I. G. Graham

V. Dominguez, I. G. Graham, T. Kim

In this paper we propose and analyse composite Filon-Clenshaw-Curtis quadrature rules for integrals of the form $I_{k}^{[a,b]}(f,g) := \int_a^b f(x) \exp(\mathrm{i}kg(x)) \rd x $, where $k \geq 0$, $f$ may have integrable singularities and $g$ may have stationary points. Our composite rule is defined on a mesh with $M$ subintervals and requires $MN+1$ evaluations of $f$. It satisfies an error estimate of the form $C_N k^{-r} M^{-N-1 + r}$, where $r$ is determined by the strength of any singularity in $f$ and the order of any stationary points in $g$ and $C_N$ is a constant which is independent of $k$ and $M$, but depends on $N$. The regularity requirements on $f$ and $g$ are explicit in the error estimates. For fixed $k$, the rate of convergence of the rule as $M \rightarrow \infty$ is the same as would be obtained if $f$ was smooth. Moreover, the quadrature error decays at least as fast as $k \rightarrow \infty$ as does the original integral $I_{k}^{[a,b]}(f,g)$. For the case of nonlinear oscillators $g$, the algorithm requires the evaluation of $g^{-1}$ at non-stationary points. Numerical results demonstrate the sharpness of the theory. An application to the implementation of boundary integral methods for the high-frequency Helmholtz equation is given.

I. G. Graham, E. A. Spence, E. Vainikko

In this paper we present an overview of recent progress on the development and analysis of domain decomposition preconditioners for discretised Helmholtz problems, where the preconditioner is constructed from the corresponding problem with added absorption. Our preconditioners incorporate local subproblems that can have various boundary conditions, and include the possibility of a global coarse mesh. While the rigorous analysis describes preconditioners for the Helmholtz problem with added absorption, this theory also informs the development of efficient multilevel solvers for the "pure" Helmholtz problem without absorption. For this case, 2D experiments for problems containing up to about $50$ wavelengths are presented. The experiments show iteration counts of order about $\mathcal{O}(n^{0.2})$ and times (on a serial machine) of order about $\mathcal{O}(n^α)$, { with $α\in [1.3,1.4]$} for solving systems of dimension $n$. This holds both in the pollution-free case corresponding to meshes with grid size $\mathcal{O}(k^{-3/2})$ (as the wavenumber $k$ increases), and also for discretisations with a fixed number of grid points per wavelength, commonly used in applications. Parallelisation of the algorithms is also briefly discussed.

J. C. H. Blake, I. G. Graham, F. Scheben et al.

We consider the classical integral equation reformulation of the radiative transport equation (RTE) in a heterogeneous medium, assuming isotropic scattering. We prove an estimate for the norm of the integral operator in this formulation which is explicit in the (variable) coefficients of the problem (also known as the cross-sections). This result uses only elementary properties of the transport operator and some classical functional analysis. As a corollary, we obtain a bound on the convergence rate of source iteration (a classical stationary iterative method for solving the RTE). We also obtain an estimate for the solution of the RTE which is explicit in its dependence on the cross-sections. The latter can be used to estimate the solution in certain Bochner norms when the cross-sections are random fields. Finally we use our results to give an elementary proof that the generalised eigenvalue problem arising in nuclear reactor safety has only real and positive eigenvalues.

I. G. Graham, S. A. Sauter

We discuss the stability theory and numerical analysis of the Helmholtz equation with variable and possibly non-smooth or oscillatory coefficients. Using the unique continuation principle and the Fredholm alternative, we first give an existence-uniqueness result for this problem, which holds under rather general conditions on the coefficients and on the domain. Under additional assumptions, we derive estimates for the stability constant (i.e., the norm of the solution operator) in terms of the data (i.e. PDE coefficients and frequency), and we apply these estimates to obtain a new finite element error analysis for the Helmholtz equation which is valid at high frequency and with variable wave speed. The central role played by the stability constant in this theory leads us to investigate its behaviour with respect to coefficient variation in detail. We give, via a 1D analysis, an a priori bound with stability constant growing exponentially in the variance of the coefficients (wave speed and/or diffusion coefficient). Then, by means a family of analytic examples (supplemented by numerical experiments), we show that this estimate is sharp