Mathematical Analysis of Localised Boundary-Domain Integral Equations for BVPs with Variable Coefficients

Research grant awarded by the Engineering and Physical Sciences Research Council (EPSRC), UK, 2010-2013

Principal Investigator: Prof. S. E. Mikhailov, Department of Mathematics, Brunel University London, UK

Visiting Researchers:
Prof. D. Natroshvili, Department of Mathematics, Georgian Technical University, Tbilisi, Georgia
Prof. O. Chkadua, A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University and Sokhumi State University, Tbilisi, Georgia

Objectives

1. To analyse the equivalence of Localised Boundary-Domain Integral and Integro-Differential Equations, LBDI(D)Es, to the original boundary value problems for partial differential equations with variable coefficients, existence and uniqueness of LBDI(D)E solutions, and invertibility of their operators.

2. To investigate spectral properties of the LBDI(D)Es of the second kind and develop iterative methods for their solution.

3. To extend the analysis to nonlinear LBDI(D)Es

Summary of project findings

The project developed 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, required a deeper analytical insight into properties of the corresponding integral and integro-differential operators, which the project provided.

The following objectives have been reached in the project.

1. The LBDI(D)Es, of linear boundary value problems for elliptic scalar PDEs and PDE systems of the second order with variable coefficients were analysed. This includes the proofs of the LBDI(D)E equivalence to the original BVPs, existence and uniqueness of LBDI(D)E solutions, and invertibility of their operators.

2. The spectral properties of the BDI(D)Es of the second kind were investigated and iterative methods for their solution, using the information about the spectral properties, were developed applied in numerical calculations.

3. The analysis was extended to some nonlinear LBDI(D)Es.

Exploitation routes

It is expected that the project results will be useful for mathematicians working in applied analysis and also mathematicians and engineers engaged in numerical solution of BVPs of science and engineering, particularly in computational solid mechanics, fluid dynamics, diffusion, electro- and magnetodynamics.
Further implementation of the results in effective and robust computer codes based on LBDI(D)Es to solve 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 project also paves the way to extend the LBDIE approach to non-elliptic PDEs of the second order, e.g. Maxwell, parabolic and hyperbolic PDE systems, as well as to higher order equations. The project results for some nonlinear LBDI(D)Es can be also essentially generalised.

Potential use in non-academic contexts

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. This will have a very definite impact in the area of numerical methods and computational mechanics both in the UK and internationally.
The project analytical results were implemented in numerical algorithms and experimental computer codes and the obtained results were 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.

Some project publications

Chkadua O., Mikhailov S.E., Natroshvili D., Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients, Integral Equations and Operator Theory (IEOT), accepted. PDF

Chkadua O., Mikhailov S.E., Natroshvili D., Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains, Analysis and Applications, Vol.11, 2013, DOI: 10.1142/S0219530513500061 (accepted, to appear). PDF

Mikhailov S.E. Solution regularity and co-normal derivatives for elliptic systems with non-smooth coefficients on Lipschitz domains. J. Math. Analysis and Appl., Vol. 400, 2013, 48-67. PDF

Grzhibovskis R., Mikhailov S., Rjasanow S. Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D, Comput. Mech., Vol. 51, 2013, 495-503, DOI: 10.1007/s00466-012-0777-8, PDF

Mikhailov S.E., Mohamed N.A., Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with variable coefficient, International J. Computer Math., Vol. 89, 2012, 1488-1503. PDF

Mikhailov S.E. Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Analysis and Appl., Vol. 378, 2011, 324-342. PDF

Chkadua O., Mikhailov S.E., Natroshvili D. Localized direct segregated boundary-domain integral equations for variable-coefficient transmission problems with interface crack, Mem. Differential Equations Math. Phys. Vol. 52, 2011, 17-64. PDF

Chkadua O., Mikhailov S.E., Natroshvili D. Analysis of segregated boundary-domain integral equations for variable-coefficient problems with cracks, Numer. Meth. for PDEs, Vol. 27, 2011, 121-140. PDF

Ayele T.G., Mikhailov S.E., Analysis of Two-Operator Boundary-Domain Integral Equations for Variable-Coefficient Mixed BVP, Eurasian Math. J. , Vol. 2, No 3, 20-41. PDF

Chkadua O., Mikhailov S.E., Natroshvili D. Analysis of some localized boundary-domain integral equations for transmission problems with variable coefficients, In: Integral Methods in Science and Engineering: Computational and Analytic Aspects. C. Constanda and P. Harris, eds., Springer (Birkhäuser): Boston, ISBN 978 0 85316 2957, 2011, 91-108. PDF

Chkadua O., Mikhailov S.E., Natroshvili D., Analysis of segregated boundary-domain integral equations for mixed variable-coefficient BVPs in exterior domains, in: Integral Methods in Science and Engineering: Computational and Analytic Aspects. C. Constanda, P. Harris, eds., Springer (Birkhäuser): Boston, ISBN 978 0 85316 2957, 2011, 109-128. PDF

