The Applied and Numerical Analysis (ANA) group is focused on the analysis of mathematical models of biological, chemical or physical processes described by differential, integral and integro-differential equations, and in the development, analysis and implementation of numerical methods for their approximate solutions.
The diverse applications considered by the group include rigorous analysis and computational modelling of problems in acoustic, elastic and electromagnetic wave propagation, modelling of fluid flows, biological processes, viscoelasticity and fracture in solids. Novel finite element and boundary element methods are developed and analysed for these applications, leading to robust algorithms and specialised software. A related research topic is approximation of orthogonal polynomials and special functions with explicit error terms.
The group research also includes analysis of nonlinear problems, like Navier-Stokes system, Gierer-Meinhardt system, and abstract bifurcation problems.
The ANA group is home of a major international collaboration on analytical and numerical methods for boundary-domain integral equations, aimed at creating a new method for analysis and solution of variable-coefficient and nonlinear PDEs.
Specific research expertise
- Analysis of partial differential equations, including nonlinear PDEs of fluid mechanics and mathematical biology (S. Mikhailov, M. Winter)
- Analysis and numerical implementation of boundary-domain integral and integro-differential equations (S. Mikhailov)
- Computational modelling of problems in solid mechanics, as well as acoustic, elastic and electromagnetic wave propagation, by Finite Element and Boundary Element methods (S. Langdon, M. Maischak, S. Shaw, M. Warby, J. Whiteman)
- Approximation of orthogonal polynomials and special functions (I. Krasikov)
- Abstract bifurcation and singularity theory (J. Furter)
- Fast solvers and preconditioners, error estimators and adaptive algorithms, high performance and scientific computing, software development (S. Langdon, M. Maischak, S. Shaw)
- Theoretical and computational modelling of fatigue, damage, durability, and fracture (S. Mikhailov)
For more detailed descriptions of research and list of individual publications, please follow the links to the web pages of individual group members.
Major external collaborators
- Prof. I. Babuska, University of Texas at Austin, USA
- Prof. O. Chkadua, Mathematical Institute, Tbilisi State University, Tbilisi, Georgia
- Prof. M. Kohr, Babeş-Bolyai University, Cluj-Napoca, Romania
- Prof. M. Lanza de Cristoforis, University of Padua, Padua, Italy
- Prof. D. Natroshvili, Georgian Institute of Technology, Tbilisi, Georgia
- Prof. S. Rjasanow, University of Saarland, Saarbrucken, Germany
- Prof. E. Stephan, University of Hannover, Hannover, Germany
- Prof. J. Wei, University of British Columbia, Vancouver, Canada
- Prof. W.L. Wendland, University of Stuttgart, Stuttgart, Germany
- Dr. T.G. Ayele, Addis Ababa University, Addis Ababa, Ethiopia
- Dr. T.T. Dufera, Adama Science and Technology University, Adama, Ethiopia
Externally funded projects
- MARM programme grant supporting collaboration with Adama University of Science and Technology, Ethiopia (London Mathematical Society, 2020-2021, S. Mikhailov)
- Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs (EPSRC, 2015-2018, S. Mikhailov).
- Acoustic Localisation of Coronary Artery Stenosis (EPSRC, 2010-2014, S. Shaw, J. Whiteman)
- Mathematical Analysis of Localised Boundary-Domain Integral Equations for BVPs with Variable Coefficients (EPSRC, 2010-2013, S. Mikhailov).
- Boundary-Domain Integral and Integro-Differential Equations: Formulation, Analysis, Localisation (Royal Society, 2006-2009, S. Mikhailov).
- Analysis of Boundary-Domain Integral and Integro-Differential Equations (LMS–IMU–AMMSI Initiative “Mentoring African Research in Mathematics” funded by the Nuffield Foundation, 2006-2008, S. Mikhailov).
Conferences
ANA group hosts triennial conference on Mathematics of Finite Elements and Applications (MAFELAP), which is one of the largest international conferences on this topic; the latest MAFELAP 2016 and MAFELAP 2019 had over 350 delegates from more than 25 countries.
We also organise mini-symposia and special sessions on Boundary-Domain Integral Equations at the two biannual international conferences: International Conference Integral Methods in Science and Engineering, IMSE, and Congress of International Society for Analysis, its Applications and Computation, ISAAC.
