High-Order Polytopic DGFEMs for Radiation Transport Problems

Paul Houston

University of Nottingham

In this talk we consider the development of high-order discontinuous Galerkin finite element methods (DGFEMs) for the discretization of the radiation transport problem on general (spatial) computational meshes consisting of polytopic elements. Our particular interest is the application to medical treatment planning in clinical radiotherapy. Here we study both the stability and a priori error analysis of the proposed scheme. The implementation is based on exploiting a nodal approximation in energy and angle, together with fast numerical integration techniques on the spatial polytopic mesh; this approach leads to a highly parallelisable algorithm whereby a large number of linear transport solves must be computed. Additionally, we consider the design of iterative solution techniques for the resulting linear system of equations; in particular, we consider both a Richardson type iteration, as well as a Richardson preconditioned GMRES solver for both monoenergetic and polyenergetic problems. For the Richardson scheme, we prove convergence, independent of the discretisation parameters h and p, while GMRES is shown (computationally) to depend only mildly on h and p. Numerical experiments are presented to highlight the accuracy and performance of the proposed solver, as well as to benchmark with more standard kinetic Monte Carlo simulations.

Biography

Education

1988 - 1991 University of Bristol. B.Sc. Mathematics.

1993 - 1996 University of Oxford. D.Phil. Mathematics.

Academic Employment

07/96-06/99 University of Oxford - Postdoc Position

07/99-03/04 University of Leicester - Lecturer in Applied Mathematics.

04/04-06/05 University of Leicester - Professor of Applied Mathematics.

07/05-Present University of Nottingham - Professor of Computational Applied Mathematics. Head of School since August 2018.

Academic Distinctions

1999 Leslie Fox Prize in Numerical Analysis (second place).

2002 EPSRC Advanced Research Fellow, 2002-2007.

2012 Leverhulme Trust Research Fellow, 2012-2013.

Current Editorial Duties

Associate Editor, International Journal of Numerical Analysis and Modeling.

Associate Editor, Numerical Mathematics: Theory, Methods and Applications.

Associate Editor, Advances in Computational Mathematics.

Associate Editor, Results in Applied Mathematics.