Parallel Methods for PDE Eigenvalue Problems 

This page is devoted to the research project

Parallel Methods for PDE Eigenvalue Problems

which was supported by  EPSRC grant GR/M59075.  The investigators were Alastair Spence (PI)    and    Ivan Graham  . 

From the grant we  employed Eero Vainikko   as a Postdoctoral Research Assistant and Joerg Berns-Mueller   as a PhD student.

Our industrial  collaborator was  Andrew Cliffe of Serco Assurance

The project  was  concerned with the computation of eigenavalue problems in  PDE,  using  Arnoldi and  inverse iteration  strategies with variable shifts. The project involves both the analysis of shift-invert eigenvalue solvers with inexact inner solves and the parallel implementation of such algorithms on bifurcation problems in fluid dynamics.

The eigenvalue solvers demand repeated solves of algebraic systems arising from discretisations of shifted  systems of PDEs. For this we have developed the  DOUG package as a fast parallel solver.  In this part of the project most of the effort has been in extending the DOUG code to the case of systems of PDEs and the design and implementation of parallel preconditioners, especially for the incompressible Navier-Stokes equations.  

Here is the    final report    for the grant (March 31st 2003). The work is continuing and further updates will be posted here in due course.

Currently there are four  papers:


I.G. Graham, A. Spence and E. Vainikko, Parallel iterative methods for Navier-Stokes equations and application to stability assessment, in Euro-Par 2002 Parallel Processing, B. Monien and R. Feldman (Eds), Lecture Notes in Computer Science 2400, Springer-Verlag, Berlin, 2002, pp705-714. ISSN 0302 9743, ISBN 3 540 44049 6. 
I.G. Graham, A. Spence and E. Vainikko, Parallel iterative methods for Navier-Stokes equations and application to eigenvalue computation, Concurrency and Computation: Practice and Experience , 15 (2003), 1151--1168.  
E. Vainikko and I.G. Graham, A parallel solver for PDE systems and application to the incompressible Navier-Stokes equations, Applied Numerical Mathematics, 49 (2004), 97-116.

J. Berns-Mueller, I.G. Graham and A. Spence,  Inexact inverse iteration for symmetric matrices, submitted to Linear Algebra and its Applications, March 2003.  Preprint,  ps format .