Group members
Krasikov
graph theory, combinatorics, coding theory, number theory and orthogonal polynomials
![Dr Ilia Krasikov]()
Dr Ilia Krasikov
Graph theory, combinatorics, coding theory, number theory and orthogonal polynomials
Langdon
i joined brunel university london in october 2019, having previously worked at the university of reading for over fifteen years, the last five as head of the department of mathematics and statistics. i served as head of the department of mathematics from october 2019 to october 2022, and then as interim executive dean of the college of engineering, design and physical sciences from november 2022 until may 2024. my research is in the area of numerical analysis, particularly the development, analysis and implementation of numerical methods for the solution of partial differential equations, and the application of such schemes to the solution of mathematical models arising from physical or biological processes such as acoustic or electromagnetic scattering, fluid flow, or tumour growth. ma2690 - professional development and project work
![Professor Stephen Langdon]()
Professor Stephen Langdon
I joined Brunel University London in October 2019, having previously worked at the University of Reading for over fifteen years, the last five as Head of the Department of Mathematics and Statistics. I served as Head of the Department of Mathematics from October 2019 to October 2022, and then as Interim Executive Dean of the College of Engineering, Design and Physical Sciences from November 2022 until May 2024. My research is in the area of Numerical Analysis, particularly the development, analysis and implementation of numerical methods for the solution of partial differential equations, and the application of such schemes to the solution of mathematical models arising from physical or biological processes such as acoustic or electromagnetic scattering, fluid flow, or tumour growth. MA2690 - Professional Development and Project Work
Maischak
elliptic boundary value and transmission problems. signorini problems/variational inequalities. boundary element and finite element methods. fast solvers and preconditioners. error estimators and adaptive algorithms. high performance and scientific computing. software development
![Dr Matthias Maischak]()
Dr Matthias Maischak
Elliptic boundary value and Transmission problems. Signorini problems/variational inequalities. Boundary Element and Finite Element Methods. Fast Solvers and Preconditioners. Error estimators and adaptive algorithms. High Performance and Scientific Computing. Software development
Mikhailov
joined brunel university london in 2006 applied analysis, solid mechanics, and computational mathematics, including: analysis of stokes, oseen, and navier-stokes pdes, especially of existence, uniqueness and regularity of solution of evolution (non-stationary) problems in sobolev spaces. boundary-domain integral and integro-differential equations. theoretical fatigue, damage, durability, and fracture mechanics. nonlinear partial integro-differential volterra equations of crack propagation in damaged media. ma2632, algebra and analysis ma1608, elements of applied mathgematics
![Professor Sergey Mikhailov]()
Professor Sergey Mikhailov
Joined Brunel University London in 2006 Applied Analysis, Solid Mechanics, and Computational Mathematics, including: Analysis of Stokes, Oseen, and Navier-Stokes PDEs, especially of existence, uniqueness and regularity of solution of evolution (non-stationary) problems in Sobolev spaces. Boundary-domain integral and integro-differential equations. Theoretical fatigue, damage, durability, and fracture mechanics. Nonlinear partial integro-differential Volterra equations of crack propagation in damaged media. MA2632, Algebra and Analysis MA1608, Elements of Applied Mathgematics
Shaw
simon shaw is a professor in the department of mathematics in the college of engineering, design and physical sciences, and belongs to the applied and numerical analysis research group. he is also a member of the structural integrity theme of our institute of materials and manufacturing, and of the centre for assessment of structures and materials under extreme conditions, and of the centre for mathematical and statistical modelling. shaw was initially a craft mechanical engineering apprentice but (due to redundancy) left this to study for a mechanical engineering degree. after graduation he became an engineering designer of desktop dental x ray processing machines, but later returned to higher education to re-train in computational mathematics. his research interests include computational simulation methods for partial differential volterra equations and, in this and related fields, he has published over thirty research papers. he is currently involved in an interdisciplinary project that is researching the potential for using computational mathematics and machine learning as a noninvasive means of screening for coronary artery disease. personal home page: computational science, engineering and mathematics: finite element and related methods. dispersive media (viscoelasticity and lossy dielectrics); deep neural nets and machine learning. finite element, and related, methods in space and time for partial differential equations arising in continuum mechanics. particularly interested in dispersive materials such as polymers and lossy dielectrics for which the constitutive laws exhibit memory effects. currently interested in using real or (from forward solves) virtual training data to solve inverse problems using machine learning, with a particular focus on deep neural networks. the motivating application for this inverse problem work is in screening for coronary artery disease.
![Professor Simon Shaw]()
Professor Simon Shaw
Simon Shaw is a professor in the Department of Mathematics in the College of Engineering, Design and Physical Sciences, and belongs to the Applied and Numerical Analysis Research Group. He is also a member of the Structural Integrity theme of our Institute of Materials and Manufacturing, and of the Centre for Assessment of Structures and Materials under Extreme Conditions, and of the Centre for Mathematical and Statistical Modelling. Shaw was initially a craft mechanical engineering apprentice but (due to redundancy) left this to study for a mechanical engineering degree. After graduation he became an engineering designer of desktop dental X Ray processing machines, but later returned to higher education to re-train in computational mathematics. His research interests include computational simulation methods for partial differential Volterra equations and, in this and related fields, he has published over thirty research papers. He is currently involved in an interdisciplinary project that is researching the potential for using computational mathematics and machine learning as a noninvasive means of screening for coronary artery disease. Personal home page: Computational Science, Engineering and Mathematics: finite element and related methods. Dispersive media (viscoelasticity and lossy dielectrics); deep neural nets and machine learning. Finite element, and related, methods in space and time for partial differential equations arising in continuum mechanics. Particularly interested in dispersive materials such as polymers and lossy dielectrics for which the constitutive laws exhibit memory effects. Currently interested in using real or (from forward solves) virtual training data to solve inverse problems using machine learning, with a particular focus on deep neural networks. The motivating application for this inverse problem work is in screening for coronary artery disease.
