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

Lead Research Organisation:
Brunel University

Department Name: Information Systems Computing and Maths

### Abstract

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.

### Planned Impact

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.

## People |
## ORCID iD |

Sergey E. Mikhailov (Principal Investigator) |

### Publications

Ayele T
(2017)

*Integral Methods in Science and Engineering, Volume 1*
Ayele T G
(2011)

*Analysis of Two-Operator Boundary-Domain Integral Equations for Variable-Coefficient Mixed BVP*in EURASIAN MATHEMATICAL JOURNAL
Ayele, T.G.
(2010)

*Two-operator boundary-domain integral equations for a variable-coefficient BVP*in Integral Methods in Science and Engineering: Analytic Methods
Chkadua O
(2013)

*Localized Boundary-Domain Singular Integral Equations Based on Harmonic Parametrix for Divergence-Form Elliptic PDEs with Variable Matrix Coefficients*in Integral Equations and Operator Theory
Chkadua O
(2011)

*Analysis of segregated boundary-domain integral equations for variable-coefficient problems with cracks*in Numerical Methods for Partial Differential Equations
Chkadua O
(2016)

*Localized boundary-domain singular integral equations of Dirichlet problem for self-adjoint second-order strongly elliptic PDE systems*in Mathematical Methods in the Applied Sciences
Chkadua O
(2011)

*Integral Methods in Science and Engineering*
Chkadua O
(2011)

*Localized direct segregated boundary-domain integral equations for variable-coefficient transmission problems with interface crack*in Memoirs on Differential Equations and Mathematical Physics
CHKADUA O
(2013)

*ANALYSIS OF DIRECT SEGREGATED BOUNDARY-DOMAIN INTEGRAL EQUATIONS FOR VARIABLE-COEFFICIENT MIXED BVPs IN EXTERIOR DOMAINS*in Analysis and Applications
Chkadua O.
(2011)

*8th UK Conference on Boundary Integral Methods*Description | 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 Route | 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. |

Sectors | Aerospace, Defence and Marine,Construction,Other |

URL | http://people.brunel.ac.uk/~mastssm/LBDEgrant.html |

Description | 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. |

Description | DTA studentship |

Amount | £0 (GBP) |

Funding ID | 1636273 |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Academic/University |

Country | United Kingdom of Great Britain & Northern Ireland (UK) |

Start | 10/2012 |

End | 09/2015 |

Description | EPSRC responsive mode |

Amount | £180,968 (GBP) |

Funding ID | EP/M013545/1 |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Academic/University |

Country | United Kingdom of Great Britain & Northern Ireland (UK) |

Start | 05/2015 |

End | 05/2018 |

Description | Ethiopia |

Organisation | Addis Ababa University |

Department | Department of Mathematics |

Country | Ethiopia, Federal Democratic Republic of |

Sector | Academic/University |

PI Contribution | Formulation and analysis of Boundary-Domain Integral Equations |

Collaborator Contribution | Extension to analysis of two-operator Boundary-Domain Integral Equations |

Impact | Dufera T.T., Mikhailov S.E. (2015) Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Dirichlet BVP in 2D, In: Integral Methods in Science and Engineering: Theoretical and Computational Advances. C. Constanda and A. Kirsh, eds., Springer (Birkhäuser): Boston, ISBN 978-3-319-16727-5, 163-175, DOI: 10.1007/978-3-319-16727-5_15. Ayele T.G., Mikhailov S.E., (2011) Analysis of Two-Operator Boundary-Domain Integral Equations for Variable-Coefficient Mixed BVP, Eurasian Math. J., Vol. 2, 2011, No 3, 20-41. |

Start Year | 2010 |

Description | Georgia |

Organisation | Georgian Technical University |

Country | Georgia |

Sector | Academic/University |

PI Contribution | Boundary-Domain Integral Equation formulation and analysis. |

Collaborator Contribution | Pseudo-Differential Equation technique for Boundary-Domain Integral Equations. |

