chapter 6 nonlinear multigrid applied to a one-dimensional

Chapter 6

Nonlinear multigrid

applied to a one-dimensional

stationary semiconductor model

P.M. de Zeeuw

Abstract

The nonlinear multigrid method is applied to a transistor problem in onedimension. A weak spot in the linearization of the well-known Scharfetter-Gummel discretization scheme is reported. Further it is shown that both theresidual transfer and the solution transfer from a �ne to a coarse grid need specialrequirements due to the rapidly varying problem coe�cients. Some modi�ca-tions are proposed which make the multigrid algorithm perform well for the hardexample problem.

Note: This chapter has been published in SIAM J. Sci. Stat. Comput. 13 (1992)512{530.

6.1 Introduction

There is a great demand for a proper numerical simulation of semiconductors inorder to reduce the costs of constructing expensive prototypes. The search for a fastand robust algorithm has proven to be a challenge. So far only a few papers haveconsidered the multigrid solution of the discrete semiconductor equations (e.g. see[1, 2, 6, 9, 13]) and therefore extensive further research is required.

In this paper we restrict ourselves on purpose to one space dimension as a prepara-tory study for the case of more space dimensions. We study a particular exampleproblem which has been put forward by W. Schilders (Philips, the Netherlands).This problem models a transistor and turns out to be much harder to solve than theforward or reversed biased diode problem. We apply the nonlinear multigrid method

120 Chapter 6

and encounter a serious di�culty due to the nonlinearity of the problem. Some mod-i�cations are proposed which signi�cantly increase the robustness of the nonlinearmultigrid method and which look promising also for the higher-dimensional case.

6.2 The problem

The behaviour of a steady semiconductor device can be described by the following setof equations (cf. [10]):

r(�"r ) = q(p� n+D); (6.1a)

rJn = +qR; (6.1b)

rJp = �qR; (6.1c)

where Jn and Jp are de�ned by

Jn = q�n(1

�rn� nr ); (6.2a)

Jp = �q�p(1

�rp+ pr ): (6.2b)

Substitution of (6.2) into (6.1) results in a system of three nonlinear partial di�erentialequations for ; n and p. In (6.1) represents the electrostatic potential, p andn describe the concentration of holes and electrons respectively. Equations (6.1b)and (6.1c) are called the continuity equations; Jn is the electron current density, Jpis the hole current density, and R is the recombination-generation rate, a function ofn and p. The doping pro�le D is a function of the space variable x. The quantities"; q; �; �n; �p represent the permittivity, the elementary charge, the inverse of thethermal voltage and the electron and hole mobility respectively.

In this paper we consider the case of only one space dimension and assume "; �; �nand �p to be constant. It is common practice to replace the variables n and p by thehole and electron quasi-Fermi potentials �n and �p de�ned by the relations:

n = nie�( ��n); (6.3a)

p = nie�(�p� ): (6.3b)

On the one hand, by this change of variables, the nonlinearity of the problem isstrongly increased, on the other hand the values assumed by ( ; �n; �p) are in a muchmore moderate range. For extensive discussions on the choice of variables see [9, 10].Using (6.3) the equations (6.1) are transformed into

�rJ = niq(e�(�p� ) � e�( ��n)) + qD; (6.4a)

�rJn = +qR; (6.4b)

�rJp = �qR; (6.4c)

Multigrid applied to a semiconductor model 121

where J is de�ned byJ = "r ; (6.5)

and Jn; Jp are now de�ned by

Jn = �ne�( ��n)r(��n); (6.6a)

Jp = �pe�(�p� )r(��p); (6.6b)

with�n =

niq�n�

; �p =niq�p�

: (6.7)

In this paper we adhere to the formulation (6.4){(6.7).

6.2.1 A particular one-dimensional model problem

We will focus our attention to a particular (hard) one-dimensional model problemwhich has been supplied by Schilders [14]. Here the problem constants are

" = 1:03591810�12AsV�1cm�1; q = 1:602110�19As;

�n = �p = 500V�1s�1cm2; ni = 1:221010cm�3; (6.8)

k = 1:3805410�23VAsK�1; T = 300K; � = q=T:

The function R is given by

R =pn� n2i

�(p + n+ 2ni); � = 10�6s .

The doping function D (in cm�3) is given by

D(x) = 61015 + 61019 exp(�(x=7:110�5)2)� 2:151018 exp(�(x=1:1510�4)2)

+1:11019 exp(�((x � 810�4)=1:310�4)2):

The equations (6.4) are de�ned on the domain = [0; 810�4 ] (cm). We have threecontacts to our semiconductor device (the one-dimensional model of a transistor): theemitter (E), the basis (B) and the collector (C) (see Figure 6.1). In Figure 6.2 thedoping function D(x) is shown after the transformation

D ! sign(D) log10(1 + jDj):

Boundary conditions at the emitter E are:

p� n+D = 0 (i.e., vanishing space charge); (6.9a)

�n = VE ; (6.9b)

Jp = 0: (6.9c)

122 Chapter 6

E B C

0 1:910�4 810�4

Figure 6.1: The contacts in the one-dimensional transistor problem.

Figure 6.2: The doping pro�le.

Boundary conditions at the basis B:

�p = VB = 0: (6.10)

Boundary conditions at the collector C:

p� n+D = 0; (6.11a)

�n = VC ; (6.11b)

�p = VC : (6.11c)

For �fteen di�erent cases, each characterized by a pair of voltages (VE ; VC), the so-lution is required (see Table 6.1). Figure 6.6 shows the solution-component for thesubsequent cases.


