# The alternate-block-factorization procedure for systems of partial differential equations

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## Abstract

The alternate-block-factorization (ABF) method is a procedure for partially decoupling systems of elliptic partial differential equations by means of a carefully chosen change of variables. By decoupling we mean that the ABF strategy attempts to reduce intra-equation coupling in the system rather than intra-grid coupling for a single elliptic equation in the system. This has the effect of speeding convergence of commonly used iteration schemes, which use the solution of a sequence of linear elliptic PDEs as their main computational step. Algebraically, the change of variables is equivalent to a postconditioning of the original system. The results of using ABF postconditioning on some problems arising from semiconductor device simulation are discussed.

## AMS subject classification

65F10## Keywords

Semiconductors simulation partial differential equations## Preview

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## References

- [1]O. Axelsson and N. Munksgaard.
*A class of preconditioned conjugate gradient methods for the solution of a mixed finite-element discretization of the biharmonic operator*. Int. J. Numer. Math. Eng., 14: 1001–1019, 1978.Google Scholar - [2]R. E. Bank, J. Bürgler, W. M. Coughran, Jr.., W. Fichtner, and R. K. Smith.
*Recent progress in algorithms for semiconductor device simulation*. In R. Bulirsch, editor,*Proceedings of the*1988*Oberwolfach Confernce on VLSI Modeling*.*Birkhäuser Verlag, Basel*, 1989.*to appear*.Google Scholar - [3]R. E. Bank, W. M. Coughran, Jr., M. A. Driscoll, R. K. Smith, and W. Fichtner.
*Iterative methods in semiconductor device simulation*. Computer Phys. Comm., 53: 210–212, 1989.Google Scholar - [4]R. E. Bank, W. M. Coughran, Jr., W. Fichtner, D. J. Rose, and R. K. Smith.
*Computational aspects of transient device simulation*. In W. L. Engl, editor,*Process and Device Simulation*, pp. 229–264. North-Holland, Amsterdam, 1986.Google Scholar - [5]R. E. Bank and D. J. Rose.
*Global approximate Newton methods*. Numer. Math., 37: 279–295, 1981.Google Scholar - [6]R. E. Bank, D. J. Rose, and W. Fichtner.
*Numerical methods for semiconductor device simulation*. IEEE Trans. Electr. Dev., ED-30: 1031–1041, 1983.Google Scholar - [7]T. F. Chan and H. C. Elman.
*Fourier analysis of iterative methods for elliptic boundary value problems*. SIAM Review, 31: 20–49, 1989.Google Scholar - [8]W. Fichtner, D. J. Rose, and R. E. Bank. Semiconductor device simulation. IEEE Trans. Electr. Dev., ED-30: 1018–1040, 1983.Google Scholar
- [9]H. K. Gummel.
*A self-consistent iterative scheme for one-dimensional steady-state transistor calculations*. IEEE Trans. Electr. Dev., ED-11: 455–465, 1964.Google Scholar - [10]T. J. Hughes, I. Levit, and J. Winget.
*An element-by-element solution algorithm for problems of structural and solid mechanics*. Comp. Meth. Appl. Mech. Eng., 36: 241–254, 1983.Google Scholar - [11]T. J. Hughes, J. Winget, I. Levit, and T. E. Tezduyar.
*New alternating direction procedures in finite element analysis based upon EBE approximate factorization*. In S. Atluri and N. Perrone, editors,*Recent Developments in Computer Methods for Nonlinear Solid and Structural Mechanics*, pp. 75–109. ASME, New York, 1983.Google Scholar - [12]J. W. Jerome.
*Consistency of semiconductor modelling: An existence/stability analysis for the stationary van Roosbroeck system*. SIAM J. Appl. Math., 45: 565–590, 1985.Google Scholar - [13]T. Kerkhoven.
*Coupled and Decoupled Algorithms for Semiconductor Simulation*. PhD thesis, Dept. of Computer Science, Yale Univ., New Haven, 1985.Google Scholar - [14]T. Kerkhoven.
*On the effectiveness of Gummel's method*. SIAM J. Sci. Stat. Comp., 9: 48–60, 1988.Google Scholar - [15]T. Kerkhoven.
*A spectral analysis of the decoupling algorithm for semiconductor simulation*. SIAM J. Numer. Anal., 25: 1299–1312, 1988.Google Scholar - [16]T. Kerkhoven and Y. Saad.
*Acceleration methods for systems of coupled nonlinear partial differential equations*. Technical report, Dept. of Computer Science, Univ. of Illinois at Urbana-Champaign, 1989.Google Scholar - [17]D. S. Kershaw.
*The incomplete Choleski-conjugate gradient method for the iterative solution of systems of linear equations*. J. Comp. Phys., 26: 43–65, 1978.Google Scholar - [18]J. A. Meijerink and H. A. van der Vorst.
*An iterative method for linear systems of which the coefficient matrix is a symmetric M-matrix*. Math. Comp., 31: 148–162, 1977.Google Scholar - [19]M. S. Mock.
*On equations describing steady-state carrier distributions in a semiconductor device*. Comm. Pure Appl. Math., 25: 781–792, 1972.Google Scholar - [20]J. M. Ortega and W. C. Rheinboldt.
*Iterative Solution of Nonlinear Equations in Several Variables*. Academic Press, New York, 1970.Google Scholar - [21]C. S. Rafferty, M. R. Pinto, and R. W. Dutton.
*Iterative methods in semiconductor device simulation*. IEEE Trans. Electr. Dev., ED-32: 2018–2027, 1985.Google Scholar - [22]T. I. Seidman.
*Steady state solutions of diffusion-reaction systems with electrostatic convection*. Nonlinear Analysis. Theory, Methods and Applications, 4: 623–637, 1980.Google Scholar - [23]S. Selberherr.
*Analysis and Simulation of Semiconductor Devices*. Springer-Verlag, Vienna, 1984.Google Scholar - [24]J. W. Slotboom.
*Computer aided analysis of bipolar transistors*. IEEE Trans. Electr. Dev., 20: 669–679, 1973.Google Scholar - [25]J. Winget and T. J. Hughes.
*Solution algorithms for nonlinear transient heat conduction analysis employing element-by-element iterative strategies*. Comp. Meth. Appl. Mech. Eng., 52: 711–815, 1985.Google Scholar - [26]G. Wittum.
*Multi-grid methods for Stokes and Navier-Stokes equations. Transforming smoothers: Algorithms and numerical results*. Numer. Math., 54: 543–563, 1989.Google Scholar