Chkadua O., Mikhailov S.E., Natroshvili D., Localized boundary-domain integral equations for Dirichlet problem for second order elliptic equations with matrix variable coefficients, in: Proceedings of the 8th UK Conference on Boundary Integral Methods, (edited by D. Lesnic), Leeds University Press, Leeds, UK , ISBN 978 0 85316 2957, (2011), 119-126. PDF

Chkadua O., Mikhailov S.E., Natroshvili D., Localized boundary-domain integral equations method for an interface crack problem, in: Proceedings of the 8th UK Conference on Boundary Integral Methods, (edited by D. Lesnic), Leeds University Press, Leeds, UK , ISBN 978 0 85316 2957, (2011), 49-56. PDF

Chkadua O., Mikhailov S.E., Natroshvili D. Localized boundary-domain integral equation formulation for mixed type problems, Georgian Math. J., Vol.17, 2010, 469-494. PDF

Some earlier publications related to the project theme

Chkadua O., Mikhailov S.E., Natroshvili D. Analysis of some localized boundary-domain integral equations, J. Integral Equations and Appl. Vol.21, 2009, 405-445. PDF

Chkadua O., Mikhailov S.E., Natroshvili D. Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, I: Equivalence and Invertibility, J. Integral Equations and Appl. Vol.21(4), 2009, 499-543. PDF

Chkadua O., Mikhailov S.E., Natroshvili D. Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, II: Solution regularity and asymptotics, J. Integral Equations and Appl. Vol.22, 2010, 19-37. PDF

Mikhailov S.E. Analysis of united boundary-domain integro-differential and integral equations for a mixed BVP with variable coefficient. Math. Methods in Applied Sciences, Vol. 29, 2006, 715-739. PDF

Mikhailov S.E. Localized direct boundary-domain integro-differential formulations for incremental elasto-plasticity of inhomogeneous body. Engineering Analysis with Boundary Elements, Vol. 30, 2006, 218-226. PDF

Mikhailov S. E. Incremental Localized Boundary-Domain Integro-Differential Equations of Elastic Damage Mechanics for Inhomogeneous Body, In: Advances in Meshless Methods (Edited by J. Sladek & V. Sladek), Tech Science Press, Forsyth, USA, ISBN: 0-9717880-2-2, 2006, 105-123. PDF

Mikhailov S.E. Will the boundary (-domain) integral equation method survive? Preface to the special issue on non-traditional boundary (-domain) integral equation methods. J. Engineering Math., Vol. 51, 2005, 197-198. PDF

Mikhailov S.E., Nakhova I.S. Mesh-based numerical implementation of the localized boundary-domain integral equation method to a variable-coefficient Neumann problem. J. Engineering Math., Vol. 51, 2005, 251-259. PDF

Mikhailov S.E. Localized direct boundary–domain integro–differential formulations for scalar nonlinear boundary-value problems with variable coefficients. J. Engineering Math. , Vol. 51, 2005, 283-302. PDF

Mikhailov S.E. Direct localized boundary-domain integro-differential formulations for physically nonlinear elasticity of inhomogeneous body. Engineering Analysis with Boundary Elements, Vol. 29, 2005, 1008–1015. PDF

Mikhailov S. E. Analysis of extended boundary-domain integral and integro-differential equations of some variable-coefficient BVP.  In: Advances in Boundary Integral Methods - Proceedings of the 5th UK Conference on Boundary Integral Methods (Edited by Ke Chen), University of Liverpool Publ., UK, ISBN 0 906370 39 6, 2005, 106-125. PDF

Mikhailov S.E. Analysis of boundary-domain integral and integro-differential equations for a Dirichlet problem with variable coefficient. In: Integral Methods in Science and Engineering: Theoretical and Practical Aspects (Edited by C.Constanda, Z.Nashed, D.Rolins), Boston-Basel-Berlin: Birkhäuser, ISBN 0-8176-4377-X, 2005, 161-176. PDF

Mikhailov S.E. Boundary-Domain Integro-Differential Equation of Elastic Damage Mechanics Model of Stationary Drilling. In: Advances in Boundary Element Techniques VI (Edited by A.P.S.Selvadurai, C.L.Tan & M.H.Aliabadi), EC Ltd., UK, ISBN 09547783-2-4, 2005, 107-114. PDF

Mikhailov S.E. Localized boundary-domain integral formulations for problems with variable coefficients. Engineering Analysis with Boundary Elements, Vol. 26, 2002, 681-690. PDF

Mikhailov S.E. Finite-dimensional perturbations of linear operators and some applications to boundary integral equations, Engineering Analysis with Boundary Elements, 1999, Vol.23, 805-813. PDF

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