Table 6.1: Subsequent voltages at the emitter and collector for which a solution isrequired.

case VE VC

0 0 0:01 0 0:22 0 0:43 0 0:64 0 0:85 0 16 �0:2 17 �0:4 18 �0:6 19 �0:7 110 �0:8 111 �0:85 112 �0:9 113 �0:95 114 �1 1

6.3 Discretization

At the outset of this section we give a short preview of its contents.In order to abide by the law of conservation we use a �nite volume technique based

on the piecewise constant approximation of J ; Jn and Jp. As a consequence we arriveat a cell-centered version of the well-known Scharfetter-Gummel scheme [4, 9, 11]. Weexamine how the nonlinear discrete operator depends on the discrete solution.

6.3.1 Box integration

The interval = (x0; xN ) is split up into disjoint boxes

Bj = (xj�1; xj); j = 1(1)N:

A point xj is called a wall, a point xj�1=2 = (xj�1 + xj)=2 is called a center. Thebasis B is at the partition-wall between two boxes. Another set of subintervals fDjgis de�ned by

D0 = (x0; x1=2);

Dj = (xj�1=2; xj+1=2); j = 1(1)N � 1;

DN = (xN�1=2; xN ):

This set is called the set of dual boxes.

124 Chapter 6

Now, by applying the Gauss divergence theorem in one dimension to the equa-tions (6.4) on the domains Bj we �nd

�J jxjxj�1

� niq

ZBj

(e�(�p� ) � e�( ��n))d = q

ZBj

Dd; (6.12a)

�Jnjxjxj�1

� q

ZBj

Rd = 0; (6.12b)

�Jpjxjxj�1

+ q

ZBj

Rd = 0; j = 1(1)N: (6.12c)

We can write (6.12) in symbolic form as

M(q) = f (6.13)

where q denotes the vector ( ; �n; �p)T ;M the nonlinear operator in the left-hand

side of (6.12) and f the right-hand side of (6.12).

6.3.2 Box discretization

We introduce the variables ( j ; �n;j ; �p;j)T , j = 1 (1)N , which are associated with the

centers xj�1=2 of the boxes Bj . Let � denote approximation by midpoint quadrature.We then de�ne

Sj � niq

ZBj

(exp(�(�p � ))� exp(�( � �n)))d; (6.14a)

Fj � q

ZBj

Dd; (6.14b)

Rj � q

ZBj

Rd; (6.14c)

j = 1(1)N:

We make the assumption that J ; Jn and Jp are piecewise constant on the dual setfDjg, (see [4, 9, 11]) and correspondingly we use the notation J ;j ; Jn;j ; Jp;j . By thisassumption and applying (6.14) we arrive at the following discrete equations:

�J ;j + J ;j�1 � Sj = Fj ; (6.15a)

�Jn;j + Jn;j�1 �Rj = 0; (6.15b)

�Jp;j + Jp;j�1 +Rj = 0; (6.15c)

with

J ;j = " j+1 � j

xj+1=2 � xj�1=2; (6.16a)


Jn;j = �nexp(��n;j+1)� exp(��n;j)

exp(�� j+1)� exp(�� j)�� j+1 � � jxj+1=2 � xj�1=2

; (6.16b)

Jp;j = �pexp(��p;j+1)� exp(��p;j)

exp(� j+1)� exp(� j)�� j+1 � � jxj+1=2 � xj�1=2

: (6.16c)

At the emitter E, basis B and collector C similar equations are obtained, for fulldetails see [16]. Thus we have obtained the cell-centered version of the well-knownScharfetter-Gummel scheme (see [4, 9, 11]). Summarizing, we have discretized theequations (6.4), together with the boundary conditions (6.9){(6.11), into a set of 3Nnonlinear equations (6.15) with the 3N variables j ; �n;j ; �p;j ; j = 1(1)N . We canwrite (6.15) in symbolic form as

Mh(qh) = fh (6.17)

where Mh denotes the nonlinear di�erence operator and fh the right-hand side.

6.3.3 Properties of the discretized operator

In this subsection we study how the Jacobian of the nonlinear discrete operatorMh

depends on the discrete solution. We assume the recombination term to be zero andcon�ne ourselves to the dependency on �p. Results for �n can be derived analogously.We freeze the solution components and �n and consider the �p-stencil, at box Bj ,de�ned by the triplet

[stp(j;�1); stp(j; 0); stp(j;+1)] with (6.18a)

stp(j; k) =@(�Jp;j + Jp;j�1)

@�p;j+k; k = �1; 0; 1: (6.18b)

We introduce the notation

�jx � xj+1=2 � xj�1=2;

�j � j+1 � j

and the function s(z) : R ! R by

s(z) �z

ez � 1: (6.19)

By straightforward computation it can be veri�ed that the following equalities hold:

stp(j;�1) = ��p exp(�(�p;j�1 � j))s(��j�1 )

�j�1x; (6.20a)

stp(j; 0) = ��p exp(�(�p;j � j))

�s(��j�1 )

�j�1x+s(��j )

�jx

�; (6.20b)

126 Chapter 6

stp(j;+1) = ��p exp(�(�p;j+1 � j))s(��j )

�jx; (6.20c)

therefore

stp(j; 0) = � (stp(j;�1) + stp(j;+1)) + � (�Jp;j + Jp;j�1) : (6.20d)

Because s(z) > 0 for all z, it follows that

stp(j;�1) < 0; stp(j; 0) > 0; stp(j;+1) < 0; (6.21)

so the �p-stencils correspond with an L-matrix. Further, at the exact discrete solution,i.e., when �Jp;j + Jp;j�1 = 0 is satis�ed, it follows from (6.20d) that

stp(j; 0) = �(stp(j;�1) + stp(j;+1));

so then the L-matrix also possesses weak diagonal dominance (provided there is atleast one stencil corresponding with a Dirichlet boundary condition, see [15]). How-ever, in the middle of some iterative process to determine the solution, we may wellhave negative residuals so that (6.20d) implies the loss of diagonal dominance. There-fore ill-conditioning and numerical di�culties can be expected.

