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Numerical treatment of a skew-derivative problem for the Laplace equation in the exterior of an open arc. (English) Zbl 1126.78018
The paper presents a numerical solution for the two-dimensional Laplace equation in the exterior of an open are with skew-derivative boundary condition. This direct problem requires solving for the electric field potential \(u\) belonging to a certain class of functions and satisfying the Laplace equation \(\nabla^2u= 0\) in the exterior of an open arc \(\Gamma\in{\mathcal C}^{2,\lambda},\lambda\in(0, 1]\) parametrized by the arc length \(s:\Gamma=\{x=x(s)=(x_1(s),x_2(s)),s\in[a,b] \}\), subject to the skew-derivative boundary condition
\[ \frac{\partial u} {\partial n}+\beta\frac{\partial u}{\partial\tau}=f\quad\text{on }\Gamma, \] where \(\beta\) is a real given constant, \(\underline n=(\sin(\alpha(s)),-\cos (\alpha(s)))=(x_2'(s),-x_1'(s))\) is the normal vector to \(\Gamma\) at \(x(s)\), \(\tau=(\cos(\alpha(s)),\sin(\alpha(s)))=(x_1'(s),x_2'(s))\) is the tangent vector to \(\Gamma\) at \(x(s)\), and \(f\in C^{0,\lambda} ([a,b])\) is a given function, and to the infinity conditions
\[ |u(x)|\leq\text{Const}., \quad |\nabla u(x)|\leq o(|x|^{-1}),\quad\text{as}\quad |x|=\sqrt {x^2_1+x^2_2}\to\infty. \] The numerical method is based on an analytic evaluation of the singular part in the single-layer potential and the use of the angular potential instead of the double-layer potential. Numerical results are presented and discussed.

MSC:
78M15 Boundary element methods applied to problems in optics and electromagnetic theory
82D37 Statistical mechanical studies of semiconductors
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