A semigroup approach to the numerical solution of parabolic differential equations

This work is concerned with the numerical solution of linear parabolic differential equations. We exploit the framework of analytic semigroups to understand the properties of the solution as well as to develop numerical algorithms. In the first chapter, we focus on the homogeneous problem. A short introduction to the theory of analytic semigroups is given, which forms the basis for the further development. We prove Besov regularity of the solution for each fixed time where the underlying spatial domain may have a non-smooth boundary and reentrant corners. Moreover, we exploit the Dunford-Cauchy representation of analytic semigroups to numerical evaluate the operator exponential. Essentially, a quadrature rule for a Banach space valued integrand over an infinite integration interval is presented. The error analysis which incorporates spatial discretisation errors also leads to an efficient algorithm which evaluates the solution to the homogeneous problem up to any prescribed target accuracy. In contrast to classical schemes (e.g. Implicit Euler) the algorithm allows us to do arbitrary large time steps and is inherently parallel. In the second chapter, the scope is extended to the inhomogeneous problem. After some results about the mapping properties of the solution operator L which maps the time dependent forcing term f to the solution u(t) = (Lf)(t) we present a new discretisation scheme. We utilise multi-wavelets to discretise the time direction. The coefficients of the wavelet decomposition become vector-valued and can in turn be discretised by a wavelet basis on the spatial domain. For this scheme to be successful, we have to show that the vector-valued coefficients decay sufficiently fast to permit a sparse approximation to the solution. This goal is accomplished under very weak regularity assumptions on the forcing term. Moreover, we outline an efficient algorithm which calculates the wavelet decomposition of the solution given the forcing term f in wavelet coordinates. This algorithm is based on the quadrature rule from the first chapter. The third chapter is devoted to applications in regularisation theory. The inverse problems to the homogeneous and the inhomogeneous problem are well-known to be ill-posed such that we have to apply some regularisation technique. After a short introduction to the classical theory, we present a modification of Tikhonov regularisation which is closely adapted to the eigenspaces of the ill-posed operator. While this eigenspaces are usually unknown, we develop an algorithm based on a contour integral which projects any given value onto certain selected eigenspaces. Moreover, this algorithm allows us to apply the inverse of the restriction of an analytic semigroup to certain eigenspaces which yields an economical regularisation scheme. Concerning the inhomogeneous problem, an adaptive discretisation is presented which proves useful in Tikhonov regularisation. In the fourth chapter numerical experiments for the algorithms from the preceding chapters as well as comparisons to classical methods are collected. The error analysis from the first chapter turns out to be rather sharp. Moreover, the algorithm based on quadrature outperforms a classical single step method of second order. The new discretisation scheme for the inhomogeneous problem leads to very few significant coefficients in the time decomposition which renders the scheme very appealing. This is exemplified in Tikhonov regularisation where we observe singularities in the solution which are resolved well by the adaptive solution scheme.

Grantor dc:publisher

Publikationsserver der RWTH Aachen University

Year dc:date

2005

Author dc:creator

• Jürgens, Markus

Contributors dc:contributor

• Dahmen, Wolfgang

• info:eu-repo/classification/ddc/510
• Lineare parabolische Differentialgleichung
• Anfangswertproblem
• Numerisches Verfahren
• Analytische Halbgruppe
• Tichonov-Regularisierung
• Mathematik

Statement dc:rights

• info:eu-repo/semantics/openAccess

Language dc:language

eng

Identifier

https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121837%22

OAI identifier oai:identifier

oai:publications.rwth-aachen.de:60111