Impact | Papers: Chkadua O., Mikhailov S.E., Natroshvili D. (2016) Localized boundary-domain singular integral equations of Dirichlet problem for self-adjoint second order strongly elliptic PDE systems, Math. Methods in Appl. Sci., DOI: 10.1002/mma.4100, 1-21. Chkadua O., Mikhailov S.E., Natroshvili D. (2013a) 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), Vol. 76, 2013, 509-547, DOI 10.1007/s00020-013-2054-4. Chkadua O., Mikhailov S.E., Natroshvili D. (20113b) Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains, Analysis and Applications, Vol.11, 2013, No 4, 1350006(1-33), DOI: 10.1142/S0219530513500061. Chkadua O., Mikhailov S.E., Natroshvili D. (2011a) Localized direct segregated boundary-domain integral equations for variable-coefficient transmission problems with interface crack, Mem. Differential Equations Math. Phys., Vol. 52, 17-64. Chkadua O., Mikhailov S.E., Natroshvili D. (2011b) Analysis of segregated boundary-domain integral equations for variable-coefficient problems with cracks, Numerical Meth. for PDEs, Vol. 27, 121-140. Chkadua O., Mikhailov S.E., Natroshvili D. (2011c) 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 978-0-8176-8237-8, 91-108. Chkadua O., Mikhailov S.E., Natroshvili D. (2010a) Localized boundary-domain integral equation formulation for mixed type problems, Georgian Math. J., Vol.17, 469-494. PDF Chkadua O., Mikhailov S.E., Natroshvili D. (2010b) 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. |

Start Year | 2010 |

Description | Georgia |

Organisation | Tbilisi State Medical University |

Country | Georgia |

Sector | Academic/University |

PI Contribution | Boundary-Domain Integral Equation formulation and analysis. |

Collaborator Contribution | Pseudo-Differential Equation technique for Boundary-Domain Integral Equations. |

Impact | Papers: Chkadua O., Mikhailov S.E., Natroshvili D. (2016) Localized boundary-domain singular integral equations of Dirichlet problem for self-adjoint second order strongly elliptic PDE systems, Math. Methods in Appl. Sci., DOI: 10.1002/mma.4100, 1-21. Chkadua O., Mikhailov S.E., Natroshvili D. (2013a) 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), Vol. 76, 2013, 509-547, DOI 10.1007/s00020-013-2054-4. Chkadua O., Mikhailov S.E., Natroshvili D. (20113b) Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains, Analysis and Applications, Vol.11, 2013, No 4, 1350006(1-33), DOI: 10.1142/S0219530513500061. Chkadua O., Mikhailov S.E., Natroshvili D. (2011a) Localized direct segregated boundary-domain integral equations for variable-coefficient transmission problems with interface crack, Mem. Differential Equations Math. Phys., Vol. 52, 17-64. Chkadua O., Mikhailov S.E., Natroshvili D. (2011b) Analysis of segregated boundary-domain integral equations for variable-coefficient problems with cracks, Numerical Meth. for PDEs, Vol. 27, 121-140. Chkadua O., Mikhailov S.E., Natroshvili D. (2011c) 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 978-0-8176-8237-8, 91-108. Chkadua O., Mikhailov S.E., Natroshvili D. (2010a) Localized boundary-domain integral equation formulation for mixed type problems, Georgian Math. J., Vol.17, 469-494. PDF Chkadua O., Mikhailov S.E., Natroshvili D. (2010b) 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. |

Start Year | 2010 |

Description | Saarbrucken |

Organisation | Saarland University |

Department | Institute of Applied Mathematics |

Country | Germany, Federal Republic of |

Sector | Academic/University |

PI Contribution | Boundary-Domain Integral Equation methodology. |

Collaborator Contribution | Numerical implementation of the Boundary-Domain Integral Equations in a computer code numerical solution of some boundary value problems. |

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

Start Year | 2012 |