Splitting schemes for nonlinear parabolic problems
Research output: Thesis › Doctoral 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 mdissipative 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 reactiondiffusion systems, nonlinear parabolic problems with delay, as well as differential Riccati equations.
Within the second theme of structure preservation, an indepth study of operatorvalued differential Riccati equations has been carried out. In such equations it is desirable for a numerical method to produce positive semidefinite approximations. Further, it is essential that an implementation can utilize the probleminherent 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.
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 mdissipative 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 reactiondiffusion systems, nonlinear parabolic problems with delay, as well as differential Riccati equations.
Within the second theme of structure preservation, an indepth study of operatorvalued differential Riccati equations has been carried out. In such equations it is desirable for a numerical method to produce positive semidefinite approximations. Further, it is essential that an implementation can utilize the probleminherent 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  

Organisations  
Research areas and keywords  Subject classification (UKÄ) – MANDATORY

Original language  English 

Qualification  Doctor 
Awarding Institution  
Supervisors/Assistant supervisor 

Award date  2015 Jun 5 
Publisher 

Print ISBNs  9789176232521 
State  Published  2015 
Bibliographic note
Defence details
Date: 20150605
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)
Related research output
Stillfjord, T. 2015 In : Preprint without journal information. 12 p.
Research output: Contribution to journal › Article
Stillfjord, T. 2015 In : IEEE Transactions on Automatic Control. 60, 10, p. 27912796
Research output: Contribution to journal › Article
Eskil Hansen & Stillfjord, T. 2014 In : SIAM Journal on Numerical Analysis. 52, 6, p. 31283139
Research output: Contribution to journal › Article