Warby
michael warby is a lecturer in mathematical sciences and he is a member of bicom (brunel institute of computational mathematics). he completed an undergraduate degree in mathematics at the university of kent in 1980 and he then moved to brunel university london to do a msc in numerical analysis which was completed in 1981. he then stayed at brunel to do a phd with the title \"bergman kernel methods and the numerical conformal mapping of simply and doubly connected domains\" which was completed in 1984. in 1984 he then joined bicom as a post doc and during the 1980s and 1990s he worked on several projects all of which have involved using the finite element method and several projects have been partly funded by companies who use the thermoforming process. work in this area continued when he became a lecturer when he was a principal investigator of the epsrc funded project ``computational modelling of thermoforming and in-mould-decoration processes\'\' during 1999--2002 which involved the company autotype. with a reasonably broad mathematical and computational background and with many years of experience with programming he has taught a wide range of modules. my main research area is computational solid mechanics usually involving the use of the finite element method. in particular i have many years of experience in the computational modelling of the thermoforming process which involves large deformations, contact and material behaviour such as hyperelastic, viscoelastic and elasto-plastic. recent activity involves considering goal orientated techniques to assess both the discretization error and the modelling error when attempting to compute some quantity of interest. other research activities have involved viscoelastic fracture and numerical techniques in conformal mapping.
![Dr Mike Warby]()
Dr Mike Warby
Michael Warby is a lecturer in Mathematical Sciences and he is a member of BICOM (Brunel institute of Computational Mathematics). He completed an undergraduate degree in Mathematics at the University of Kent in 1980 and he then moved to Brunel University London to do a MSc in Numerical Analysis which was completed in 1981. He then stayed at Brunel to do a PhD with the title \"Bergman kernel methods and the numerical conformal mapping of simply and doubly connected domains\" which was completed in 1984. In 1984 he then joined BICOM as a Post Doc and during the 1980s and 1990s he worked on several projects all of which have involved using the finite element method and several projects have been partly funded by companies who use the thermoforming process. Work in this area continued when he became a lecturer when he was a principal investigator of the EPSRC funded project ``Computational Modelling of Thermoforming and In-Mould-Decoration Processes\'\' during 1999--2002 which involved the company Autotype. With a reasonably broad mathematical and computational background and with many years of experience with programming he has taught a wide range of modules. My main research area is computational solid mechanics usually involving the use of the finite element method. In particular I have many years of experience in the computational modelling of the thermoforming process which involves large deformations, contact and material behaviour such as hyperelastic, viscoelastic and elasto-plastic. Recent activity involves considering goal orientated techniques to assess both the discretization error and the modelling error when attempting to compute some quantity of interest. Other research activities have involved viscoelastic fracture and numerical techniques in conformal mapping.
Winter
phd stuttgart 1993, habilitation stuttgart 2003; postdoctoral research fellow, institute for advanced study, princeton, 1993-94; postdoctoral research fellow, heriot-watt university, edinburgh, 1994-96; wissenschaftlicher mitarbeiter/wissenschaftlicher assistent, stuttgart, 1996-2005; lecturer/senior lecturer, brunel, 2005- mathematical biology, pattern formation, infectious diseases. phase transitions, micromagnetics, microstructure. nonlinear partial differential equations. nonlinear functional analysis. calculus of variations. dynamical systems.
![Dr Matthias Winter]()
Dr Matthias Winter
PhD Stuttgart 1993, Habilitation Stuttgart 2003; Postdoctoral Research Fellow, Institute for Advanced Study, Princeton, 1993-94; Postdoctoral Research Fellow, Heriot-Watt University, Edinburgh, 1994-96; Wissenschaftlicher Mitarbeiter/Wissenschaftlicher Assistent, Stuttgart, 1996-2005; Lecturer/Senior Lecturer, Brunel, 2005- Mathematical Biology, Pattern Formation, Infectious Diseases. Phase Transitions, Micromagnetics, Microstructure. Nonlinear Partial Differential Equations. Nonlinear Functional Analysis. Calculus of Variations. Dynamical Systems.