An a posteriori analysis of C\u3csup\u3e0\u3c/sup\u3e interior penalty methods for the obstacle problem of clamped Kirchhoff plates

We develop an a posteriori analysis of C interior penalty methods for the displacement obstacle problem of clamped Kirchhoff plates. We show that a residual based error estimator originally designed for C 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 interior penalty methods. 0 0

Finite Element Methods for Fourth Order Variational Inequalities

In this work we study finite element methods for fourth order variational inequalities. We begin with two model problems that lead to fourth order obstacle problems and a brief survey of finite element methods for these problems. Then we review the fundamental results including Sobolev spaces, existence and uniqueness results of variational inequalities, regularity results for biharmonic problems and fourth order obstacle problems, and finite element methods for the biharmonic problem. In Chapter 2 we also include three types of enriching operators which are useful in the convergence analysis. In Chapter 3 we study finite element methods for the displacement obstacle problem of clamped Kirchhoff plates. A unified convergence analysis is provided for $C^1$ finite element methods, classical nonconforming finite element methods and $C^0$ interior penalty methods. The key ingredient in the error analysis is the introduction of the auxiliary obstacle problem. An optimal $O(h)$ error estimate in the energy norm is obtained for convex domains. We also address the approximations of the coincidence set and the free boundary. In Chapter 4 we study a Morley finite element method and a quadratic $C^0$ interior penalty method for the displacement obstacle problem of clamped Kirchhoff plates with general Dirichlet boundary conditions on general polygonal domains. We prove the magnitudes of the errors in the energy norm and the $L^{\infty}$ norm are $O(h^{\alpha})$, where $\alpha \u3e 1/2$ is determined by the interior angles of the polygonal domain. Numerical results are also presented to illustrate the performance of the methods and verify the theoretical results obtained in Chapter 3 and Chapter 4. In Chapter 5 we consider an elliptic optimal control problem with state constraints. By formulating the problem as a fourth order obstacle problem with the boundary condition of simply supported plates, we study a quadratic $C^0$ interior penalty method and derive the error estimates in the energy norm based on the framework we introduced in Chapter 3. The rate of convergence is derived for both quasi-uniform meshes and graded meshes. Numerical results presented in this chapter confirm our theoretical results

[Formula presented] interior penalty methods for an elliptic state-constrained optimal control problem with Neumann boundary condition

We study [Formula presented] interior penalty methods for an elliptic optimal control problem with pointwise state constraints on two dimensional convex polygonal domains. The approximation of the optimal state is obtained by solving a fourth order variational inequality and the approximation of the optimal control is computed by a post-processing procedure. We prove the convergence of numerical solutions with rates in the [Formula presented]-like energy error by using the complementarity form of the variational inequality. Furthermore, we develop an a posteriori analysis for a residual based error estimator and introduce an adaptive algorithm. Numerical experiments are provided to gauge the performance of the proposed methods

A partition of unity method for the displacement obstacle problem of clamped Kirchhoff plates

A partition of unity method for the displacement obstacle problem of clamped Kirchhoff plates is considered in this paper. We derive optimal error estimates and present numerical results that illustrate the performance of the method. © 2013 Elsevier B.V. All rights reserved

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

Finite element methods for a model elliptic distributed optimal control problem with pointwise state constraints are considered from the perspective of fourth order boundary value problems

A Morley finite element method for the displacement obstacle problem of clamped Kirchhoff plates

We study a Morley finite element method for the displacement obstacle problem of clamped Kirchhoff plates on polygonal domains. Error estimates are derived in the energy norm and the L norm. The performance of the method is illustrated by numerical experiments. © 2013 Elsevier B.V. All rights reserved.

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods