Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs

The proposal is aimed at developing rigorous mathematical backgrounds of an emerging new family of computational methods for solution of nonlinear Partial Differential Equations (PDEs). The approach is based on reducing the original nonlinear boundary value problems for PDEs to global or localised Boundary-Domain Integral or Integro-Differential Equations, BDI(D)Es, which after mesh-based or mesh-less discretisation lead to nonlinear systems of algebraic equations. In case of localised BDI(D)Es, the matrices of corresponding algebraic equations will be sparse.
Nonlinear PDEs arise naturally in mathematical modelling of nonlinear physical processes, e.g. of nonlinear heat transfer in materials with the thermo-conductivity coefficients depending on the point temperature and coordinate, materials with damage-induced inhomogeneity, elasto-plastic materials, nonlinear equation of stationary potential compressible flow, nonlinear flows trough porous media, nonlinear electromagnetics and other areas of physics and engineering.
The main ingredient for reducing a boundary-value problem for a linear 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 linear PDEs with variable coefficients and for nonlinear PDEs. Developing ideas of Levi and Hilbert, one can use in this case a parametrix (Levi function) either to the original nonlinear PDE or to another, linear, 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 generally reduces the nonlinear boundary value problem not to a boundary integral equation but to a global nonlinear boundary-domain integro-differential equation.
A discretisation of a global nonlinear BDIDE system leads to a system of nonlinear algebraic equations of the similar size as in the finite element method (FEM), however the matrix of the system is not sparse. The Localised Boundary-Domain Integro-Differential Equations, LBDIDEs, for nonlinear problems, emerged recently addressing this deficiency and making them competitive with the FEM for such problems. The LBDIDE method employs specially constructed localised parametrices to reduce nonlinear BVPs with variable coefficients to LBDIDEs. After employing a locally supported mesh-based or mesh-less discretisation, this leads to sparse systems of nonlinear algebraic equations efficient for computations.
However implementation of this idea requires a deeper analytical insight into properties of the corresponding nonlinear integral and integro-differential operators. Such analysis is available in the applicants publications for the global and localised BDIEs in the linear case, and for some global indirect non-linear BDIEs. The project is intended to make a leap from these results to the analysis of much more general nonlinear global and localised BDIDEs.
Further development of the project concerns the iterative algorithms to solve the global or localised nonlinear BDIDEs, particularly based on the fixed-point theorems. It is also expected that the project analytical results will be implemented in numerical algorithms and computer codes developed under the PI supervision by PhD students.

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 nonlinear heat transfer, nonlinearly elastic or elasto-plastic stress analysis of structure elements, including elastic shells under large deformations, nonlinear filtration through inhomogeneous rocks and Navier-Stokes problem of fluid flow. Through application in engineering and design, these computer codes will increase efficiency and reliability of engineering structures and machines, leading to the material and energy savings and increased competitiveness.
The PhD students 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.