My research is in Numerical Analysis, mainly on PDEs and applications. Here are some current, recent and not so recent journal articles. A more complete list can be found here

Efficient Solvers for discretizations of frequency-domain wave problems :

• J. Galkowski, S. Gong, I.G. Graham, D. Lafontaine, E.A. Spence Convergence of overlapping domain decomposition methods with PML transmission conditions applied to nontrapping Helmholtz problems, 2 April 2024 preprint
• S. Gong, I. G. Graham and E.A. Spence, Convergence of Restricted Additive Schwarz with impedance transmission conditions for discretised Helmholtz problems, Math. Comput. 92, 175–215 (2023). final preprint
• N. Bootland, V. Dolean, I. G. Graham, C. Ma and R. Scheichl, Overlapping Schwarz methods with GenEO coarse spaces for indefinite and non-self-adjoint problems, IMA J.Numer. Anal. 43, 1899–1936 (2023). final preprint
• S. Gong, M.J. Gander, I.G. Graham, D. Lafontaine and E.A. Spence, Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation, Numer. Math. 152, 259–306 (2022). Published paper    Final Preprint.
• S. Gong, I.G. Graham and E.A. Spence, Domain decomposition preconditioners for high-order discretisations of the heterogeneous Helmholtz equation, IMA J. Numer. Anal. 41, 2139–2185 (2021). final preprint    Final published version
• I.G. Graham, E.A. Spence and J. Zou, Domain Decomposition with local impedance conditions for the Helmholtz equation, SIAM J. Numer. Anal. 58, pp 2515–2543 (2020) final preprint    Final published version
  
• M. Bonazzoli, V. Dolean, I.G. Graham, E. A. Spence, P.-H. Tournier, Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption, Math. Comp. 88 (2019), 2559-2604 Final preprint    Final published version
  
• I.G. Graham, E.A. Spence and E. Vainikko, Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption. Math. Comp. 86 (2017), 2089-2127    Final preprint    Final published version
  
• M.J. Gander, I.G. Graham, E.A. Spence, How should one choose the shift for the shifted Laplacian to be a good preconditioner for the Helmholtz equation?, Numerische Mathematik, 2015. DoI: 10.1007/s00211-015-0700-2, preprint

The Helmholtz equation at high frequency :

• I.G. Graham and S.A. Sauter, Stability and error analysis for the Helmholtz equation with variable coefficients. Math. Comp. 89 (2020), 105-138 Final preprint    Final published version
  
• I.G. Graham, O.R. Pembery and E.A. Spence The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances, Journal of Differential Equations 266 (2019) 2869-2923. Final preprint    Final published version
  

Analysis of methods for solving PDEs with random coefficients:

• I.G. Graham, O.R. Pembery, E.A. Spence, Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification, Advances in Computational Mathematics(2021) 47:68 final preprint    published paper
  
• I.G. Graham, M.J. Parkinson, R. Scheichl, Error Analysis and Uncertainty Quantification for the Heterogeneous Transport Equation in Slab Geometry, IMA Journal of Numerical Analysis 2020. final preprint    Final published version
  
• I. G. Graham, F. Y. Kuo, D. Nuyens, R. Scheichl, and I. H. Sloan, Circulant embedding with QMC -- analysis for elliptic PDE with lognormal coefficients, Numer. Math. (2018). https://doi.org/10.1007/s00211-018-0968-0 final preprint    Final published version
  
• I.G. Graham, R. Scheichl and E. Ullmann, Mixed Finite Element Analysis of Lognormal Diffusion and Multilevel Monte Carlo Methods, Stochastic Partial Differential Equations, Analysis and Computation, 4, 41–75 (2016)    Final preprint    Final published version
• I.G. Graham, F.Y. Kuo, J.A. Nicholls, R. Scheichl, Ch. Schwab and I.H. Sloan, Quasi-Monte Carlo Finite Element Methods for Elliptic PDEs with Log-normal Random Coefficients, Numer. Math. 131 (2), 329-368 (2015).    Final preprint    Final published version
• I. G. Graham, F. Y. Kuo, D. Nuyens, R. Scheichl, and I. H. Sloan, Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications, J. Comp. Phys. 230 (10), 3668-3694 (2011) Final preprint
  Final published version

Algorithms for sampling random fields:

