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An integral equation-based numerical method for scattering from multi-dielectric cylinders is presented. Electromagnetic fields are represented via layer potentials in terms of surface densities with physical interpretations. The existence of null-field representations then adds superior flexibility to the modeling. Local representations are used for fast field evaluation at points away from their sources. Partially global representations, constructed as to reduce the strength of kernel singularities, are used for near-evaluations. A mix of local- and partially global representations is also used to derive the system of integral equations from which the physical densities are solved. Unique solvability is proven for the special case of scattering from a homogeneous cylinder under rather general conditions. High achievable accuracy is demonstrated for several examples found in the literature. • Fast and accurate direct solver applied at triple junctions. • Mix of local and global integral representations for efficient integral equation modeling. • Unique solutions guaranteed in special situations, also for negative permittivity ratios. • Reduced strength of kernel singularities gives accurate field evaluations close to boundaries. [ABSTRACT FROM AUTHOR]

A particular mix of integral equations and discretization techniques is suggested for the solution of a planar Helmholtz transmission problem with relevance to the study of surface plasmon waves. The transmission problem describes the scattering of a time-harmonic transverse magnetic wave from an infinite dielectric cylinder with complex permittivity and sharp edges. Numerical examples illustrate that the resulting scheme is capable of obtaining total magnetic and electric fields to very high accuracy in the entire computational domain. [ABSTRACT FROM AUTHOR]

The magnetic field integral equation for axially symmetric cavities with perfectly conducting piecewise smooth surfaces is discretized according to a high-order convergent Fourier–Nyström scheme. The resulting solver is used to accurately determine eigenwavenumbers and normalized electric eigenfields in the entire computational domain. [ABSTRACT FROM AUTHOR]

Abstract: A high-order accurate, explicit kernel-split, panel-based, Fourier–Nyström discretization scheme is developed for integral equations associated with the Helmholtz equation in axially symmetric domains. Extensive incorporation of analytic information about singular integral kernels and on-the-fly computation of nearly singular quadrature rules allow for very high achievable accuracy, also in the evaluation of fields close to the boundary of the computational domain. [Copyright &y& Elsevier]

Abstract: An efficient integral equation based solver is constructed for the electrostatic problem on domains with cuboidal inclusions. It can be used to compute the polarizability of a dielectric cube in a dielectric background medium at virtually every permittivity ratio for which it exists. For example, polarizabilities accurate to between five and ten digits are obtained (as complex limits) for negative permittivity ratios in minutes on a standard workstation. In passing, the capacitance of the unit cube is determined with unprecedented accuracy. With full rigor, we develop a natural mathematical framework suited for the study of the polarizability of Lipschitz domains. Several aspects of polarizabilities and their representing measures are clarified, including limiting behavior both when approaching the support of the measure and when deforming smooth domains into a non-smooth domain. The success of the mathematical theory is achieved through symmetrization arguments for layer potentials. [Copyright &y& Elsevier]

Abstract: An integral equation based scheme is presented for the fast and accurate computation of effective conductivities of two-component checkerboard-like composites with complicated unit cells at very high contrast ratios. The scheme extends recent work on multi-component checkerboards at medium contrast ratios. General improvement include the simplification of a long-range preconditioner, the use of a banded solver, and a more efficient placement of quadrature points. This, together with a reduction in the number of unknowns, allows for a substantial increase in achievable accuracy as well as in tractable system size. Results, accurate to at least nine digits, are obtained for random checkerboards with over a million squares in the unit cell at contrast ratio 106. Furthermore, the scheme is flexible enough to handle complex valued conductivities and, using a homotopy method, purely negative contrast ratios. Examples of the accurate computation of resonant spectra are given. [Copyright &y& Elsevier]

Abstract: An algorithm is presented for the fast and accurate solution of the electrostatic equation on multi-component random checkerboards. It relies on a particular choice of integral equation, extended as to separate ill-conditioning due to singular fields in corners from ill-conditioning due to interaction of clusters of well-conducting squares at large distances. Two separate preconditioners take care of the two separate phenomena. In a series of numerical examples, effective conductivities are computed for random checkerboards containing up to 104 squares with conductivity ratios of up to 106. The achievable relative precision in these examples is on the order of 10−11. [ABSTRACT FROM AUTHOR]

