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Existence and uniqueness of solutions of certain systems of algebraic equations with off-diagonal nonlinearity
Applied Numerical Mathematics 37 (2001) 359–370
Existence and uniqueness of solutions of certain systems of algebraic equations with off-diagonal nonlinearity Jürgen Fuhrmann Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
Abstract Using inverse positivity properties and Brouwer’s fixed point theorem, we derive existence and uniqueness results for certain nonlinear systems of equations with off diagonal nonlinearity. Systems of the type considered arise, e.g., from stable finite volume discretizations of viscous nonlinear conservation laws. Though connected to such an important field of applications, results of this type are seldom used. 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Systems of algebraic equations; Positive matrices; Viscous conservation laws; Finite volume methods; Nonlinear parabolic PDEs
1. Introduction Existence and uniqueness results for nonlinear systems of equations in Rn with arbitrary right-hand sides can be given only under structural restrictions to the system. One well-understood class of systems in this respect has the structure B(u) + Mu = f,
(1.1)
where M is an M-matrix and B : Rn → Rn is an isotone diagonal mapping [9]. This type of systems arises, e.g., from the nonlinear Poisson equation in semiconductor device simulation when discretized with finite volume methods [7,13]. More general nonlinear systems of equations are established when numerically treating nonlinear heat transfer [4] or porous media flow problems [6,10]. When discretized following certain rules, within each time-step of the solution algorithm for these problems we have to solve a system of the form B(u) + A(u) + Q(u) = B(u0 )
(1.2)
with B being an isotone diagonal operator, Q(u) bounded and A(u) being an operator with off-diagonal nonlinearities and certain structural properties. E-mail address: [email protected] (J. Fuhrmann). 0168-9274/01/$ – see front matter 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 2 7 4 ( 0 0 ) 0 0 0 5 2 - 0
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In this paper, under natural assumptions about isotonicity properties for A and B, we prove existence and uniqueness results for (1.2), also for cases with degenerating B or A. The basic idea of the proof is the usage of estimates of inverse matrix norms from the Perron–Frobenius theory in Brouwer’s fixed point theorem. We describe the structural properties of the discrete system which allow to establish these results without referring to the continuous operator. A similar idea has been used in [12] to establish existence results for the finite difference discretization of quasi-linear elliptic problems. In addition to the considerations in [12], we consider discrete problems corresponding to the doubly nonlinear parabolic case with possible degenerations. Further, existence and uniqueness results based on column sum estimates and L1 -norms are provided. As we do not want to overload the present paper, for the motivation from the application point of view we provide just a simple example of a discretization of a doubly nonlinear parabolic problem in one space dimension. The treatment of the higher dimensional unstructured mesh case together with results about stability and maximum principles based on the estimates given in the present paper the reader can find together with numerical examples in [5]. So far, we did not consider solution algorithms from the theoretical point of view. The elliptic case can be tackled in a similar way replacing weak diagonal dominance with irreducible diagonal dominance. It is subject of the forthcoming paper [3]. The paper is organized as follows. In Section 2 after introducing the notation of a Dirichlet problem in a finite-dimensional setting, we prove a basic existence result using Brouwer’s fixed point theorem. Section 3 contains existence results based on L∞ -estimates and weakly rowwise diagonal dominance. Section 4 is devoted to existence and uniqueness considerations in the case of L1 -estimates and weakly columnwise diagonal dominance. To illustrate which application we have in mind, Section 5 contains an example of a finite difference scheme for a nonlinear parabolic problem in a one-dimensional domain. Appendix A explains our notations, and for the convenience of the reader, collects some well-known definitions and implications of the Perron–Frobenius theory.
2. Introductory results
Throughout the paper, by C(Rn , Rn ), respectively C(Rn , M(n)), we denote the spaces of componentwise continuous vector-, respectively matrix-, valued functions depending on vectors. C 1 denotes the corresponding spaces of continuously differentiable functions. Further, I denotes the identity operator in Rn or any of its subspaces. Having in mind the application of our theory in the framework of numerical methods for the solution of initial boundary value problems for partial differential equations, we introduce here the notation of a Dirichlet problem. We do this strictly algebraically, as it can be implemented in numerical codes as well. In this case, a Dirichlet problem is posed just by fixing the values of a subset D of unknowns and regarding only the equations corresponding to the complementary subset of unknowns I = {1, . . . , n} \ D.
