William McLean

Associate Professor

Office:

Room 2085, Red Centre

Phone:

+61 2 9385 7045

Fax:

+61 2 9385 7123

Brief History

Research Interests

• Partial Differential Equations
• Numerical Analysis
• Integral Equations

Teaching for 2016

• Session 1:
  • Math5295 (weeks 1-6)
  • Math2111 (weeks 7-12)
• Session 2
  • Math2221 Higher theory and applications of differential equations (lecture slides)

Administration

• Associate Editor, SIAM Journal on Numerical Analysis
• Associate Editor, Journal of Integral Equations and Applications
• Manager of the Student Record System for the School of Maths and Stats.

Selected Publications

• Shev MacNamara, Bruce Henry and William McLean, Fractional Euler limits and their applications SIAM J. Appl. Math. 77: 447-469, 2017.
• Kim Ngan Le, William McLean and Bishnu Lamichhane Finite element approximation of a time-fractional diffusion problem in a non-convex polygonal domain arXiv:1602.00040, 2016; SupplementaryMaterials.tgz (146KiB).
• Kim Ngan Le, William McLean and Kassem Mustapha, Numerical solution of the time-fractional Fokker-Planck equation with general forcing SIAM J. Numer. Anal. 54: 1763-1784, 2016; Supplementary_Materials.tgz (177KiB).
• William McLean and Kassem Mustapha, Time-stepping error bounds for fractional diffusion problems with non-smooth initial data, J. Comp. Phys. 293: 201-217, 2015.
• Quoc Thong Le Gia and William McLean, Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions, Adv. Comput. Math. 40: 353-375, 2014.
• Kai Wang, William McLean and Henner Kempwerth, Transient photoluminescence from silicon wafers: Finite element analysis, J. Appl. Phys. 114, 163105, 2013.
• Kassem Mustapha and William McLean, Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations, SIAM J. Numer. Anal. 51: 516-525, 2013.
• William McLean, Fast summation by interval clustering for an evolution equation with memory, SIAM J. Sci. Comput. 34: A3039--A3056, 2012.
• Kassem Mustapha and William McLean, Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation, IMA J. Numer. Anal. 32: 906-925, 2012.
• William McLean and Vidar Thomee, Iterative methods for shifted positive definite linear systems and time discretization of the heat equation, ANZIAM J. 53: 134--155, 2011.
• Quoc Thong Le Gia and William McLean, Numerical solution of a parabolic equation on the sphere using Laplace transforms and radial basis functions, ANZIAM J. 52: C89-C102, 2011.
• William McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J. 52: 123-138, 2010.
• William McLean and Vidar Thomee, Numerical solution via Laplace transforms of a fractional order evolution equation, J. Integral Equations Appl 22: 57-94, 2010.
• William McLean and Kassem Mustapha, Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation, Numer. Algorithms 52:69-88, 2009.
• Kassem Mustapha and William McLean, Discontinuous Galerkin method for an evolution equation with a memory term of positive type Math. Comp. 78: 1975-1995, 2009.
• William McLean and Kassem Mustapha, A second-order accurate numerical method for a fractional wave equation, Numer. Math. 105:481-510, 2007.
• Ivan Graham and William McLean, Anisotropic mesh refinement: the conditioning of Galerkin boundary element matrices and simple preconditioners, SIAM J. Numer. Anal. 44: 1487-1513, 2006.
• William McLean, Ian H. Sloan and Vidar Thomee, Time discretization via Laplace transformation of an integro-differential equation of parabolic type, Numer. Math. 102: 497-522, 2006.
• Youngmok Jeon and William McLean, A new boundary element method for the biharmonic equation with Dirichlet boundary conditions, Adv. Comput. Math. 102: 497-522, 2003.
• Mark Ainsworth and William McLean, Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes, Numer. Math. 93:387-413, 2003.
• William McLean, Strongly Elliptic Systems and Boundary Integral Equations Cambridge University Press, 2000. (errata file)
• W. McLean and O. Steinback, Boundary element preconditioners for a hypersingular integral equation on an interval", Adv. Comput. Math. 11: 271-286, 1999.
• Mark Ainsworth, William McLean and Thanh Tran, The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling, SIAM J. Numer. Anal. 36: 1901-1932, 1999.
• W. McLean and S. Proessdorf, Boundary element collocation methods using splines with multiple knots, Numer. Math. 74: 419-451, 1996.
• W. McLean and I. H. Sloan, A fully discrete and symmetric boundary element method, IMA J. Numer. Anal. 14: 311-345, 1994.
• William McLean, Numerical evaluation of some trigonometric series, Math. Comp. 63: 271-275, 1994.
• W. McLean, S. Proessdorf and W. L. Wendland, A fully-discrete trigonometric collocation method, J. Integral Equations Appl. 5: 103-129, 1993.
• W. McLean and V. Thomee, Numerical solution of an evolution equation with a positive-type memory term, J. Austral. Math. Soc Ser. B 35: 23-70, 1993.
• Martin Costabel and William McLean, Spline collocation for strongly elliptic equations on the torus, Numer. Math. 62: 511-538, 1992.
• William McLean, Local and global descriptions of periodic pseudodifferential equations, Math. Nachr. 150: 151-161, 1991.
• W. McLean, Asymptotic error expansions for numerical solutions of integral equations, IMA J. Nuerm. Anal. 9: 373-384, 1989.
• William McLean, Solutions in Hölder spaces of singular integral equations on Lipschitz contours", Technical Report No.~266, Department of Mathematics, University of Tasmania, 1988.
• W. McLean and D. Elliott, On the p-norm of the truncated Hilbert trasform, Bull. Austral. Math. Soc. 38: 413-420, 1988.
• W. McLean, A spectral Galerkin method for a boundary integral equation, Math. Comp. 47: 597-607, 1986.