6.4 The Newton method and expedients

An obvious way of solving the set of nonlinear equations (6.17) is application of theNewton method. Because the Newton method is not globally convergent and theoperator Mh is strongly nonlinear in the variables ( ; �n; �p) we use two additionaltools which are considered subsequently in this section:

1. Correction transformation.

2. Smoothing of the Newton-iterates.

It turns out that these expedients make the Newton method well applicable. Othermodi�cations of the Newton method including inexact line searches and related tech-niques have been found to be reliable elsewhere (see [4]).

In two or more space dimensions direct application of the Newton method to (6.17)would involve large storage requirements and the solution of large linear systems.If well designed, a nonlinear multigrid algorithm holds out a prospect of both acomputational complexity which is linear in the number of grid-points and low storagerequirements even for the case of two or more space dimensions. Therefore we wantto apply the Newton method only for very coarse grids and we restrict the use of theNewton method as a coarsest grid solver for multigrid methods (Section 6.5).

6.4.1 Correction transformation

The correction transformation, introduced by Schilders [10] is a device to trans-form the Newton-correction (d ; d�n; d�p), computed by linearization with respect


to ( ; �n; �p), into the correction for these very variables that would be obtained iflinearization were applied with respect to ( ; n; p). Because the system in terms of( ; n; p) is much less nonlinear, a much better convergence behaviour of the Newtonmethod can be expected. By performing the calculations in terms of ( ; �n; �p) andapplying a transformation afterwards, we avoid complications due to the extremelywide range of values of n and p. In this way we take advantage of the bene�ts of bothvariable-sets [9, 10, 14].

6.4.2 Smoothing

In Section 6.4.1 we pointed out a technique to improve the global convergence be-haviour of the Newton method. Even yet di�culties are encountered when we applythe improved Newton method. As an example consider Figure 6.3 which shows sub-sequent Newton iterates for case 12 starting from the solution for case 11. The dips

Figure 6.3: Subsequent Newton iterates of �p.

in the iterates are attended with very small pivot numbers while solving the linearsystems. Section 6.3.3 explains the ill-conditioning whenever there is a large residualsomewhere. Arti�cially increasing the main diagonal of the Jacobian turned out to benot e�cient. Simply cutting o� the correction at certain points is hardly justi�ablebecause of lack of a more or less general criterion to do so. A more appropriate way ofhandling the phenomenon sketched above is to apply relaxation or smoothing sweepsat the beginning of the Newton process (cp. [9]). As a smoother the collective Sym-

128 Chapter 6

metric Gauss-Seidel relaxation (CSGS) can be used. It is called collective because ateach box we solve collectively the three nonlinear equations which arise (employingNewton's method).

We will present here some numerical results to show the e�ect of smoothing. Thegrid is more or less uniform and satis�es xN=4 = B. The set of voltages f(VE ; VC)gfor which a solution is required is de�ned in Table 6.1. For each case > 0 the solutionof the previous case serves as a starting solution; in case 0 we start with �n;j =�p;j = 0, for all j, and is determined by assuming space charge neutrality. Weuse the correction transformation. For the solution of the linear systems we applyrow-scaling followed by row-pivotting. Table 6.2 shows the number of Newton sweepsrequired to reach a correction with absnorm < 10�12, and the smallest pivot-numberencountered during the solution process. Table 6.2 also contains the results for thecase when in addition a CSGS-sweep is applied each time after a Newton-correctionfor which the in�nity norm of the correction was larger than 0.1. This method willhenceforth be referred to as Newton-CSGS. We observe that in the di�cult cases

Table 6.2: Number of Newton- and CSGS-sweeps used, and smallest pivot-numbers;N = 32.

No smoothing applied Smoothing appliedSmallest Smallest

case Newton- pivot- Newton- CSGS- pivot-sweeps number sweeps sweeps number

0 6 3E � 3 4 1 5E � 11 5 2E � 4 5 2 5E � 12 6 1E � 9 5 2 5E � 13 5 2E � 7 5 2 5E � 14 5 1E � 7 5 2 3E � 15 5 4E � 5 5 2 3E � 16 5 3E � 3 5 2 2E � 17 5 3E � 3 5 2 2E � 18 5 2E � 4 5 2 2E � 19 5 4E � 7 5 2 3E � 110 6 1E � 7 6 3 3E � 111 9 4E � 9 7 3 2E � 112 15 5E � 13 8 4 4E � 213 13 2E � 11 7 3 1E � 114 10 5E � 10 7 3 1E � 1

11�14 the application of smoothing sweeps has a positive e�ect on the e�ciency androbustness of the Newton method. When smoothing is applied the smallest pivot-numbers encountered keep a substantial distance from zero which shows that then


the Jacobians generated within the Newton method are far from being singular (e.g.,compare for case 12 in Table 6.2) and therefore no large dips in the Newton-correctionsdo occur. Experiments for N = 16; 64; 128 show results similar to Table 6.2.

