Mathematical analysis of Localised Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems

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 boundary-domain integral or integro-differential equations, which after mesh-based or mesh-less 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 non-homogeneous linear and nonlinear media (e.g. functionally graded materials, materials with damage-induced inhomogeneity or elastic shells) in solid mechanics, electromagnetics, thermo-conductivity, fluid flows trough porous media, and other areas of physics and engineering. The main ingredient for reducing a boundary-value 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 boundary-domain 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 non-linear heat transfer, elasticity or elastic shells under large deformations) by boundary-domain integral equation method. The Localised Boundary-Domain 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 non-linear BVPs with variable coefficients to Localised Boundary-Domain Integral or Integro-Differential Equations, LBDI(D)Es. After a locally-supported mesh-based or mesh-less 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 integro-differential 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.

Although this is mainly mathematical analysis project, in a longer run its results can be implemented in effective and robust computer codes for solving problems of heat transfer and stress analysis of structure elements made of functionally graded materials, variable-curvature inhomogeneous elastic shells, filtration through inhomogeneous rocks etc, will have a very definite impact in the area of numerical methods and computational mechanics both in the UK and internationally. The PhD students already working in the PI research group will do some experimental prototype computer implementation, in parallel to the project, and its results will be informed to the prospective users through journal and conference publications and the project web-site, as well as through individual contacts with prospective users in computational mechanics. If the experimental numerical implementation proves to be successful, a commercial software can stem from it in 5-10 year period. This would then benefit the software developers and numerous users in mechanical, structural, civil, marine, and aerospace engineering including design.