Alan Demlow

Alan Demlow
Ph.D. (2002) Cornell University

First Position

Visiting Assistant Professor at Cornell University

Dissertation

Estimate for and Properties of Mixed Finite Element Methods for Elliptic Problems

Advisor:


Research Area:
Numerical Analysis of PDEs

Abstract: We consider two mixed finite element methods for a general second-order linear elliptic problem on a domain $\Omega\subset R^n$. These methods originate from two different forms of the same partial differential equation, and they sometimes behave in substantially different ways. The choice of element space used in these methods also has an effect on their behavior. In this thesis we consider the effects of different choices of elements and methods on the optimality and localization properties of the resulting approximations.

In the "divergence'' form method, the vector variable in the mixed method approximates $– A \nabla u$, while in the "conservation'' form method, the vector variable approximates $–(A \nabla u – \vec{b} u)$. Here u solves the given elliptic scalar problem, A is a matrix of coeffients, and $\vec{b}$ is a vector of coefficients. We demonstrate via general L_2 error estimates that, in the divergence form method, the errors in the vector and scalar variables are weakly coupled, while in the conservation form method they are strongly coupled. The strong coupling in the latter case leads to suboptimal convergence when members of the Brezzi-Douglas-Marini (BDM) family of elements for simplicial spaces are used, a fact which we demonstrate computationally. The well-known Raviart-Thomas-Nedelec family of spaces, in contrast, always give optimal convergence.

Using pointwise error estimates which generalize previously known almost-best-approximation maximum-norm estimates, Schatz has recently shown that standard finite element methods for elliptic problems yield errors which are in a certain sense mostly local in character, except in the lowest-order piecewise linear case. We carry out a similar pointwise error analysis for the mixed methods described above. Our estimates indicate that localization occurs in both of these mixed methods except when the lowest order BDM simplicial elements are used, and we again confirm the sharpness of our theoretical results via computational examples.