• M. Bachmayr, I.G. Graham, V. K. Nguyen and R. Scheichl, Unified Analysis of Periodization-Based Sampling Methods for Matérn Covariances, SIAM J. Numer. Anal. 58(5), 29532980 (2020) final preprint
  
• I. G. Graham, F. Y. Kuo, D. Nuyens, R. Scheichl, and I. H. Sloan, Analysis of circulant embedding methods for sampling stationary random fields, SIAM J. Numer. Anal. 56(3), 1871–1895 (2018). Final preprint    Final published version

• I.G. Graham, M. Loehndorf, J.M. Melenk and E.A. Spence, When is the error in the h-BEM for solving the Helmholtz equation bounded independently of k? BIT Num. Math., vol. 55, no. 1, 171-214 (2015) final preprint
• S.N. Chandler-Wilde, I.G. Graham, S. Langdon, E.A. Spence, Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering, Acta Numerica, vol. 21, 89--305 (2012) local (official) copy
• E.A. Spence, S. N. Chandler-Wilde, I. G. Graham, V. P. Smyshlyaev, A new frequency-uniform coercive boundary integral equation for acoustic scattering, Communications on Pure and Applied Mathematics 64(10) (2011) 1384-1415. final preprint
• T. Betcke, S.N. Chandler-Wilde, I.G. Graham, S. Langdon, M. Lindner, Condition number estimates for combined potential operators in acoustics and their boundary element discretisation, Numerical Methods for PDEs 27 (2011), 31-69      final preprint

Numerical Methods for highly oscillatory integrals :

• Z. Wu, I. G. Graham, D. Ma, Z. Zhang A Filon-Clenshaw-Curtis-Smolyak rule for multi-dimensional oscillatory integrals with application to a UQ problem for the Helmholtz equation, Math. Comp., to appear preprint
• V. Dominguez, I. G. Graham and T. Kim, Filon-Clenshaw-Curtis rules for highly-oscillatory integrals with algebraic singularities and stationary points. http://arxiv.org/abs/1207.2283 SIAM J. Numerical Analysis 51(3): 1542-1566 (2013)
• V. Dominguez, I.G. Graham and V.P. Smyshlyaev, Stability and error estimates for Filon-Clenshaw-Curtis rules for highly-oscillatory integrals, final version, appeared in IMA J. Numer. Anal. 2011

Multiscale Methods for PDEs and fast solvers for PDEs modelling flow in heterogeneous media :

• I.G. Graham, T.Y. Hou, O. Lakkis and R. Scheichl (Editors) Numerical Analysis of Multiscale Problems , Springer Lecture Notes in Computational Science and Engineering 83, 2011.
• C.-C. Chu, I.G.Graham and T.-Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp. 79 (2010) 1915-1955. Final preprint
• I.G. Graham. P.O. Lechner and R. Scheichl, Domain Decomposition for Multiscale PDEs, Numerische Mathematik 106 (2007), 589-626 .    final preprint

Numerical analysis of PDE eigenvalue problems:

• F. Scheben and I. G. Graham, Iterative methods for neutron transport eigenvalue problems, SIAM Journal on Scientific Computing, 33 (2011), 2785-2804. Final Preprint     
• S. Giani and I. G. Graham, Adaptive finite element methods for computing band gaps in photonic crystals. Numerische Mathematik 121(1), 31-64, 2012. DOI 10.1007/s00211-011-0425-9 Final preprint
• S. Giani and I.G. Graham, A convergent adaptive method for elliptic eigenvalue problems      SIAM J Numer Anal, 47 (2009), 1067-1091. DOI: 10.1137/070697264      Final preprint

Older papers on Nystroem and collocation methods for boundary integral equations on corner domains:

• K.E. Atkinson and I.G. Graham, Iterative solution of linear systems arising from the boundary integral method. SIAM J. Sci. Statist. Comput. 13 (1992), no. 3, 694–722.
• I.G. Graham and G.A. Chandler, High-order methods for linear functionals of solutions of second kind integral equations. SIAM J. Numer. Anal. 25 (1988), no. 5, 1118–1137. 65R20
• G.A. Chandler and I.G. Graham, Product integration-collocation methods for noncompact integral operator equations. Math. Comp. 50 (1988), no. 181, 125–138.
• G.A. Chandler and I.G. Graham, The convergence of Nyström methods for Wiener-Hopf equations. Numer. Math. 52 (1988), no. 3, 345–364.