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We discuss the stability theory and numerical analysis of the Helmholtz equation with variable and possibly nonsmooth 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 a 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 the stability constant growing exponentially in the variance of the coefficients (wave speed and/or diffusion coefficient). Then, by means of a family of analytic examples (supplemented by numerical experiments), we show that this estimate is sharp. [ABSTRACT FROM AUTHOR]

This paper rigorously analyses preconditioners for the timeharmonic Maxwell equations with absorption, where the PDE is discretised using curl-conforming finite-element methods of fixed, arbitrary order and the preconditioner is constructed using additive Schwarz domain decomposition methods. The theory developed here shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters and overlap are chosen appropriately, then the classical two-level overlapping additive Schwarz preconditioner (with PEC boundary conditions on the subdomains) performs optimally--in the sense that GMRES converges in a wavenumber-independent number of iterations--for the problem with absorption. An important feature of the theory is that it allows the coarse space to be built from low-order elements even if the PDE is discretised using high-order elements. It also shows that additive methods with minimal overlap can be robust. Numerical experiments are given that illustrate the theory and its dependence on various parameters. These experiments motivate some extensions of the preconditioners which have better robustness for problems with less absorption, including the propagative case. At the end of the paper we illustrate the performance of these on two substantial applications; the first (a problem with absorption arising from medical imaging) shows the empirical robustness of the preconditioner against heterogeneity, and the second (scattering by a COBRA cavity) shows good scalability of the preconditioner with up to 3,000 processors. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze composite Filon-Clenshaw-Curtis quadrature rules for integrals of the form Ik[a,b](f, g): = ∫ab f(x) exp (ikg(x))dx, where k ≥ 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 CNk-rM-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 CN 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 → ∞ is the same as would be obtained if f was smooth. Moreover, the quadrature error decays at least as fast as k → ∞ as does the original integral Ik[a,b] (f, g). For the case of nonlinear oscillators g, the algorithm requires the evaluation of g-1 at nonstationary 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. [ABSTRACT FROM AUTHOR]

In this paper we obtain new results on Filon-type methods for computing oscillatory integrals of the form ∫−11f(s)exp(iks)ds. We use a Filon approach based on interpolating f at the classical Clenshaw–Curtis points cos(jπ/N),j=0,…,N. The rule may be implemented in O(NlogN) operations. We prove error estimates that show explicitly how the error depends both on the parameters k and N and on the Sobolev regularityof f. In particular we identify the regularity of f required to ensure the maximum rate of decay of the error as k → ∞. We also describe a method for implementing the method and prove its stability both when N ≤ k and N > k. Numerical experiments illustrate both the stability of the algorithm and the sharpness of the error estimates. [ABSTRACT FROM PUBLISHER]

Vasculitis of the ovary is a rare condition that can occur as an isolated finding or in association with systemic vasculitis. We describe a case of a 36-year-old female with vasculitis involving the left ovary on a background of severe active systemic lupus erythematosus (SLE). Despite a florid histopathological picture of ovarian vasculitis, the clinical and imaging findings were nonspecific. We have compared the current case to the literature on ovarian vasculitis, including relating to SLE. Ovarian vasculitis in SLE may be an underestimated entity as it may not be looked for routinely in the context of vasculitic involvement of other organs. [ABSTRACT FROM AUTHOR]

In this paper we consider the numerical solution of discrete boundary integral equations on polyhedral surfaces in three dimensions. When the solution contains typical edge singularities, highly stretched meshes are preferred to uniform meshes, since they reduce the number of degrees of freedom needed to obtain a fixed accuracy. The classical panel-clustering method can still be applied in the presence of such highly stretched meshes. However, we will show that the savings in computation time and storage become suboptimal because the nearfield matrix arising in the panel-clustering algorithm is no longer as sparse as it is in the case of uniform meshes. Hence, a natural question arises as to whether a new enhanced panel-clustering algorithm can be designed which performs efficiently even in the presence of highly stretched meshes. The main result of this paper is to formulate such an enhanced version of the panel-clustering algorithm. The key features of the algorithm are (i) the employment of partial analytic integration in the direction of stretching, yielding a new kernel function on a one-dimensional manifold where the influence of high aspect ratios in the stretched elements is removed, and (ii) the introduction of a generalized admissibility condition with respect to the partially integrated kernel, which ensures that certain stretched clusters which are inadmissible in the classical sense now become admissible. In the context of a model problem, we prove that our algorithm yields an accurate (up to the discretization error) matrix-vector multiplication which requires ...(NlogκN) operations, where N is the number of degrees of freedom and k is small and independent of the aspect ratio. The generalized admissibility condition can be viewed as an addition to the classical method which may be useful in general when stretched meshes are present. We also have performed a numerical experiment which shows that the sparsity of the nearfield matrix for the enhanced panel-clustering method is not negatively affected by stretched elements, and the method will perform optimally. [ABSTRACT FROM AUTHOR]

When a high-frequency acoustic or electromagnetic wave is scattered by a surface with a conical point, the component of the asymptotics of the scattered wave corresponding to diffraction by the conical point can be represented as an asymptotic expansion, valid as the wave number $k \rightarrow \infty$. The diffraction coefficient is the coefficient of the principal term in this expansion and is of fundamental interest in high-frequency scattering. It can be computed by solving a family of homogeneous boundary value problems for the Laplace--Beltrami--Helmholtz equation (parametrized by a complex wave number--like parameter $\nu$) on a portion of the unit sphere bounded by a simple closed contour $\ell$, and then integrating the resulting solutions with respect to $\nu$. In this paper we give the numerical analysis of a method for carrying out this computation (in the case of acoustic waves) via the boundary integral method applied on $\ell$, emphasizing the practically important case when the conical scatterer has lateral edges. The theory depends on an analysis of the integral equation on $\ell$, which shows its relation to the corresponding integral equation for the planar Helmholtz equation. This allows us to prove optimal convergence for piecewise polynomial collocation methods of arbitrary order. We also discuss efficient quadrature techniques for assembling the boundary element matrices. We illustrate the theory with computations on the classical canonical open problem of a trihedral cone. [ABSTRACT FROM AUTHOR]