6.5 The multigrid method

More advanced ways of solving a set of nonlinear equations are the Full ApproximationScheme (FAS) [5], and the Nonlinear Multigrid Method (NMGM) [7]. Both multigrid

methods are very similar although the NMGM is more general. The multigrid methodhas already found many speci�c applications in the �elds of elliptic, parabolic andhyperbolic equations and integral equations as well. Recently, also in the �eld ofsemiconductor equations research on multigrid methods has been initiated (see [1, 2,6, 9, 13]). If well applied, a multigrid method can be optimal in the sense that therate of convergence is independent of the mesh-size. An important advantage of theFAS/NMGM-method is that no large linear systems need to be stored and solved.The subsequent stages of a usual FAS-method, applied to (6.17), are:

1. Apply p nonlinear relaxation sweeps; thus we get an approximation qh of thesolution which has a smooth residual dh � fh �Mh(qh).

2. Transfer qh and dh from h to a coarser grid H by means of the respectiverestriction operators RH and RH .

3. Solve (approximately) on H the equation

MH(qH ) =MH(RHqh) + RHdh:

4. Interpolate the correction, computed on H , onto h and add the correction toqh.

5. Apply q nonlinear relaxation sweeps.

The combination of stage 2,3 and 4 is called the coarse grid correction (CGC). Stage3 may be obtained by applying a number of � FAS-cycles on the coarser grid. Inthis way a recursive procedure is obtained in which a sequence of increasingly coarsergrids is used. In this paper we use p = q = � = 1 throughout. In the subsections tocome we will de�ne precisely the coarse grid correction and the grid transfer operatorsinvolved. In Section 6.6 a signi�cant improvement of the CGC will be introduced. Itconsists of a solution-dependent adjustment of the restriction of the residual dh.

6.5.1 Nested boxes

Let a coarse grid H , a discretization of , be given by the set of boxes fBH;jgj=1(1)N .

From H we construct the next �ner grid h = fBh;jgj=1(1)2N by division of each

BH;j into two disjoint boxes Bh;2j�1 and Bh;2j . By repetition we obtain thus a

130 Chapter 6

sequence of increasingly �ner grids. By de�nition all boxes are nested. Of course, thecorresponding dual boxes are not nested. For all our numerical experiments in thispaper we assume in addition that Bh;2j�1 and Bh;2j have equal size.

6.5.2 Restriction operators

For the problem (6.13) on , let S denote the domain and V the range of nonlinearoperatorM. For each discretization on h, we have the spaces Sh and Vh, the discreteanalogues of S and V .

Let the restriction operator for right-hand side functions

Rh : V ! Vh (6.22)

be de�ned by

Rhf = fh; (6.23a)

fh;j =

ZBh;j

fd; 8j at h: (6.23b)

It follows for the next coarser grid that

(RHf)j = (Rhf)2j�1 + (Rhf)2j ; 8j at H : (6.24)

By this equality, RH can be de�ned also on Vh:

RH : Vh ! VH ; (6.25a)

(RHfh)j = fh;2j�1 + fh;2j ; 8j at H : (6.25b)

The restriction operator for solutions

Rh : S ! Sh (6.26)

may be de�ned by the well-known fullweighting operator [5].

6.5.3 Prolongation/interpolation

A prolongation transfers a solution from a coarse grid to a �ner one:

Ph : SH ! Sh (6.27)

A common and simple choice for the prolongation should be linear interpolation.However, two objections against this choice do arise. Firstly, by the use of linearinterpolation it is implicitly assumed that the solution behaves like a smooth functionon H . Because of the exponential behaviour of the solution in some areas, this is


e e H (coarse)

L M R

e e e e h (�ne)

L0 R0

Figure 6.4: Staggering of a coarse and �ne grid.

only true on an unfeasibly �ne grid. Secondly, linear interpolation does not satisfyhere the so-called Galerkin condition

RHMh(PhsH) =MH(sH) (6.28)

which is a condition that ascertains the reduction of low frequency components in theresidual after a CGC. Hemker [9] has introduced a prolongation which is based onthe assumption of smoothness of uxes, and which satis�es (6.28) for the simpli�edcase that all Sj and Rj are zero, see (6.15). Here, we use the same assumption but wechoose a short and convenient formulation in order to handle also the situation nearthe inner boundary point B. Figure 6.4 depicts how the dual box [L,R] is dividedinto the boxes [L,M] and [M,R]. The assumption reads that J ; Jn; Jp are constanton [L,R]. Given the values of the variables ( ; �n; �p) at L and R we wish to computethe values at L0 and R0. From (6.16a) it follows that jL0 and jR0 can be computedby linear interpolation. For �p we �rst determine the value at the wallM . If we write

� = jR � jL;��p = �pjR � �pjL (6.29)

then we derive that:

�pjM = (6.30)

if � < 0 (1)

then if� 2 ��p < 0 (2)

then �pjR + z(� 2 ; � 2 ��p) (3)

else �pjL +� 2 + z(� 2 ;��

2 +��p) (4)end if (5)

else if� 2 ��p < 0 (6)

then �pjR �� 2 + z(��

2 ; � 2 ��p) (7)

else �pjL + z(�� 2 ;��

2 +��p) (8)end if (9)

end if (10)

132 Chapter 6

where the function z : R2! R is de�ned by

z(u; v) =1

�log

�exp(�v) + 1

exp(�u) + 1

�: (6.31)

Note that in (6.30) the function z is used with only non-positive arguments and

jz(u; v)j �log(2)

�for u � 0; v � 0: (6.32)

By repeating the interpolation procedure, we can compute �pjL0 from �pjL and �pjM ,and �pjR0 from �pjM and �pjR. In the particular case that the wall M is the basis B,we do not �rst determine �pjM by interpolation, but simply state that

�pjM � �pjB = VB (6.33)

Analogously, we can derive a formula for �njM .

