On mixed finite element methods for non-linear second-order elliptic problems
Abstract
The main objective of this thesis is to develop the mixed finite element method to approximate the solution of the Dirichlet problem for the most general quasi-linear second-order elliptic operator in divergence form. As a first step toward our goal, we treat a strongly nonlinear elliptic equation whose divergence term is a vector function of the gradient of the solution. As an example, minimal surface equations are considered. As a next step, Hamilton-Jacobi-Bellman type equations, the case of a linear divergence term and arbitrarily nonlinear lower order terms, is treated, which arises, for example, in optimal stochastic control. In each case existence and uniqueness are demonstrated and optimal order error estimates are derived in $L\sp{q},2\le q\le + \infty,$ through negative Sobolev norm error estimates using inverse type estimates. An $L\sp{\infty}$-error bound for the vector unknown is also derived for a strongly nonlinear case using weighted $L\sp2$-norms. Finally, we consider the most general case, using RTN elements. But our argument also works for other elements, such as BDM, BDFM, DW. We formulate discrete mixed weak forms which lead to systems of nonlinear algebraic equations for the approximate solutions, for which existence, uniqueness, and error estimates are necessary. To show existence, we use a fixed point argument which requires linearization of the error equations, and a duality lemma for the linearized problems. Then, uniqueness of the approximation is proved, and optimal order error estimates in $L\sp2$ are demonstrated for both the scalar and vector functions approximated by the method using a bootstrapping argument with minimal regularity requirements in almost all cases. Error estimates are also derived in $L\sp{q},2\le q\le + \infty.$ Some numerical experiments were performed, and their results are included.
Degree
Ph.D.
Advisors
Milner, Purdue University.
Subject Area
Mathematics
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