EPSRC Reference: 
EP/H043519/1 
Title: 
Adaptive Multiscale Methods for Approximation and Preconditioning 
Principal Investigator: 
Graham, Professor I 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematical Sciences 
Organisation: 
University of Bath 
Scheme: 
Standard Research 
Starts: 
01 May 2010 
Ends: 
31 July 2011 
Value (£): 
16,027

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Multiscale modelling is ubiquitous in science, technology and social science. Models with multiple time or length scales arise, for example, in modelling pollutant transport in groundwater, turbulent fluid flow in reactor cooling systems, highfrequency waves in radar and sonar and atomisticcontinuum models in material science. Because of the highly varying scales involved in multiscale problems, the accurate modelling of all scales is often outside the reach of even the largest supercomputers. A suitable goal of computation is then to compute a solution on the finest computationally affordable grid, in such a way that the accuracy is not polluted by the fine scales which remain unresolved. More precisely, the aim is to solve the multiscale problem on a coarse mesh in such a way that the error is of the same order (with respect to the number of degrees of freedom) as if the problem were smooth, and moreover, the accuracy does not degrade if the finest scale present in the model decreases. Such computational methods are often called ``robust''. There are two paradigms for the construction of robust numerical methods for multiscale problems. The first is to replace the multiscale problem with a (nearby) smooth problem and then solve the latter numerically. Examples of this approach include upscaling in porous medium flow and transport or the use of geometric theory of diffraction and raytracing in highfrequency wave propagation problems. The basic difficulty with this approach is that these approximations tend to be valid only when the fine scale small parameter is sufficiently small (in order to make some sort of averaging valid), and, moreover, their rigorous analysis requires simplifying assumptions (such as periodicity and scale separation in homogenization theory).An alternative approach is to devise problemadapted numerical methods which are targeted to the type of multiscale behaviour arising in the particular application, and are capable of resolving it robustly on a coarse mesh. This usually involves replacing the (piecewise) polynomial approximations at the heart of classical numerical methods with problemadapted bases which are better able to reflect the solution behaviour on coarse meshes. Examples of this type of approach in application areas include subgrid scale modelling in large eddy simulation, and the modelling of localised convective storms in largescale weather prediction software. This is a new collaboration between the PI and the proposed VF which will not take place without the requested EPSRC support. We will produce new results on methods of the second type. Our methods will be adaptive (i.e. the nonpolynomial bases will be computed automatically, rather than designed in detail by the practitioner) and they will work well both in the presence of small lengthscale (small wavelength of data) as well as large contrast (large amplitude of data). We will test our methods on systems arising from problems with random data with small lengthscale and large variance (leading to small wavelength and large amplitude). We will also investigate the application of the same ideas in the design of robust preconditioners for conventional discretisations of multiscale problems including those which approximate equations describing high frequency wave phenomena.

Key Findings 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Potential use in nonacademic contexts 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Impacts 
Description 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk 
Summary 

Date Materialised 


Sectors submitted by the Researcher 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Project URL: 

Further Information: 

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