Abstract: Laplace’s equation with mixed boundary conditions, that is, Dirichlet conditions on parts of the boundary and Neumann conditions on the remaining contiguous parts, is solved on an interior planar domain using an integral equation method. Rapid execution and high accuracy is obtained by combining equations which are of Fredholm’s second kind with compact operators on almost the entire boundary with a recursive compressed inverse preconditioning technique. Then an elastic problem with mixed boundary conditions is formulated and solved in an analogous manner and with similar results. This opens up for the rapid and accurate solution of several elliptic problems of mixed type. [Copyright &y& Elsevier]

Abstract: New techniques allow for more efficient boundary integral algorithms to compute the Dirichlet–Neumann map for Laplace’s equation in two-dimensional exterior domains. Novelties include a new post-processor which reduces the need for discretization points with 50%, a new integral equation which reduces the error for resolved geometries with a factor equal to the system size, systematic use of regularization which reduces the error even further, and adaptive mesh generation based on kernel resolution. [Copyright &y& Elsevier]

• Method for integral equations with singular and nearly singular right-hand sides. • Universal method independent of the nature of the right-hand side singularities. • Strong right-hand side singularities treated in full machine precision. • Optimal number of discretization points and superior stability. • Speedup for the linearized BGKW kinetic equation for steady Couette flow. A numerical scheme is presented for the solution of Fredholm second-kind boundary integral equations with right-hand sides that are singular at a finite set of boundary points. The boundaries themselves may be non-smooth. The scheme, which builds on recursively compressed inverse preconditioning (RCIP), is universal as it is independent of the nature of the singularities. Strong right-hand-side singularities, such as 1 / | r | α with α close to 1, can be treated in full machine precision. Adaptive refinement is used only in the recursive construction of the preconditioner, leading to an optimal number of discretization points and superior stability in the solve phase. The performance of the scheme is illustrated via several numerical examples, including an application to an integral equation derived from the linearized BGKW kinetic equation for the steady Couette flow. [ABSTRACT FROM AUTHOR]

Abstract: We take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points. Use of non-standard integral equations, graded meshes, interpolatory quadrature, and compressed inverse preconditioning are techniques that are explored, developed, mixed, and tested on some familiar problems in materials science. The recursive compressed inverse preconditioning, the major novelty of the paper, turns out to be particularly powerful and, when it applies, eliminates the need for mesh grading completely. In an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of . In another example, five times as large but with a conductivity ratio of only a hundred, we achieve an estimated relative error of . [Copyright &y& Elsevier]

Abstract: When solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel. Standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources. Close to the sources this is not so. Cancellation and nearly singular kernels may cause serious degradation. This paper presents a new scheme based on a mix of composite polynomial quadrature, layer density interpolation, kernel approximation, rational quadrature, high polynomial order corrected interpolation and differentiation, temporary panel mergers and splits, and a particular implementation of the GMRES solver. Criteria for which mix is fastest and most accurate in various situations are also supplied. The paper focuses on the solution of the Dirichlet problem for Laplace’s equation in the plane. In a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close-to-touching contours. Densities are computed and potentials are evaluated, rapidly and accurate to almost machine precision, at points that lie arbitrarily close to the boundaries. [Copyright &y& Elsevier]

Abstract: Mikhlin’s method for solving Laplace’s equation in domains exterior to a number of closed contours is discussed with particular emphasis on the Dirichlet–Neumann map. In the literature there already exist two computational modes for Mikhlin’s method. Here a new mode is presented. The new mode is at least as stable as the previous modes. Furthermore, its computational complexity in the number of closed contours is better. As a result, highly accurate solutions in domains exterior to tens of thousands of closed contours can be obtained on a simple workstation. [Copyright &y& Elsevier]

The outstanding problem of finding a simple Muskhelishvili-type integral equation for stress problems on multiply connected domains is solved. Complex potentials are represented in a way which allows for the incorporation of cracks and inclusions. Several numerical examples demonstrate the generality and extreme stability of the approach. The stress field is resolved with a relative error of less than 10−10 on a large, yet simply reproducible, setup with a loaded square plate containing 4096 holes and cracks. Comparison with previous results in the literature indicates that general-purpose finite-element software may perform better than many special-purpose codes. [Copyright &y& Elsevier]