Applied Mathematics

Mark Ainsworth

Francis Wayland Professor of Applied Mathematics, on Sabbatical
Room 313, 182 George Street
Office Hours: By appointment


Professor Ainsworth obtained his PhD from the University of Durham, United Kingdom in 1989. He is currently Professor of Applied Mathematics at Brown University, and also holds a Joint Faculty Appointment with the Mathematics and Computer Science Group at Oak Ridge National Laboratory, Tennessee. 

Research Interests

Ainsworth's original research interests are in the numerical approximation of partial differential equations. He is, together with J.T. Oden, the co-author of a monograph on,  A Posteriori Error Estimation in Finite Element Analysis.  

Ainsworth has worked in a range of areas relating to the numerical solution of partial differential equations including a posteriori error estimation and adaptive solution of PDEs using high order finite element methods, multiigrid and domain decomposition methods for the solution of large scale linear algebra problems, the analysis of dispersive and dissipative behaviour of numerical methods for wave propagation, hierarchical and multiscale modelling.  

Ainsworth is currently researching in several new areas: 

  • resiliency of state of the art numerical algorithms such as the multigrid method on emerging architectures such Exascale machines
  • numerical analysis and modelling using fractional partial differential equations
  • compression of scientific data arising from very large scale simulations on leadership computing facilities
  • the use of techniques from Computer Aided Geometric Design and in particular, Bernstein-Bezier polynomials for the analysis and efficient implementation of high order finite element methods

Academic Career

2013-present: Joint Faculty Appointment (Oak Ridge National Laboratory)

2012-present:  Professor of Applied Mathematics (Brown University)

2004-2012: The 1825 Chair Professor of Mathematics (Strathclyde University)

1998-2004: Professor of Applied Mathematics (Strathclyde University)

1996-98: Reader in Mathematics (Leicester University)


2014 Elected to Fellowship of the Society for Industrial and Applied Mathematics (SIAM Fellow)

2010 Elected to Fellowship of the Institute of Mathematics and Its Applications (FIMA)

2004 Awarded the Whitehead Prize for Mathematics from  the London Mathematical Society

2004 Awarded the J. L. Lions Prize for Computational Mathematics from ECCCOMAS

2003 Elected Fellow of the Royal Society of Edinburgh (Scottish National Academy of Science and Letters, FRES)

2002 Leverhulme Research Fellowship

Editorial Duties

Member of editorial board of SIAM Journal on Numerical Analysis (2013-present).

Member of editial board of IMA Journal on Numerical Analysis (2004-present).

Member of editorial board of SIAM Journal on Scientific Computing 2001-2016 (Served five terms).

Senior Editor of Applied Numerical Mathematics (2003-2013). Associate editor (1998-2003).

Member of editorial board of Journal on Scientific Computing 2012-2014.

Member of editorial board of International Journal for Numerical Methods in Engineering (2003-present).

Member of editorial board of International Journal of Numerical Analysis and Modeling (2004-2012).

Member of editorial board of Numerical Methods for Partial Differential Equations (2004-present).

Member of editorial board of International Journal for Numerical Methods in Fluids (2005-present).

Member of editorial board of Proceedings of Royal Society of Edinburgh. Section A: Mathematics (2003-2007).  Invited to become Editor-in-Chief 2008 (declined).

Recent and Upcoming Plenary Lectures 

Plenary speaker 14th US National Congress on Computational Mechanics held at Montreal, Canada from 17th-21st July 2017.

Plenary speaker Congresso Nacional de Matemática Applicada e Computacional (CNMAC2016) held at Gramado, Brazil from 5th-9th September 2016.

Plenary speaker at Mathematics of Finite Elements and Applications MAFELAP 2017 held in London, UK from 13th-17th June 2016.

Plenary speaker at European Conference on Numerical Mathematics. ENUMATH 2015 held in Ankara, Turkey from 14th-18th September 2015.

Plenary speaker at 2014 AARMS-CRM Workshop on Adaptive Methods for PDEs held in St. John’s, NL, Canada from 17-22nd August 2014.

Plenary speaker at International Workshop on Finite Element and Spectral Methods held at Shanghai Normal University, Shanghai, China from 16-18th May 2014.

Plenary speaker at Adaptive Multiscale Methods for the Atmosphere and Ocean held at Isaac Newton Institute, Cambridge UK from 22 - 24 August 2012.

Plenary speaker at International Conference on Computational Science held at Shanghai Normal University, Shanghai, China from 16-20 July 2012.

Plenary speaker at High-Order Numerical Approximation for Partial Differential Equations held at Hausdorff Centre for Mathematics, Bonn from 6-10 February 2012.

Selected Past Plenary Lectures

Plenary speaker at Theory of Highly Oscillatory Problems held at Isaac Newton Institute for Mathematical Sciences, Cambridge, 26th-30th March, 2007.

Plenary speaker at Seventeenth International Conference on Domain Decomposition Methods (DDM XVII) held in St. Wolfgang, Austria, 3rd-7th July 2006.

Plenary speaker at conference on Mathematics of Finite Elements and Applications (MAFELAP 2006) held at Brunel University, UK, 13th-16th June 2006.

Plenary speaker at International Conference on High Order and Spectral Methods (ICOSAHOM), held at Brown University, Providence USA from 21st–24th June 2004.