6.5.4 Coarse grid correction

Let qoldn and qoldH be given approximations to the solution on h and H , respectively.The CGC is de�ned by

compute dH = RH(fh �Mh(qoldh )); (6.34a)

solve MH(qnewH ) =MH(q

oldH ) + dH ; (6.34b)

compute qnewh = qoldh + (PhqnewH � Phq

oldH ); (6.34c)

where RH and Ph are the grid transfer operators de�ned in the previous subsections.Note that qnewH in (6.34b) may be approximated by applying a number of � FAS-cycleson the grid H with qoldH as an initial approximation. The approximation qoldH maybe given by means of full-weighting:

qoldH = RHqoldh ; (6.35a)

qoldH;j = 12q

oldh;2j�1 +

12q

oldh;2j ;8j at H : (6.35b)

Another possibility is to take qoldH equal to qnewH obtained from the last of previousCGCs. The solution e�ciency of many nonlinear problems is not in uenced by eitherchoice of qoldH , in our case however it is (see Section 6.7).

6.5.5 Full Multi-Grid

The full multigrid (FMG) algorithm provides the e�cient construction of an initialapproximation to the solution on a �ne grid, once a solution on a coarse grid has beencomputed (cf. [5, 7]). Let coarse be the coarsest grid and �ne be the �nest one.Intermediate grids are denoted by l 2 N . Operators and grid functions do now havel as a subscript instead of h or H . Here we introduce an improvement of the usualFMG in a quasi-Algol description:


procedure BOX-FMG ("M�ne(q�ne) = f�ne"; input : f�ne; output : q�ne)begin

for l from �ne �1 by �1 to coarse (1)do fl = Rlfl+1 (2)end do (3)SOLVE ("Mcoarse(qcoarse) = fcoarse"; input : fcoarse; output : qcoarse) (4)for l from coarse +1 to �ne (5)do ql = Plq�1 (6)

to (7)do FAS ("Ml(ql) = fl"; input : fl; in=output : ql) (8)end do (9)

end do (10)end procedure

where Rl is de�ned by (6.25). The improvement is in the lines (1)-(3) of the procedure.The grid-function fl is independent of ql, the components represent

fl;j =

ZBj

Dd (6.36)

i.e., the dope-function integrated over box Bj . By means of (1){(3) we compute theintegral as a Riemann-sum over a larger number of subintervals. This is more accuratebecause D is a rapidly varying function. In the numerical experiments to come weuse = 1 throughout. For SOLVE( ) we use the techniques of Section 6.4.

6.6 Adaptation of the Coarse Grid Correction

Hemker [9] successfully applied box-centered multigrid FAS iteration to the forwardand the reverse biased diode problem. A key feature in his application is the pro-longation based on locally constant uxes. This prolongation has been reformulatedand made suitable for the transistor problem in Section 6.5. Application of the sameMG-algorithm to the transistor problem, gives rise to a complication in the CGC dueto drastically varying problem coe�cients. This complication and possible remediesare the topics of this section.

6.6.1 Improper solution transfer

The �rst attempt of applying multigrid to our speci�c problem was done by employingBOX-FMG with only two grids. The coarse grid problem (6.34b), within the CGCof FAS, was to be solved up to machine-accuracy by means of Newton-CSGS. Forseveral cases of our test-problem it turned out that the two-grid-algorithm gets stuckprecisely at stage (6.34b) of the CGC. This is remarkable because Newton-CSGS wasshown in Section 6.4.2 to be successful for MH(qH) = fH even for rather coarsegrids. Apparently fH is within an appropriate range ofMH while the right-hand side

134 Chapter 6

of (6.34b) may be outside such a proper range of MH . The computational di�cultyoccurs in CSGS on the coarse grid exactly where one or more of the three solutioncomponents depicts a steep gradient. Consider two adjacent boxes BhL and BhR onthe �ne grid which together constitute a box BH on the coarse grid. Because of thesteep gradient it may well occur that the problem coe�cients, i.e., the entries of theJacobian ofMh, show a quite di�erent order of magnitude on BhL and BhR respectively.Grid-function dH , the restriction of the residual, is dominated by the �ne grid boxwith the large coe�cients. On the other hand, the operatorMH is generated by theparticular choice of qoldH . This particular choice may be full weighting applied to qoldh ,

or the last qH available, etc. Because of the steep gradient in qoldh there is a large

range of possible values for qoldH at BH . Depending on the choice of qoldH the operatorMH may have either large or small coe�cients at box BH due to the exponentialbehaviour of the entries in the Jacobian as a function of the solution. In the case ofsmall coe�cients, the right-hand side of (6.34b) may become out of the appropriaterange for MH (dH does not depend on the particular choice of qoldH ) and the twogrid-algorithm gets stuck.

We will now con�rm the foregoing by considering our discretized problem in moredetail. Consider the center of the �p-stencil given by (6.20b) and let us suppose that is monotonous on [xj�3=2; xj�1=2]; then either s(��j�1 ) � 1 or s(��j ) � 1.If both j�j�1 j and j�j j are su�ciently small then stp(j; 0) is approximated by

stp(j; 0) � ��p exp(�(�p;j � j)) � (1

�j�1x+

1

�jx):

If both j�j�1 j and j�j j are su�ciently large then stp(j; 0) is approximated by

stp(j; 0) � ��p exp(�(�p;j � j)) �

8<:

��j�1 �j�1x

if �j�1 � 0;

��j �jx

if �j � 0

These approximations show that indeed the �p-stencil is extremely sensitive to thedi�erence (�p;j � j). Hence the �p-stencil on the coarse grid is sensitive to how �p;jand j on the coarse grid are determined from their counterparts on the �ne grid. If

qoldH is determined by applying full weighting (linear interpolation) to qoldh then

stp(j=2; 0) � exp��

2j�j(�p � )j

��max fstp(j � 1; 0); stp(j; 0)g

where stp(j � 1; 0); stp(j; 0) (j even) are de�ned at the �ne grid h and stp(j=2) atthe coarse grid H . If (�p � ) shows a steep gradient then indeed

stp(j=2)� max fstp(j � 1; 0); stp(j; 0)g :

Note. The possible occurrence of the above sketched phenomenon has already beennoti�ed (for general nonlinear problems) by A. Brandt [5, p. 279], where he discusseshow the transferred solution (i.e., qoldH ) implicitly determines the problem coe�cientson the coarse grid.


e H

M

he e

L R

Figure 6.5: Nested boxes.

