreover, the operator E h maps V h ∩ H 1 0 (Ω) into H 2 (Ω) ∩ H 1 0 (Ω). Enriching operators that satisfy (1.4) and (1.5) are useful for a priori and a posteriori error analyses for fourth order elliptic problems [8,17,6,7,9], and they also play an important role in fast solvers for fourth order problems [4,5,10]. A recent application to Hamilton-Jacobi-Bellman equations can be found in [21].…”

A stabilised finite element method for the plate obstacle problem

“…Moreover, only a few articles exist on the a posteriori error analysis of fourth-order obstacle problems (cf. [17,4]) and none on conforming C 1 -continuous finite elements, most probably due to the limited regularity of the underlying continuous problem. Here, we consider a stabilised method based on a saddle point formulation which introduces the contact force as an additional unknown (Lagrange multiplier).…”

“…Numerical approximation of fourth-order obstacle-type problems has been previously studied in [14,15,21,7,5,6,4]. In [14,15] the authors considered mixed finite element methods and presented general convergence theorems without convergence rates.…”

“…Brenner et al [7] made a unified a priori error analysis for classical conforming and non-conforming (C 1 -continuous and C 0 -continuous) finite element methods (see, e.g., [11]) as well as for the C 0 interior penalty methods and showed O(h) convergence rate for all methods in convex domains, see also [5,6] for some generalisations. The only existing a posteriori analyses on fourth-order obstacle-type problems are due to Brenner et al [4] and Gudi and Porwal [17], both performed on the C 0 interior penalty methods. In [17], the authors also derive a priori error estimates with minimal regularity assumptions using the techniques developed by Gudi in [16] much in the same spirit as we do here, see also [23,18].…”