Doctoral thesis, 2012

This thesis is concerned with numerical approximation of linear stochastic partial
differential equations driven by additive noise. In the first part, we develop a
framework for the analysis of weak convergence and within this framework we
analyze the stochastic heat equation, the stochastic wave equation, and the linearized
stochastic Cahn-Hilliard, or the linearized Cahn-Hilliard-Cook equation.
The general rule of thumb, that the rate of weak convergence is twice the rate of
strong convergence, is confirmed.
In the second part, we investigate various ways to approximate the driving
noise and analyze its effect on the rate of strong convergence. First, we consider
the use of frames to represent the noise. We show that if the frame is chosen in a
way that is well suited for the covariance operator, then the number of elements
of the frame needed to represent the noise without effecting the overall convergence
rate of the numerical method may be quite low. Second, we investigate the
use of finite element approximations of the eigenpairs of the covariance operator.
It turns out that if the kernel of the operator is smooth, then the number of basis
functions needed may be substantially reduced.
Our analysis is done in a framework based on operator semigroups. It is performed
in a way that reduces our results to results about approximation of the
respective (deterministic) semigroup.

Weak convergence

Stochastic partial differential equation

Wiener process

Additive noise

Finite element

Cahn-Hilliard-Cook equation

Error estimate

Hyperbolic equation

Truncation

Strong convergence

Rational approximation

Parabolic equation

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

Numerical Algorithms,; Vol. 53(2010)p. 309-320

**Journal article**

BIT Numerical Mathematics,; Vol. 53(2013)p. 497-525

**Journal article**

BIT (Copenhagen),; Vol. 52(2012)p. 85-108

**Journal article**

Journal of Computational and Applied Mathematics,; Vol. 235(2011)p. 3554-3570

**Journal article**

Mathematics

Computational Mathematics

Basic sciences

C3SE (Chalmers Centre for Computational Science and Engineering)

978-91-7385-787-1

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 3468

Pascal

Opponent: Professor Klaus Ritter, Fachbereich Mathematik, Technische Universität Kaiserslautern, Tyskland.