Doctor of Philosophy (PhD)



Document Type



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 $\alpha > 1/2$ 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.



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Committee Chair

Brenner, Susanne