Plenary speaker at Mathematical and Numerical Aspects of Wave Propagation (WAVES 2003) held in Jyväskylä, Finland from 30th June-4th July 2003.

Plenary speaker at Multiscale Modeling, Multi-resolution and Adaptivity held at Newton Institute, Cambridge UK from 7th-11th April 2003.

Keynote speaker at Numerical and Applied Mathematics held at Mathematical Sciences Research Institute, Berkeley, 1st-28th April 2000.

Plenary speaker at Eleventh International Conference on Domain Decomposition Methods (DDM XI) held in Greenwich, 20th-24th July 1998.

Distinguished Scholarly Activity

London Mathematical Society Prize Committee 2014–2016

Director of Centre for Numerical Algorithms and Intelligent Software, 2011-2012. £5M multi-disciplinary centre formed by Universities of Strathclyde, Edinburgh, Herriot-Watt and Edinburgh Parallel Computing Centre. Ainsworth was one of four founding members of the centre.

Mathematical Sciences Research Institute (Berkeley): Programme on Numerical and Applied Mathematics (6th March-28th April 2000). Co-organizer along with Professors Babuška, Bank and Verfürth, of MSRI workshop on a posteriori error estimation and adaptive approaches in the finite element method supported by MSRI and NSF.

Isaac Newton Institute for Mathematical Sciences (Cambridge): Programme on Computational Challenges in Partial Differential Equations held in Cambridge, January-July 2003. Co-organizer with Professors C.M. Elliot and E. Süli.

EMS Prize Committee: Member of prize committee of Edinburgh Mathematical Society 2004-2012.

Leslie Fox Prize Committee: Member of prize committee for Leslie Fox Prize in Numerical Analysis. 2008-2013. Chair of adjudicating panel 2013.

Current Research Awards

2015 ARO MURI: Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics and Applications (Co-PI).  October 2015-October 2020.

2015 ASCR SIRIUS: Science Driven Data Management for Multi-tiered Storage (Co-PI).

2013 UT Batelle Joint Faculty Appointment ORNL (PI). 

Previous Graduate Students

David Alan Kay (PhD, 1997). The p- and hp- Finite Element Method Applied to a Class of Non-Linear Elliptic PDEs. Supported EPSRC Research Studentship. David now holds a lectureship at Oxford University.

William Carl Senior (PhD, 1998). Parallel Data Structures and Implementation of h-p Finite Element Algorithms.  Supported by EPSRC Research Studentship. Bill is now based in the Netherlands where he heads a computational reservoir simulation team at Shell.

Mark Edward Arnold (PhD, 1999). A Posteriori Error Estimation and Hierarchical Modelling. Supported by EPSRC Earmarked Studentship. Mark went on to model spread of foot and mouth disease for the Veterinary Laboratories Agency, UK.

Patrick William Coggins (PhD, 2000). Adaptive Finite Element Methods for Stokes and Oseen Equations. Supported by EPSRC Earmarked Studentship. Pat went on to develop software and climate models at the UK Meteorological Office.

David James Blacker (PhD, 2002). Robust Non-conforming Finite Element Methods for Nearly IncompressibleElasticity. Supported by EPSRC Studentship. David went on to model gas networks for Transco UK.

Richard Rankin (PhD 2008). Fully computable a posteriori error bounds for non-conforming and discontinuous Galerkin finite element approximations. Supported by EPSRC Studentship. Richard went on to work at Rice University.

Hafiz Abdul Wajid (PhD, 2009) Analysis and Application of High Order Reduced Integration Schemes for Wave Propagation. Supported by scholarship from COMSATS Institute of Technology, Pakistan. Hafiz is now an Associate Professor at COMSATS.

Alejandro Allendes (PhD, 2012) A posteriori error estimation and adaptivity for Stokes and Navier-Stokes equations. Supported by Faculty scholarship and Chilean Government. Alejandro now has a position at Universidad Tecnia Federico Santa Maria, Chile.

Gaelle Andriamaro (PhD, 2013) Bernstein-Bezier methods for finite element approximation. Supported by ORSAS and Faculty scholarship Gaelle is a researcher at Vector Fields, Cobham UK.

Omer Riaz (PhD, 2014) Algorithmic skeletons and auto-tuning of finite elements for high performance computing.

Hongrui Wang (PhD, 2016) Error and Stability Analysis for High Order B-Spline Finite Element Methods.

Christian Glusa (Current) Resilience of Multigrid Solvers at Exascale.

Ben Whitney (Current) Compression of Large-Scale Scientific Data: Algorithms and Analysis.


M. AINSWORTH AND C. GLUSA, Is the multigrid method fault tolerant? The multilevel case, SIAM J. Sci. Comp., (Submitted for publication).  

Is the multigrid method fault tolerant? the two-grid case, SIAM J. Sci. Comp., (Submitted for publication). 

M. AINSWORTH, S. KLASKY, AND B. WHITNEY, Compression using lossless decimation: Analysis and application, SIAM J. Sci. Comp., (Submitted for publication).

M. AINSWORTH AND Z. MAO, Analysis and approximation of a fractional cahn-hilliard equation, SIAM J. Numer. Anal., (Submitted for publication).

Publications in Refereed Journals