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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.

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
Full Text: DOI
[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
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