6.6.2 Possible remedies

Let L and R be the centers of the two adjacent boxes BhL and BhR on the �ne grid hwhich together constitute a coarse grid box BHM with centerM on the coarse grid H(see Figure 6.5). Let us assume that @

@x = c is constant on BhL [ BhR. The centers of

the �p-stencils at L;R are then determined by the coe�cients

ahL � exp(�(�p � )jL)c; ahR � exp(�(�p � )jR)c

respectively and the center of the �p-stencil at M by

aHM � exp(�(�p � )jM )c

with c = �2�pjcj (see Section 6.6.1). Let �M (�p � ) denote the variation

�M (�p � ) = (�p � )jR � (�p � )jL:

The solution at M on the coarse grid somehow relates to the solution at L and R onthe �ne grid (for instance by means of the full weighting restriction). If �M (�p � )is small (a smooth solution) then obviously aHM does not di�er much from either ahLor ahR. If �M (�p � ) is large (a steep gradient in the solution) then aHM may di�erorders of magnitude from both ahL and ahR, and therefore the MG-algorithm mayget stuck as was pointed out in the previous subsection. A radical remedy to meetthis situation is to prevent �M (�p � ) from getting large, i.e., to introduce localre�nement of the mesh just where the solution has a large variation �M (�p � ),e.g., by means of equidistributing the variation. However, we want to be able to�nd solutions without much re�nement, in order to apply coarse grids in our MG-algorithm. Besides, a solution without much resolution can serve as a guide for wherea local mesh re�nement should take place. For this reasons we resort to anotherremedy. Let us consider the CGC (6.34). Let dh(L); dh(R) be the residuals at L;R(e.g., for the third equation (6.15c) only). At M the di�erence between qnewH and qoldHmay have the order of magnitude dH(M)=aHM with dH(M) = dh(L)+dh(R). Becauseof (6.34c) at either L;R, or both L and R a correction with order of magnitude

136 Chapter 6

dH(M)=aHM is added to the solution qoldh . Assume that (because of a steep gradientin the solution) the inequality

aHM � max�ahL; a

hR

holds. Therefore

dH(M)=aHM � (dh(L) + dh(R))=max�ahL; a

hR

;

which implies that the correction that will be transferred to the �ne grid becomesfar too large and the solution qoldh gets spoiled. A way to prevent this situation is tomultiply the restricted residual dH with

�HM �aHM

max�ahL; a

hR

; 0 < �HM � 1; (6.37)

at each center M . For a smooth part of the solution this fraction will be near one,for a rapidly varying part of the solution it will be near zero, so that the solution qoldhwill be preserved. The foregoing is the motivation for the following modi�cation ofthe FAS-algorithm (MFAS) using the notation of Section 6.5.5:

Procedure MFAS ("Ml(ql) = fl"; input : fl; in=output : ql)begin

if l = coarse (1)then SOLVE ("Mcoarse(qcoarse) = fcoarse" ,

input : fcoarse; in=output : qcoarse) (2)else RELAX ("Ml(ql) = fl"; input : fl; in=output : ql) (3)

dl�1 := Rl�1(fl �Ml(ql)) (4)ql�1 := Rl�1ql (optional !) (5)dl�1 := �l�1(Ml�1;Ml)dl�1 (6)dl�1 := dl�1 +Ml�1(ql�1) (7)sl�1 := ql�1 (8)to � (9)do MFAS ("Ml�1(ql�1) = dl�1"; input : dl�1; in=output : ql�1) (10)end do (11)ql := ql + Plql�1 � Plsl�1 (12)RELAX ("Ml(ql) = fl"; input : fl; in=output : ql) (13)

end if (14)end procedure

The modi�cation is in line (6). Here �l�1 represents a diagonal matrix R3N(l�1) !

R3N(l�1) (where N(l�1) denotes the number of boxes at l�1). It is de�ned by

�l�1dl�1 = (�l�1;1dl�1;1; � � � ; �l�1;jdl�1;j ; � � � ; �l�1;N(l�1)dl�1;N(l�1))T


with dl�1;j 2 R3; �l�1;j : R

3! R

3and

�l�1;j �

0B@

�l�1;j;1 0 0

0 �l�1;j;2 0

0 0 �l�1;j;3

1CA

(�l�1;j;k 2 R ; k = 1; 2; 3):

For our particular semiconductor problem the �l�1;j;k are de�ned by:

�l�1;j;1 = 1; (6.38a)

�l�1;j;2 = min f2�l�1;j ; 1g ; �l�1;j 2 R ; (6.38b)

�l�1;j;3 = min f2�l�1;j ; 1g ; �l�1;j 2 R ; (6.38c)

where

�l�1;j � exp(�( l�1j � �l�1n;j ))= maxi=0;�1

exp(�( l2j+i � �ln;2j+i)); (6.39a)

�l�1;j � exp(�(�l�1p;j � l�1j ))= maxi=0;�1

exp(�(�lp;2j+i � l2j+i)): (6.39b)

The superscripts l � 1; l refer to l�1;l respectively.The �rst component of the restricted residual needs not to be adjusted. The de�ni-

tion originates from the evaluation of expression (6.37). By means of (6.38b){(6.38c)the numbers �l�1;j;2 and �l�1;j;3 are rounded o� upwards to 1 when �l�1;j ; �l�1;jare � 1

2 . Summarizing, we observe the following from (6.38) and the body of MFAS:

1. Where ql is smooth, dl�1 will not be suppressed.

2. Where ql depicts a steep gradient, dl�1 may be strongly suppressed.

3. Let 0 be some �xed grid, then, for l ! 1, the matrix �l�1 becomes asymp-totically the identity matrix.

4. By a proper local mesh re�nement the suppression of dl�1 will decrease.

The performance of the modi�ed FAS-algorithm will be shown and discussed in Sec-tion 6.7.

