Reviving the Method of Particular Solutions

Fox, Henrici and Moler made famous a "Method of Particular Solutions" for computing eigenvalues and eigenmodes of the Laplacian in planar regions such as polygons. We explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties. The crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant

Computations of eigenvalue avoidance in planar domains

The phenomenon of eigenvalue avoidance is of growing interest in applications ranging from quantum mechanics to the theory of the Riemann zeta function. Until now the computation of eigenvalues of the Laplace operator in planar domains has been a difficult problem, making it hard to compute eigenvalue avoidance. Based on a new method this paper presents the computation of eigenvalue avoidance for such problems to almost machine precision

Adaptive BEM with optimal convergence rates for the Helmholtz equation

We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a residual error estimator and does not rely on any a priori information that the underlying meshes are sufficiently fine. We prove convergence of the error estimator with optimal algebraic rates, independently of the (coarse) initial mesh. As a technical contribution, we prove certain local inverse-type estimates for the boundary integral operators associated with the Helmholtz equation

The Factorization method for three dimensional Electrical Impedance Tomography

The use of the Factorization method for Electrical Impedance Tomography has been proved to be very promising for applications in the case where one wants to find inhomogeneous inclusions in a known background. In many situations, the inspected domain is three dimensional and is made of various materials. In this case, the main challenge in applying the Factorization method consists in computing the Neumann Green's function of the background medium. We explain how we solve this difficulty and demonstrate the capability of the Factorization method to locate inclusions in realistic inhomogeneous three dimensional background media from simulated data obtained by solving the so-called complete electrode model. We also perform a numerical study of the stability of the Factorization method with respect to various modelling errors.Comment: 16 page

Bempp-cl: A fast Python based just-in-time compiling boundary element library

Summary The boundary element method (BEM) is a numerical method for approximating the solution of certain types of partial differential equations (PDEs) in homogeneous bounded or unbounded domains. The method finds an approximation by discretising a boundary integral equation that can be derived from the PDE. The mathematical background of BEM is covered in, for example, Steinbach (2008) or McLean (2000). Typical applications of BEM include electrostatic problems, and acoustic and electromagnetic scattering. Bempp-cl is an open-source boundary element method library that can be used to assemble all the standard integral kernels for Laplace, Helmholtz, modified Helmholtz, and Maxwell problems. The library has a user-friendly Python interface that allows the user to use BEM to solve a variety of problems, including problems in electrostatics, acoustics and electromagnetics. Bempp-cl began life as BEM++, and was a Python library with a C++ computational core. The ++ slowly changed into pp as functionality gradually moved from C++ to Python with only a few core routines remaining in C++. Bempp-cl is the culmination of efforts to fully move to Python, and is an almost complete rewrite of Bempp. For each of the applications mentioned above, the boundary element method involves approximating the solution of a partial differential equation (Laplace’s equation, the Helmholtz equation, and Maxwell’s equations respectively) by writing the problem in boundary integral form, then discretising. For example, we could calculate the scattered field due to an electromagnetic wave colliding with a series of screens by solving ∇ x ∇ x E - k2E = 0; v x E = 0 on the screens; where E is the sum of a scattered field Es and an incident field Einc, and � is the direction normal to the screen. (Additionally, we must impose the Silver–Müller radiation condition to ensure that the problem has a unique solution.) This problem is solved, and the full method is derived, in one of the tutorials available on the Bempp website (Betcke & Scroggs, 2020a). The solution to this problem is shown below

Computed eigenmodes of planar regions

Recently developed numerical methods make possible the high-accuracy computation of eigenmodes of the Laplacian for a variety of "drums" in two dimensions. A number of computed examples are presented together with a discussion of their implications concerning bound and continuum states, isospectrality, symmetry and degeneracy, eigenvalue avoidance, resonance, localization, eigenvalue optimization, perturbation of eigenvalues and eigenvectors, and other matters.\ud \ud Timo Betcke was supported by a Scatcherd European Scholarship

An OSRC Preconditioner for the EFIE

The Electric Field Integral Equation (EFIE) is a well-established tool to solve electromagnetic scattering problems. However, the development of efficient and easy to implement preconditioners remains an active research area. In recent years, operator preconditioning approaches have become popular for the EFIE, where the electric field boundary integral operator is regularised by multiplication with another convenient operator. A particularly intriguing choice is the exact Magnetic-to-Electric (MtE) operator as regulariser. But, evaluating this operator is as expensive as solving the original EFIE. In work by El Bouajaji, Antoine and Geuzaine, approximate local Magnetic-to-Electric surface operators for the time-harmonic Maxwell equation were proposed. These can be efficiently evaluated through the solution of sparse problems. This paper demonstrates the preconditioning properties of these approximate MtE operators for the EFIE. The implementation is described and a number of numerical comparisons against other preconditioning techniques for the EFIE are presented to demonstrate the effectiveness of this new technique

Boundary element methods for Helmholtz problems with weakly imposed boundary conditions

We consider boundary element methods where the Calder\'on projector is used for the system matrix and boundary conditions are weakly imposed using a particular variational boundary operator designed using techniques from augmented Lagrangian methods. Regardless of the boundary conditions, both the primal trace variable and the flux are approximated. We focus on the imposition of Dirichlet and mixed Dirichlet--Neumann conditions on the Helmholtz equation, and extend the analysis of the Laplace problem from the paper \emph{Boundary element methods with weakly imposed boundary conditions} to this case. The theory is illustrated by a series of numerical examples.Comment: 27 page