EPSRC Reference: 
EP/H020497/1 
Title: 
Mathematical analysis of Localised BoundaryDomain Integral Equations for VariableCoefficient Boundary Value Problems 
Principal Investigator: 
Mikhailov, Professor SE 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Information Systems Computing and Maths 
Organisation: 
Brunel University London 
Scheme: 
Standard Research 
Starts: 
03 April 2010 
Ends: 
02 April 2013 
Value (£): 
202,144

EPSRC Research Topic Classifications: 
Mathematical Analysis 
Nonlinear Systems Mathematics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 
Panel Date  Panel Name  Outcome 
03 Dec 2009

Mathematics Prioritisation Panel

Announced


Summary on Grant Application Form 
The proposal is aimed at developing a rigorous mathematical backgrounds of an emerging new family of computational methods for solution of partial differential equations (PDEs) of science and engineering. The approach is based on reducing the original linear or nonlinear boundary value problems for PDEs to localised boundarydomain integral or integrodifferential equations, which after meshbased or meshless discretisation lead to systems of algebraic equations with sparse matrices. This is especially beneficial for problems with variable coefficients, where no fundamental solution is available in an analytical and/or cheaply calculated form, but the approach employs a widely available localised parametrix instead. PDEs with variable coefficients arise naturally in mathematical modelling nonhomogeneous linear and nonlinear media (e.g. functionally graded materials, materials with damageinduced inhomogeneity or elastic shells) in solid mechanics, electromagnetics, thermoconductivity, fluid flows trough porous media, and other areas of physics and engineering. The main ingredient for reducing a boundaryvalue problem for a PDE to a boundary integral equation is a fundamental solution to the original PDE. However, it is generally not available in an analytical and/or cheaply calculated form for PDEs with variable coefficients or PDEs modelling complex media. Following Levi and Hilbert, one can use in this case a parametrix (Levi function) to the original PDE as a substitute for the fundamental solution. Parametrix is usually much wider available than fundamental solution and correctly describes the main part of the fundamental solution although does not have to satisfy the original PDE. This reduces the problem not to boundary integral equation but to boundarydomain integral equation. Its discretisation leads to a system of algebraic equations of the similar size as in the finite element method (FEM), however the matrix of the system is not sparse as in the FEM and thus less efficient for numerical solution. Similar situation occurs also when solving nonlinear problems (e.g. for nonlinear heat transfer, elasticity or elastic shells under large deformations) by boundarydomain integral equation method. The Localised BoundaryDomain Integral Equation method emerged recently addressing this deficiency and making it competitive with the FEM for such problems. It employs specially constructed localised parametrices to reduce linear and nonlinear BVPs with variable coefficients to Localised BoundaryDomain Integral or IntegroDifferential Equations, LBDI(D)Es. After a locallysupported meshbased or meshless discretisation this leads to sparse systems of algebraic equations efficient for computations. Further development of the LBDI(D)Es, particularly exploring the idea that they can be solved by iterative algorithms needing no preconditioning, due to their favourable spectral properties, requires a deeper analytical insight into properties of the corresponding integral and integrodifferential operators, which the project is aimed to provide. The project analytical results will be implemented in numerical algorithms and computer codes developed under the PI supervision by two PhD students, who are supported from other sources and thus are not included the proposal.

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Summary 

Date Materialised 


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Project URL: 
http://people.brunel.ac.uk/~mastssm/LBDEgrant.html 
Further Information: 

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