Seminar Presentations and other Talks

• Semidiscrete finite element approximation of a fractional Fokker-Planck equation, Workshop on Numerical Methods for Fractional-derivative Problems: Singularities and Fast Algorithms, Beijing Computational Science Research Center, 19-20 May, 2017.
• Subdiffusion in a nonconvex polygon, Monash Workshop on Numerical PDEs, February 2016, and MAFELAP, June 2016, Brunel University.
• Numerical solution of fractional PDEs, (1.8MB) 10 lectures at the Beijing Computational Mathematics Research Center, November 2015.
• A simple finite element code written in Julia, Australia and New Zealand Mathematics Convention 2014, University of Melbourne, 8-12 December, 2014.
• Error bounds for time stepping fractional diffusion problems with non-smooth initial data, Department of Mathematics and Statistics, University of Otago, Dunedin, 6 May, 2014.
• Julia: a programming language for scientific computing, Computational Maths Seminar, UNSW, 19 November, 2013. (Code examples)
• Discontinuous Galerkin time-stepping and fast summation for fractional diffusion and wave equations, International Symposium on Fractional PDEs, Newport, 3-5 June, 2013.
• Fast summation for an evolution equation with memory, SCPDE11, Hong Kong, 5-9 December, 2011.
• Iterative methods for positive definite linear systems with a complex shift, Workshop on Laplace transform methods and their applications, NIMS, Daejeon, Korea, 1-5 November, 2011.
• Discontinuous Galerkin methods for fractional diffusion problems, Max Planck Institute fur Mathematik in den Naturwissenschaften, Leipzig, 07/10/2010.
• Numerical Solution of a Fractional Diffusion Equation via Laplace Transformation (PRIMA Congress, Sydney, 10/07/2009).
• Numerical Solution of Fractional Diffusion Equations (Conference to Celebrate Bob Anderssen's 70th Birthday, ANU, 23/01/2009).
• The Conjugate Gradient Algorithm (16/09/2008) and its convergence theory (23/09/2008), expository talks for the Computational Maths Seminar.
• Discontinuous Galerkin method for a nonlocal evolution equation ( Computation Techniques and Applications 2008, Canberra, 13-16 July, 2008).
• A second-order accurate scheme for a fractional wave equations (NA07, 22nd Biennial Conference on Numerical Analysis, University of Dundee, Scotland, 26-29 June, 2007).
• Python for scientific computing with example programs in pyscicomp_progs.tgz. (Talk for masters students, Dept. of Mathematical Sciences, University of Bath, 21 March, 2007.)
• Numerical solution of a fractional diffusion equation (Numerical analysis seminar, Dept. of Mathematical Sciences, University of Bath, 2 February, 2007.)
• Numerical methods for some fractional-order partial differential equations (International Symposium on Fractional Calculus, Dunedin, 9-13 January, 2006)
• Conditioning of Boundary Element Equations (Symposium in honour of Mike Osborne, ANU, Canberra, 21-23 September, 2005)
• Lectures on Boundary Integral Equations (Boundary Integral Methods and Applications, Reading, 14-18 September, 2004)
• Quadrature for Boundary Element Methods.

Software

• fosc.m is a Matlab GUI illustrating forced oscillations.
• sake-0.3.tgz is a Fortran 95 library that generates special-purpose, 4D quadrature rules used in boundary element computations for integral equations on surfaces.
• gaussquad-2.5.tgz provides a Fortran 95 module that generates all of the classical Gauss quadrature rules, including the Radau and Lobatto variants. The package includes test programs and bindings for C and Python. (The latter require a fortran compiler, such as g95, that supports a few of the interoperability features in the f2003 standard.)
• libnpy-0.5.tgz is a small library for writing a C or Fortran array to a data file, using the NumPy binary format. A Python script can easily load such a .npy file to create a NumPy array.
• blocktstsolver-0.1.tgz provides an FFT based solver for block tridiagonal symmetric Toeplitz matrices.