arXiv subjects

I. G. Graham

Publications and source records attributed to I. G. Graham.

5 recordsLinked to original sources

arXiv2020

Domain Decomposition with local impedance conditions for the Helmholtz equation with absorption

We consider one-level additive Schwarz preconditioners for a family of Helmholtz problems with absorption and increasing wavenumber $k$. These problems are discretized using the Galerkin method with nodal conforming finite elements of any (fixed) order on meshes with diameter $h = h(k)$, chosen to maintain accuracy as $k$ increases. The action of the preconditioner requires solution of independent (parallel) subproblems (with impedance boundary conditions) on overlapping subdomains of diameter $H$ and overlap $δ\leq H$. The solutions of these subproblems are linked together using prolongation/restriction operators defined using a partition of unity. In numerical experiments (with $δ\sim H$) for a model interior impedance problem, we observe robust (i.e. $k-$independent) GMRES convergence as $k$ increases. This provides a highly-parallel, $k-$robust one-level domain decomposition method. We provide supporting theory by studying the preconditioner applied to a range of absorptive problems, $k^2\mapsto k^2+ \mathrm{i} \varepsilon$, with absorption parameter $\varepsilon$. Working in the Helmholtz ``energy'' inner product, and using the underlying theory of Helmholtz boundary-value problems, we prove a $k-$independent upper bound on the norm of the preconditioned matrix, valid for all $\vert \varepsilon\vert \lesssim k^2$. We also prove a strictly-positive lower bound on the distance of the field of values of the preconditioned matrix from the origin which holds when $\varepsilon/k$ is constant or growing arbitrarily slowly with $k$. These results imply robustness of the preconditioner for the corresponding absorptive problem as k increases and give theoretical support for the observed robustness of the preconditioner for the pure Helmholtz problem.

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arXiv2019

The radiative transport equation with heterogeneous cross-sections

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.

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arXiv2018

Stability and error analysis for the Helmholtz equation with variable coefficients

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

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arXiv2016

Recent Results on Domain Decomposition Preconditioning for the High-frequency Helmholtz Equation using Absorption

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 $\mathcal{O}(n^{\alpha})$, { with $\alpha \in [1.3,1.4]$} 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.

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arXiv2012

Filon-Clenshaw-Curtis rules for highly-oscillatory integrals with algebraic singularities and stationary points

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.

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