Ivan Graham : Research Interests 
My research is in Numerical Analysis, mainly on PDEs and
applications.
Here are some current and recent projects, with some papers. A more
complete list can be found here
Efficient Solvers for discretizations of
frequencydomain wave problems :
 S. Gong, I.G. Graham and E.A. Spence,
Domain decomposition preconditioners for highorder discretisations of the heterogeneous Helmholtz equation, IMA Journal of Numerical Analysis 2020.
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
highfrequency
timeharmonic Maxwell equations with absorption, Math. Comp. 88 (2019), 25592604
Final preprint
Final published version

I.G. Graham, E.A. Spence and E. Vainikko,
Domain decomposition preconditioning for highfrequency Helmholtz
problems with absorption. Math. Comp. 86 (2017), 20892127
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/s0021101507002,
preprint
The Helmholtz equation at high frequency :

I.G. Graham, O.R. Pembery and E.A. Spence The Helmholtz equation in heterogeneous media: a priori bounds, wellposedness, and resonances, Journal of Differential Equations 266 (2019) 28692923.
Final preprint
Final published version

I.G. Graham and S.A. Sauter, Stability and error analysis for the
Helmholtz equation with variable coefficients. Math. Comp. 89 (2020), 105138
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,
submitted 27th May 2020
preprint
 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/s0021101809680
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, QuasiMonte Carlo Finite Element Methods for Elliptic PDEs
with Lognormal Random Coefficients, Numer. Math. 131 (2), 329368
(2015).
Final preprint
Final published version

I. G. Graham, F. Y. Kuo, D. Nuyens, R. Scheichl, and I. H. Sloan,
QuasiMonte Carlo methods for elliptic PDEs
with random coefficients and applications, J. Comp. Phys. 230 (10), 36683694 (2011)
Final preprint
Final published version
Algorithms for sampling random fields:

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
 M. Bachmayr, I.G. Graham, V. K. Nguyen and R. Scheichl,
Unified Analysis of PeriodizationBased Sampling Methods for Matérn
Covariances, submitted 21st June 2019 to appear in SIAM J. Numer. Anal.
final preprint
Hybrid NumericalAsymptotic Methods in
HighFrequency Scattering:

I.G. Graham, M. Loehndorf, J.M. Melenk and E.A. Spence,
When is the error in the hBEM for solving the Helmholtz equation
bounded independently of k? BIT Num. Math., vol. 55, no. 1, 171214
(2015)
preprint

S.N. ChandlerWilde, I.G. Graham, S. Langdon, E.A. Spence,
Numericalasymptotic boundary integral methods in highfrequency
acoustic scattering, Acta Numerica, vol. 21, 89305 (2012)
local
(official) copy

E.A. Spence, S. N. ChandlerWilde, I. G. Graham,
V. P. Smyshlyaev,
A new frequencyuniform coercive boundary integral equation for
acoustic scattering, Communications on Pure and Applied
Mathematics 64(10) (2011) 13841415.
preprint

T. Betcke, S.N. ChandlerWilde, 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), 3169
preprint
Numerical Methods for highly oscillatory
integrals :

V. Dominguez, I. G. Graham and T. Kim,
FilonClenshawCurtis rules for highlyoscillatory integrals with
algebraic singularities and stationary points.
http://arxiv.org/abs/1207.2283
SIAM J. Numerical Analysis 51(3): 15421566 (2013)

V. Dominguez, I.G. Graham and V.P. Smyshlyaev,
Stability and error estimates for FilonClenshawCurtis rules for
highlyoscillatory 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 highcontrast elliptic interface problems, Math. Comp. 79 (2010)
19151955.
Final Version

I.G. Graham. P.O. Lechner and R. Scheichl,
Domain Decomposition for Multiscale PDEs,
Numerische Mathematik 106 (2007), 589626 .
Details
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),
27852804.
Preprint, February 18th 2011

S. Giani and I. G. Graham,
Adaptive finite element methods for computing band gaps in photonic
crystals. Numerische Mathematik 121(1), 3164, 2012. DOI 10.1007/s0021101104259
Revised version, dated 21st January 2011.

S. Giani and I.G. Graham, A convergent adaptive method for elliptic eigenvalue
problems SIAM J Numer Anal, 47 (2009), 10671091.
DOI: 10.1137/070697264
Details