×

zbMATH — the first resource for mathematics

Sobolev preconditioning for the Poisson-Boltzmann equation. (English) Zbl 0960.82035
Summary: This paper presents an overview of least-squares steepest descent using Sobolev gradients for several prototype differential equations. In the linear case, the method is viewed as a very effective preconditioning strategy for the basic iterative method which arises from steepest descent, in particular, it acts to smooth the Euclidean gradient. Results are given for the one-dimensional Poisson-Boltzmann equation from semiconductor device modeling.

MSC:
82D37 Statistical mechanical studies of semiconductors
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] R. Amantea The general solution of the Poisson-Boltzmann equation, Technical Report PRRL-81-TR-158, RCA laboratories 1981
[2] Berger, M.S., Some nonlinear analytic aspects of VLSI semiconductor device modeling, IEEE trans electron devices ED-30, 1181-1183, (1983)
[3] A. Bjørck Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA, 1996
[4] Cai, Z.; Lazarov, R.; Manteuffel, T.; McCormick, S., First-order system least squares for second-order partial differential equations, SIAM J. num. anal., part I, 31, 1785-1802, (1994) · Zbl 0813.65119
[5] Cai, Z.; Manteuffel, T.; McCormick, S., First-order system least squares for second-order partial differential equations, SIAM J. num. anal., part II, 34, 425-454, (1997) · Zbl 0912.65089
[6] G.F. Carey, W.B. Richardson, C.S. Reed, B.J. Mulvaney, Circuit Device and Process Simulation Mathematical and Numerical Aspects, Wiley, Chichester, 1996
[7] Eason, E.D., A review of least-squares methods for solving partial differential equations, Int. J. num. meth. engr., 10, 1021-1046, (1976) · Zbl 0327.65079
[8] R.F. Feynman, F.B. Leighton, M. Sands, The Feynman Lectures on Physics Vol. II, Addison, Reading, MA, Chap. 7, 8-10 · Zbl 0131.38703
[9] A.S. Grove, Physics and Technology of Semiconductor Devices, Wiley, Chichester, 1967
[10] Gummel, H.K., Self-consistent iterative scheme for one-dimensional steady state transistor calculations, IEEE trans. electron devices ED-11, 455-465, (1964)
[11] M. Hestenes, Conjugate Direction Methods in Optimization, Springer, New York, 1980 · Zbl 0439.49001
[12] Jiang, B.N.; Carey, G.F., A stable least-squares finite element method for non-linear hyperbolic problems, Int. J. numer. methods fluids, 8, 933-942, (1988) · Zbl 0666.76087
[13] R.B. Kellogg, G.R. Shubin, A.B. Stephens, Uniqueness and the cell Reynolds number, SIAM, J. Number Anal., 17 (1980) 733-739 · Zbl 0463.76069
[14] Mayergoyz, I.D., Solution of the nonlinear Poisson equation of semiconductor device theory, J. appl. phys., 59, 195-199, (1986)
[15] Neuberger, J.W., Steepest descent for general systems of linear differential equations in Hilbert space, Springer lecture notes, 1032, 390-406, (1983) · Zbl 0534.35002
[16] J.W. Neuberger, Sobolev Gradients and Differential Equations, Springer, Berlin, 1997 · Zbl 0935.35002
[17] R.J. Renka, J.W. Neuberger, Minimal surfaces and Sobolev gradients, SIAM, J. Sci. Comp. 16 (1995) 1412-1427 · Zbl 0857.35004
[18] J.W. Neuberger, R.J. Renka, Sobolev gradients and the Ginzburg-Landau functional, SIAM J. Sci. Comp. 20 (1998) 582-590 · Zbl 0920.35059
[19] W.B. Richardson Jr., Nonlinear Boundary Conditions in Sobolev Spaces, North Texas State University, 1986
[20] Richardson, W.B., Steepest descent and the least C for Sobolev’s inequality, Bull. London math. soc., 18, 478-484, (1986) · Zbl 0585.46026
[21] P.C. Rosenbloom, The Method of Steepest Descent, Proc. Symp. Appl. Math. VI Amer. Math. Soc. (1961) 127-176
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.