In the nonlinear multigrid algorithm as proposed by Hackbusch [7, p. 187], therestricted residual dl�1 is divided by a global parameter s � 1 and the resulting coarsegrid correction is multiplied by s. The division by an appropriate s ensures that theright-hand side of the coarse grid equation is within an appropriate range of the coarsegrid operator Ml�1. There are two main di�erences with our approach. First, thesame number s is used at each di�erent box. Second, within our class of problems wehave to omit the multiplication of the correction by s. Such a multiplication wouldresult in a far too large correction and thereby a dip or peak in the �ne grid solution.

138 Chapter 6

In recent work of Hackbusch and Reusken [8] a global parameter is proposed bywhich the coarse grid correction should be damped. For a limited class of problemsan appropriate can be computed. Important di�erences with our approach are thefollowing:

1. is a damping parameter for the correction, instead of the residual.

2. is a global parameter, i.e. the same is used at each di�erent box.

3. After su�cient FAS-sweeps the damping parameter converges to 1, the pa-rameters �l�1;j;2, �l�1;j;3 do not and should not converge to 1.

4. The -parameter is meant to enlarge the domain of guaranteed convergenceon the analogy of the damping parameter in the Newton method; the �l�1-operator is meant to deal with discrepancies between the operators Ml�1 andMl due to rapidly varying problem coe�cients.

6.7 Numerical results

In this section we investigate the performance of our nonlinear multigrid-algorithm.We focus our attention on the e�ects of local suppressing of the restricted residualand the choice of the coarse grid solution. The residual norm (k � kres) that we useis the maximum norm of the scaled residual. At level l the said scaling is done bymultiplying the residual at each box with the inverse of the 3� 3-matrix

@Mljxj�1=2

@( lj ; �ln;j ; �

lp;j)

T

!:

The performance of the MFAS-algorithm is shown in Table 6.3. In the heading of thetable we use the following abbreviations:

Case See Table 6.1.

qoldH : De�ned by : : : See Section 6.5.4.Without � No local suppression of the restricted residual is

applied.With � Local suppression of the restricted residual is applied

on all coarser grids.# MFAS, 10�1 red. The average number of MFAS-sweeps necessary to

obtain an additional reduction factor 10�1 of theresidual-norm after the application of BOX-FMG.

After FMG The last column shows the scaled norm of theresidual, after application of BOX-FMG (see Sec-tion 6.5.5, = 1). In each case > 0 we obtain astarting approximation of the solution on the coars-est grid by means of continuation and application ofNewton-CSGS (see Section 6.4.2).


For Table 6.3 the multigrid procedures are applied with 3 grids, with N = 16, 32,64, respectively. In the event of no convergence the symbol ? is written. We observe

Table 6.3: Performance of MFAS; use of 3 grids: N = 16; 32; 64; respectively.

qoldH : De�ned by qoldH : De�ned by qnewH

full-weighting of previous CGCWithout With Without With With

� � � � �Case

(#MFAS, (#MFAS, (#MFAS, (#MFAS, (After10�1 red.) 10�1 red.) 10�1 red.) 10�1 red.) FMG)

0 0.80 0.80 0.66 0.66 4:1E � 41 1.07 1.07 0.88 0.73 8:8E � 42 ? 1.15 ? 0.85 1:2E � 33 ? ? ? 1.03 1:2E � 34 ? ? ? 1.06 7:1E � 45 ? ? ? 0.89 5:7E � 46 ? ? ? 0.89 5:7E � 47 ? 1.38 ? 0.89 5:7E � 48 1.40 1.37 0.89 0.89 5:6E � 49 1.54 1.37 0.87 0.90 5:7E � 410 2.59 2.52 0.88 0.82 1:1E � 311 1.22 1.22 1.25 1.25 1:3E � 312 1.75 1.74 2.01 2.01 9:1E � 413 4.66 4.66 2.19 2.19 1:9E � 314 2.44 2.43 1.76 1.77 2:5E � 3

that the use of the �-operator, combined with a proper choice of the coarse gridsolution, gives convergence for all cases. In the cases 3{6 the use of the �-operator isessential for convergence. The use of the �-operator does not slow down convergencein the cases where it is not needed (case 0{2 and 7{14). We observe further thatapparently the full weighting approximation of the �ne grid solution on the coarsegrid may be a poor one. Experiments for more and �ner grids showed almost identicalresults for Table 6.3. Further we observe that mere application of BOX-FMG, withoutfurther MFAS-sweeps, already gives fairly accurate results which may be well enoughfor practical purposes.Figure 6.6 shows the electrostatic potential as computed on a grid of 128 boxes.

