Error Analysis and Smoothing Properties of Discretized Deterministic and Stochastic Parabolic Problems

Doctoral thesis, 2003

In this thesis we consider smoothing properties and approximation of time derivatives for parabolic equations and error estimates for stochastic parabolic partial differential equations approximated by the finite element method. In the first two papers, we study smoothing properties and approximation of the time derivative in time discretization schemes with constant and variable time steps for an abstract homogeneous linear parabolic problem. The time stepping schemes are based on using rational functions r (z) which are A(θ)-stable for suitable θ ∈ [0, π/2] and satisfy |r(∞)| <1, and the approximations of the time derivative are based on using difference quotients in time. In the third paper, we consider the smoothing properties and time derivative approximations in multistep backward difference methods for a nonhomogeneous parabolic equation. In the fourth paper, as an application of the error estimates for the time derivative developed in the previous three papers, we study a postprocessed finite element method for semilinear parabolic equations. In the last two papers, we consider the finite element method for both linear and nonlinear stochastic parabolic partial differential equations. Using appropriate nonsmooth data error estimates for deterministic finite element problems, we prove error estimates for both space and time discretization.