Corner asymptotics of the magnetic potential in the eddy-current model

TL;DR

This paper derives explicit corner asymptotic expansions for the magnetic potential in a 2D eddy-current model near a conducting corner, introducing methods to compute singular functions and coefficients, supported by finite element validation.

Contribution

It provides a novel explicit asymptotic expansion for the magnetic potential near corners, including new methods for calculating singular functions and coefficients in the eddy-current context.

Findings

Explicit corner asymptotic expansion derived

Two methods for computing singular coefficients introduced

Finite element computations validate theoretical results

Abstract

In this paper, we describe the magnetic potential in the vicinity of a corner of a conducting body embedded in a dielectric medium in a bidimensional setting. We make explicit the corner asymptotic expansion for this potential as the distance to the corner goes to zero. This expansion involves singular functions and singular coefficients. We introduce a method for the calculation of the singular functions near the corner and we provide two methods to compute the singular coefficients: the method of moments and the method of quasi-dual singular functions. Estimates for the convergence of both approximate methods are proven. We eventually illustrate the theoretical results with finite element computations. The specific non-standard feature of this problem lies in the structure of its singular functions: They have the form of series whose first terms are harmonic polynomials and further terms are genuine non-smooth functions generated by the piecewise constant zeroth order term of the operator.