For two typical cases, case 4 and case 12, we investigate the grid-dependence of themultigrid convergence. In Figure 6.7 we show the 10-logarithm of the scaled residualnorms after subsequent FAS-sweeps, starting from the result obtained by BOX-FMG.The coarsest grid contains 16 boxes; for the �nest grid we take 32, 64, 128 and 256

140 Chapter 6

Figure 6.6: The electrostatic potential on a 128-grid.


boxes, respectively; � is applied (without application of � case 4 persistently depictsdivergence). We observe that the multigrid convergence becomes grid-independentwhen the mesh-size of the �nest grid decreases. This indicates that the semiconductorproblem has been treated correctly at each multigrid stage. Hereby it is shown thateven for the strongly nonlinear (and particular hard) problem it is possible to composea multigrid method with optimal multigrid e�ciency. Of course, considered in onespace dimension only, competitive methods are available. However, the multigridmethod as developed in this paper o�ers several clues for the foundation of a MG-algorithm which solves the semiconductor problem also in more space dimensions witha computational complexity that is linear in the number of gridpoints. In order to give

Figure 6.7: Multigrid convergence histories; the coarsest grid numbers 16 boxes: left:(a) case 4; right: (b) case 12.

some insight into the behaviour of the �-operator, we show in Figure 6.8 the solution-components and �n for case 4 on a 64-grid and a graph of �2;j;2 (see (6.38b)). Weobserve the typical behaviour that �2;j;2 equals 1 almost everywhere, except for someisolated points.

6.8 Conclusions

We �nd, by deriving explicit expressions for the entries of the Jacobian, that thelinearization of the Scharfetter-Gummel discretization scheme contains a weak spot.When applying full multigrid followed by FAS/NMGM-iterations to our one-dimen-sional transistor-problem, we �nd a serious lack of robustness which is explained bythe strong nonlinearity of the discretized problem. This di�culty is met by adap-tation of the coarse grid correction, which looks to be equally applicable for thehigher-dimensional case. A proper choice of the coarse grid solutions is of importancetoo, e.g., the full weighting approximation is not satisfactory. Furnished with the im-provements as proposed, we obtain a robust multigrid algorithm with a convergence

142 Chapter 6

Figure 6.8: The components and �n of the solution for case 4 and the corresponding�.

which is independent of the mesh-size.

Acknowledgments

The author wishes to thank P.W. Hemker and P. Wesseling for their useful comments,and Ms. M. Middelberg for typing the original version of the manuscript.

References

[1] R.E. Bank, J.W. Jerome and D.J. Rose, Analytical and numerical aspects ofsemiconductor device modelling, in: R. Glowinski and J.L. Lions, eds., Comput-

ing Methods in Applied Sciences and Engineering (North-Holland, Amsterdam,1982) 593{597.

[2] R.E. Bank, H.D. Mittelmann, Continuation and multi-grid for nonlinearelliptic systems, in: W. Hackbusch and U. Trottenberg, eds., Multigrid Methods

II, Lecture Notes in Mathematics 1228 (Springer Verlag, Berlin, 1986) 23{37.

[3] R.E. Bank, D.J. Rose, Analysis of a multilevel iterative method for nonlinear�nite element equations, Math. Comp. 39 (1982) 453{465.


[4] R.E. Bank, D.J. Rose and W. Fichtner, Numerical methods for semicon-ductor device simulation, SIAM J. Sci. Statist. Comput. 4 (1983) 416{435.

[5] A. Brandt, Guide to multigrid development, in: W. Hackbusch and U. Trot-tenberg, eds., Multigrid Methods, Lecture Notes in Mathematics 960 (Springer,Berlin, 1982) 220{312.

[6] S.P. Gaur and A. Brandt, Numerical solution of semiconductor transportequations in two dimensions by multi-grid method, in: R. Vichnevetsky, ed.,Advances in Computer Methods for Partial Di�erential Equations II (IMACS,New Brunswick, 1977) 327{329.

[7] W. Hackbusch, Multi-Grid Methods and Applications (Springer, Berlin, 1985).

[8] W. Hackbusch, A. Reusken, On global multigrid convergence for nonlinearproblems, in: W. Hackbusch, ed., Robust Multigrid Methods, Notes on NumericalFluid Mechanics 23 (Vieweg Verlag, Braunschweig, 1989).

[9] P.W. Hemker, A nonlinear multigrid method for one-dimensional semiconduc-tor device simulation: results for the diode, J. Comput. Appl. Math. 30 (1990)117{126.

[10] S.J. Polak, C. Den Heijer, W.H.A. Schilders and P.A. Markowich,Semiconductor device modelling from the numerical point of view, Internat. J.Numer. Methods Engrg. 24 (1987) 763{838.

[11] D. Scharfetter and H.K. Gummel, Large-signal analysis of a silicon Readdiode oscillator, IEEE Trans. Electron. Devices 16 (1969) 64{77.

[12] S. Selberherr, Analysis and Simulation of Semiconductor Devices (Springer-Verlag, New York, 1984).

[13] A.S.L. Shieh, Solution of coupled systems of PDEs by the transistorized multi-grid method, in: W. Fichtner, D.J. Rose, eds., Joint Special Issue on Numerical

solutions of VLSI devices (IEEE, New York, 1985).

[14] B.P. Sommeijer, W.H. Hundsdorfer, C.T.H. Everaars, P.J. van der

Houwen, J.G. Verwer, A numerical study of a one-dimensional stationarysemiconductor model, Note NM-8702, CWI, Amsterdam (1987).

[15] D.M. Young, Iterative Solution of Large Linear Systems (Academic Press, NewYork and London, 1971).

[16] P.M. de Zeeuw, Nonlinear multigrid applied to a 1D stationary semiconductormodel, Report NM-R8905, CWI, Amsterdam (1989).

144 Chapter 6

chapter 6 nonlinear multigrid applied to a one-dimensional

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