Ph.D. (2004) Cornell University
Abstract: In this thesis we consider the standard Galerkin finite element approximation to the solution of a parabolic partial differential equation given on a bounded domain Ω ⊂ R^n.
The error between the exact solution and the approximate solution naturally splits into two parts. The first part is the so called semidiscrete error and the second part is the error between approximate solution and semidiscrete approximation. A big part of the material of this thesis is devoted to obtaining pointwise weighted estimates for the semidiscrete error. These results extend those obtained by A.Schatz in 1998, in the case of stationery elliptic problems.
Then we will give an example, which shows that the localization does not always hold for time discretization.
The analysis of the fully discrete error is very different and requires a completely different technique. In all known results, in order to show an optimal order estimate in terms of data, it is necessary to make some artificial assumptions. In the last chapter, we show that, using a certain "trick," those artificial assumptions are not necessary.