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Sobolev gradient preconditioning for the electrostatic potential equation. (English) Zbl 1098.65113

This paper deals with the numerical solution of the following problem \[ \begin{aligned} -\Delta u + e^{u} = 0, &\\ u| _{\partial\Omega} = 0, \end{aligned} \] on a bounded domain \(\Omega\subset \mathbb{R}^{3}\) being \(C^{2}\)-diffeomorphic to a convex one. The function \(u\) describes the electrostatic potential in \(\Omega\). The authors propose a Sobolev gradient type preconditioning for the stated problem, suitable regularity leads to a constructive representation of the gradients that involves Laplacian preconditioners in the iteration. Moreover, convergence is verified for the corresponding sequence in Sobolev space, which provides mesh independent convergence results for the discretized problems.
The second part of the paper is devoted to a special radially symmetric problem, a direct realization is developed which is independent on a particular discretization. The main advantage of the presented method is its simplicity as for the corresponding algorithm. The straightforward coding of the algorithm as well as the obtained fast linear convergence in a numerical test example are enclosed.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
65F35 Numerical computation of matrix norms, conditioning, scaling
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
78A30 Electro- and magnetostatics
78M30 Variational methods applied to problems in optics and electromagnetic theory
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