In this paper we obtain a range of inverse-type inequalities which are applicable to finite-element functions on general classes of meshes, including degenerate meshes obtained by anisotropic refinement. These are obtained for Sobolev norms of positive, zero and negative order. In contrast to classical inverse estimates, negative powers of the minimum mesh diameter are avoided. We give two applications of these estimates in the context of boundary elements: (i) to the analysis of quadrature error in discrete Galerkin methods and (ii) to the analysis of the panel clustering algorithm. Our results show that degeneracy in the meshes yields no degradation in the approximation properties of these methods. [ABSTRACT FROM PUBLISHER]

This paper concerns the iterative solution of symmetric elliptic problems with piece- wise constant coefficients in two or three space dimensions discretized by linear finite element methods on unstructured triangular or tetrahedral meshes. The effect of the discontinuous coefficients is studied by first postulating that there are d fixed regions of the domain where the coefficient takes constant positive values a = (a1,…,ad) and then considering certain (positive) coefficient sequences {a(m)} in which some of these values approach 0 or ∞ as m → ∞. We consider the performance of additive Schwarz domain decomposition preconditioners constructed from local solves on automatically generated subdomains together with a global solve on some coarser grid. Assuming no relationship between the regions on which the coefficient function is constant and either the subdo- mains or the coarse grid, we show that the preconditioned conjugate gradient method converges (in both the energy and the Euclidean norms) with a number of iterations which grows only logarithmically in the size of the maximum jump J(m) := max{ak(m)/al(m): k,l = 1,…,d}, as m → ∞ . The result is obtained by a careful analysis of the preconditioned matrix. It cannot be obtained by the usual procedure of estimating condition numbers: a simple example is given in which the condi- tion number of both the original and the preconditioned stiffness matrices degrade linearly in J(m). Recent results of Chan, Smith, and Zou have shown that by using this preconditioner together with the conjugate gradient method, the number of iterations can be bounded independently of the mesh diameter provided the subdomains have overlap commensurate with the size of the coarse mesh. Our results now show that this method is also highly resilient to discontinuous coefficients, even if no attention is paid to the coefficient discontinuity in the construction of the solver. [ABSTRACT FROM AUTHOR]

We examine the piecewise-constant collocation method, with collocation points the mid-points of subintervals, for first-kind integral equations with logarithmic kernels on polygonal boundaries. Previously this method had been shown to converge subject to certain restrictions on the angles at the corners of the polygon. Here, by considering a slightly modified collocation method, we are able to remove any restrictions on these angles, and to generalise slightly the meshes which may be used. Moreover, the modification leads to new results on the convergence of preconditioned two-(or multi-) grid methods for solving the resultant linear systems. [ABSTRACT FROM PUBLISHER]

We wish to correct a minor error in the recent paper [2]. That paper was concerned with an integral equation defined on a closed polygon Γ with r corners at the points x0, x2, …, x2r = x0. We parameterized Γ using a mapping γ:[−π,π] → Γ defined as follows. For each l, introduce the mid-point x2l−1 of the side joining x2l—2 to x2l. Then introduce 2r + 1 points in parameter spacewith the property that for each j = 1, …, 2rwhere mj are integers and . Then γ(s) is defined byfor j = 1, …, 2r. The {Sj} are then the preimages of the {xj} under γ. Moreover, in view of (1), a family of uniform meshes can be constructed on [−π, π] which include {Sj} as the break-points. Then γ maps these to meshes which are uniform on each segment joining xj−1 to xj (which we denote Γj). These meshes are used to discretize the integral equation. [ABSTRACT FROM PUBLISHER]

In this paper we examine spatio-temporal pattern formation in reaction-diffusion systems on the surface of the unit sphere in 3D. We first generalise the usual linear stability analysis for a two-chemical system to this geometrical context. Noting the limitations of this approach (in terms of rigorous prediction of spatially heterogeneous steady-states) leads us to develop, as an alternative, a novel numerical method which can be applied to systems of any dimension with any reaction kinetics. This numerical method is based on the method of lines with spherical harmonics and uses fast Fourier transforms to expedite the computation of the reaction kinetics. Numerical experiments show that this method efficiently computes the evolution of spatial patterns and yields numerical results which coincide with those predicted by linear stability analysis when the latter is known. Using these tools, we then investigate the rĵle that pre-pattern (Turing) theory may play in the growth and development of solid tumours. The theoretical steady-state distributions of two chemicals (one a growth activating factor, the other a growth inhibitory factor) are compared with the experimentally and clinically observed spatial heterogeneity of cancer cells in small, solid spherical tumours such as multicell spheroids and carcinomas. Moreover, we suggest a number of chemicals which are known to be produced by tumour cells (autocrine growth factors), and are also known to interact with one another, as possible growth promoting and growth inhibiting factors respectively. In order to connect more concretely the numerical method to this application, we compute spatially heterogeneous patterns on the surface of a growing spherical tumour, modelled as a moving-boundary problem. The numerical results strongly support the theoretical expectations in this case. Finally in an appendix we give a brief analysis of the numerical method.