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Register a new userAn Integral Equation Formulation of the $N$-body Dielectric Spheres Problem. Part II: Complexity Analysis
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Added by
Muhammad Hassan
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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This article is the second in a series of two papers concerning the mathematical study of a boundary integral equation of the second kind that describes the interaction of $N$ dielectric spherical particles undergoing mutual polarisation. The first article presented the numerical analysis of the Galerkin method used to solve this boundary integral equation and derived $N$-independent convergence rates for the induced surface charges and total electrostatic energy. The current article will focus on computational aspects of the algorithm. We provide a convergence analysis of the iterative method used to solve the underlying linear system and show that the number of liner solver iterations required to obtain a solution is independent of $N$. Additionally, we present two linear scaling solution strategies for the computation of the approximate induced surface charges. Finally, we consider a series of numerical experiments designed to validate our theoretical results and explore the dependence of the numerical errors and computational cost of solving the underlying linear system on different system parameters.
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This article deals with the efficient and accurate computation of the electrostatic forces between charged, spherical dielectric particles undergoing mutual polarisation. We use the spectral Galerkin boundary integral equation framework developed by Lindgren et al. (J. Comput. Phys. 371 (2018): 712-731) and subsequently analysed in two earlier contributions of the authors to propose a linear scaling in cost algorithm for the computation of the approximate forces. We establish exponential convergence of the method and derive error estimates for the approximate forces that do not explicitly depend on the number of dielectric particles $N$. Consequently, the proposed method requires only $mathcal{O}(N)$ operations to compute the electrostatic forces acting on $N$ dielectric particles up to any given and fixed relative error.
For the first time, we develop a convergent numerical method for the llinear integral equation derived by M.M. Lavrentev in 1964 with the goal to solve a coefficient inverse problem for a wave-like equation in 3D. The data are non overdetermined. Convergence analysis is presented along with the numerical results. An intriguing feature of the Lavrentev equation is that, without any linearization, it reduces a highly nonlinear coefficient inverse problem to a linear integral equation of the first kind. Nevertheless, numerical results for that equation, which use the data generated for that coefficient inverse problem, show a good reconstruction accuracy. This is similar with the classical Gelfand-Levitan equation derived in 1951, which is valid in the 1D case.
We propose and analyse an augmented mixed finite element method for the Oseen equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and homogeneous Dirichlet boundary condition for the velocity. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition, and we show that it satisfies the hypotheses of the Babuvska-Brezzi theory. Repeating the arguments of the continuous analysis, the stability and solvability of the discrete problem are established. The method is suited for any Stokes inf-sup stable finite element pair for velocity and pressure, while for vorticity any generic discrete space (of arbitrary order) can be used. A priori and a posteriori error estimates are derived using two specific families of discrete subspaces. Finally, we provide a set of numerical tests illustrating the behaviour of the scheme, verifying the theoretical convergence rates, and showing the performance of the adaptive algorithm guided by residual a posteriori error estimation.
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