Splitting schemes for nonlinear parabolic problems

Research output: ThesisDoctoral Thesis (compilation)

Abstract

This thesis is based on five papers, which all analyse different aspects of splitting schemes when applied to nonlinear parabolic problems.
These numerical methods are frequently used when a problem has a natural decomposition into two or more parts, as the computational cost may then be significantly decreased compared to other methods.
There are two prominent themes in the thesis; the first concerns convergence order analysis, while the second focuses on structure preservation.

To motivate the first theme, we note that even if a method has been shown to converge it might be that the speed of convergence is arbitrarily slow. As such a method is unusable in practice we see that it is essential to prove convergence orders. However, those studies that present such error analyses in the fully nonlinear setting typically assume more regularity of the solution than what should be expected.
In this context, we present a convergence order analysis for a class of splitting schemes which, importantly, does not require any artificial regularity assumptions. This analysis is carried out in the setting of m-dissipative operators, which includes a large number of interesting problem classes. As demonstrated by the first three papers, the theory can be applied to such diverse problems as nonlinear reaction-diffusion systems, nonlinear parabolic problems with delay, as well as differential Riccati equations.

Within the second theme of structure preservation, an in-depth study of operator-valued differential Riccati equations has been carried out. In such equations it is desirable for a numerical method to produce positive semi-definite approximations. Further, it is essential that an implementation can utilize the problem-inherent property of low rank. As shown in the last three papers, both these features are readily satisfied for various splitting schemes. Since these are additionally less costly than existing comparable methods, they constitute a particularly competitive choice for such problems.

Details

Authors
  • Tony Stillfjord
Organisations
Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Mathematics
Original languageEnglish
QualificationDoctor
Awarding Institution
Supervisors/Assistant supervisor
Award date2015 Jun 5
Publisher
  • Centre for Mathematical Sciences, Lund University
Print ISBNs978-91-7623-252-1
StatePublished - 2015
Publication categoryResearch

Bibliographic note

Defence details Date: 2015-06-05 Time: 13:15 Place: MH:C, Centre for Mathematical Sciences, Sölvegatan 18, Lund, Sweden External reviewer(s) Name: Emmrich, Etienne Title: Professor Dr. Affiliation: Technische Universität Berlin --- The information about affiliations in this record was updated in December 2015. The record was previously connected to the following departments: Numerical Analysis (011015004)

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Related research output

Stillfjord, T., 2015, In : IEEE Transactions on Automatic Control. 60, 10, p. 2791-2796

Research output: Contribution to journalArticle

Eskil Hansen & Stillfjord, T., 2014, In : SIAM Journal on Numerical Analysis. 52, 6, p. 3128-3139

Research output: Contribution to journalArticle

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