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.


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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[1] Křiek, M.; Neittaanmäki, P., Mathematical and numerical modelling in electrical engineering: theory and applications, (1996), Kluwer Academic Publishers
[2] D’yakonov, E.G., On an iterative method for the solution of finite difference equations, Dokl. akad. nauk SSSR, 138, 522-525, (1961), (in Russian)
[3] Gunn, J.E., The numerical solution of ∇ · a∇u = ƒ by a semi-explicit alternating direction iterative method, Numer. math., 6, 181-184, (1964) · Zbl 0131.14902
[4] Concus, P.; Golub, G.H., Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations, SIAM J. numer. anal., 10, 1103-1120, (1973) · Zbl 0245.65043
[5] Greenbaum, A., Diagonal scalings of the Laplacian as preconditioners for other elliptic differential operators, SIAM J. matrix anal. appl., 13, 826-846, (1992) · Zbl 0754.65042
[6] Faber, V.; Manteuffel, T.; Parter, S.V., On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differential equations, Adv. in appl. math., 11, 109-163, (1990) · Zbl 0718.65043
[7] Neuberger, J.W., ()
[8] Richardson, W.B., Sobolev gradient preconditioning for PDE applications, (), 223-234
[9] Faragó, I.; Karátson, J., The gradient finite-element method for elliptic problems, Computers math. applic., 42, 8/9, 1043-1053, (2001) · Zbl 0987.65121
[10] Lóczi, L., The gradient-Fourier method for nonlinear elliptic partial differential equations in Sobolev space, Annales univ. sci. ELTE, 43, 139-149, (2000) · Zbl 1004.65115
[11] Karátson, J., The gradient method for non-differentiable operators in product Hilbert spaces and applications to elliptic systems of quasilinear differentialequations, J. appl. anal., 3, 2, 205-217, (1997)
[12] Karátson, J., Sobolev space preconditioning of strongly nonlinear 4th order elliptic problems, (), 459-466 · Zbl 0978.65105
[13] Faragó, I.; Karátson, J., Numerical solution of nonlinear elliptic problems via preconditioning operators: theory and applications, () · Zbl 1030.65117
[14] Kadlec, J., On the regularity of the solution of the Poisson problem on a domain with boundary locally similar to the boundary of a convex open set, Czechosl. math. J., 89, 14, 386-393, (1964) · Zbl 0166.37703
[15] Struwe, M., Variational methods. applications to nonlinear partial differential equations and Hamiltonian systems, (1990), Springer-Verlag New York · Zbl 0746.49010
[16] Dorr, F.W., The direct solution of the discrete Poisson equation on a rectangle, SIAM rev., 12, 248-263, (1970) · Zbl 0208.42403
[17] Swarztrauber, P.N., The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson’s equation on a rectangle, SIAM rev., 19, 3, 490-501, (1977) · Zbl 0358.65088
[18] Rossi, T.; Toivanen, J., A parallel fast direct solver for block tridiagonal systems with separable matrices of arbitrary dimension, SIAM J. sci. comput. (electronic), 20, 5, 1778-1796, (1999), electronic · Zbl 0931.65020
[19] Vassilevski, P.S.; Lazarov, R.D.; Margenov, S.D., Vector and parallel algorithms in iteration methods for elliptic problems, (), 40-51, (Albena, 1989)
[20] Boyd, J.P., Chebyshev and Fourier spectral methods, (2001), Dover Publications Sofia · Zbl 0987.65122
[21] Funaro, D., Polynomial approximation of differential equations, lecture notes in physics, new series, monographs 8, (1992), Springer Mineola
[22] Börgers, C.; Widlund, O.B., On finite element domain imbedding methods, SIAM J. numer. anal., 27, 4, 963-978, (1990) · Zbl 0705.65078
[23] Gidas, B.; Ni, W.N.; Nirenberg, L., Symmetry and related properties via the maximum principle, Commun. math. phys., 68, 209-243, (1979) · Zbl 0425.35020
[24] (), 437-442
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