EPSRC Reference: 
GR/S74072/01 
Title: 
The spectrum of the SchmidHenningson block operator 
Principal Investigator: 
Marletta, Professor M 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Mathematics 
Organisation: 
Cardiff University 
Scheme: 
Standard Research (PreFEC) 
Starts: 
01 September 2004 
Ends: 
31 August 2006 
Value (£): 
4,204

EPSRC Research Topic Classifications: 
Mathematical Analysis 
Numerical Analysis 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Describe the proposed research using (about 200) words geared to the nonspecialist reader.The SchmidHenningson problem, which arises in the analysis of the stability of pipe flow, is an operator pencil problem (Nlambda P)y = 0, in which lambda is the eigenvalue/spectral parameter and N, P are 2x2 matrices having ordinary differential operators of order up to 4 as their elements. These operators have singular behaviour at at least one boundary point. Schmid and Henningson have carried out computational work on this problem over the last 1015 years, because of its importance in flow applications. They have also stated 'boundary conditions' at the singular point r=0, for some ranges of the physical parameters appearing in N and P, with good physical justifications.However the last ten years have also seen the development of a rich operator/spectral theory for block operator matrix problems, block operator pencils and singular nonselfadjoint systems of differential equations. The proposed Visiting Fellow, Professor Christiane Tretter of the University of Bremen, has been at the forefront of these developments, and significant work has also been done by the proposer and others in Cardiff. By applying this theory we believe it will be possible to obtain a deeper understanding of a whole class of singular block operator pencil problems of which SchmidHenningson is an example: for instance, we hope to understand how the existence and structure of the essential spectrum (Does it divide the complex plane? Into how many components?) depend on the physical parameters, and to know which boundary conditions are mathematically necessary and which are automatically satisfied.These results will have important consequences for the approximation of the spectrum of the SchmidHenningson and similar operators which, as singular nonselfadjoint operators, cannot immediately be expected to have nice properties such as spectral exactness when they are approximated and/or discretized.

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Summary 

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Organisation Website: 
http://www.cf.ac.uk 