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So, assume a partition of the index set {1, . . . , n} = D ∪ I into the sets of “Dirichlet nodes” and “interior nodes”. For a given nonlinear operator A : Rn → Rn , the splitting Rn = RD ⊕ RI generated by the coordinate projections induces a splitting of an operator equation A(u) = f into the system AD (uD , uI ) = fD ,
(2.1)
AI (uD , uI ) = fI ,
where for any vector u ∈ Rn , uD ∈ RD and uI ∈ RI denote the coordinate projections. The following definition we introduce mainly for convenience and better understanding of our notation. Definition 2.1. For an operator A : Rn → Rn , a given right-hand side f ∈ Rn and a “boundary value” ud ∈ RD , we define the Dirichlet problem
A(u) = f, u|D = ud
(2.2)
to be the system of equations uD = ud ,
(2.3)
AI (uD , uI ) = fI .
Here, we state a mean value theorem which several times is applied in this paper. We are indebted to L. Recke for the hint to use this type of mean value problem within our considerations. Theorem 2.1. Let A : Rn → Rn be continuously differentiable with the derivative A . Then for any v) defined by u, v ∈ Rn , the operator A(u, A(u, v) =
1
A v + θ(u − v) dθ
(2.4)
0
satisfies A ∈ C(Rn × Rn , M(n)) and v)(u − v). A(u) − A(v) = A(u,
(2.5)
v) If A (u) independently of u belongs to any of the subsets of M(n) defined in Appendix A, then A(u, does.
Proof. The existence of A satisfying (2.5) is proven in [9]. The nonzero pattern, the sign pattern and the row/column-sum conditions are straightforward. ✷ The following theorem is the central building block of the theory developed in this paper. It basically states that the equation u + K(u)u = Q(u)
(2.6)
has a solution if Q is bounded and continuous, and K is a Z-matrix continuously depending on u which is row or column weakly diagonally dominant.
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Theorem 2.2. Let K ∈ C(Rn , M(n)) and Q ∈ C(Rn , Rn ). For any u ∈ Rn assume either K(u) ∈ Z+ r (n) and
Q(u)
Q+ ∞
(2.7a)
K(u) ∈ Z+ c (n) and
Q(u) Q+ . 1 1
(2.7b)
∞
or
with either Then Eq. (2.6) has a solution u
u∞ Q+ ∞
(2.8a)
u1 Q+ 1,
(2.8b)
or
respectively. Proof. The solution u would be a fixed point of the mapping
L(u) = I + K(u)
−1
Q(u).
For ξ ∈ {1, ∞} Theorems A.1, A.2 give us I + K(u) −1 1, ξ
hence
L(u) Q+ . ξ ξ
(2.9)
The entries of (I + K(u))−1 are rational functions of the entries of I + K(u). Inverse positivity bounds them below by zero, and the norm estimate together with the explicit norm representation (A.2) + bounds them from above. Hence for K ∈ Z+ r (n), respectively K ∈ Zc (n), these rational functions cannot have poles and need to be continuous. Thus the mapping L is continuous as well. The closed ball {u ∈ Rn | uξ Q+ ξ } is convex and compact, and by the estimate (2.9) it is mapped onto itself. So we can apply Brouwer’s fixed point theorem which ensures the existence of the desired fixed point which at the same time admits the estimates (2.8a) and (2.8b), respectively. ✷ 3. Nonnegative row sums and L∞ -estimates In this section, based on row-sum and L∞ -estimates, we generalize Theorem 2.2 to the case corresponding to doubly nonlinear parabolic equations. An introductory example for this case one finds in Section 5. To be applicable to a wider range of problems, we prove our results for the case of Dirichlet boundary conditions. For this notation, see Definition 2.1. Theorem 3.1. Let K ∈ C(Rn , M(n)) and Q ∈ C(Rn , Rn ). For any u ∈ Rn assume and
K(u) ∈ Z+ r (n)
(3.1a)
Q(u)
(3.1b)
∞
Q+ .
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Further, let B ∈ C 1 (Rn , Rn ) be a diagonal operator with B (u)
∞
B − > 0.
(3.1c)
Then for any u0 ∈ Rn , for any (possibly empty) subset D ⊂ {1, . . . , n} and any ud ∈ RD , the Dirichlet problem
B(u) + K(u)u = Q(u) + B(u0 ), u|D = ud
(3.2)
has a solution u with Q+ + max u0,I ∞ , ud ∞ , − B assuming ud ∞ = 0 for D = ∅.
u∞
(3.3)
Proof. According to Definition 2.1, problem (3.2) rewrites as BI (u) + KII a(u)uI = QI (u) + BI (u0 ) − KID (u)ud ,
(3.4a)
where for any K(u) ∈ C(Rn , M(n)),
K=
KDD (u) KID (u)
KDI (u) KII (u)
is the block splitting corresponding to the subdivision of the set of unknowns into the “Dirichlet” and the “interior” parts. Let BI (u, u0 ) be the diagonal matrix defined from BI : RI → RI by the mean value theorem 2.1. It allows us to rewrite Eq. (3.4a) by BI (u, u0 )uI + KII (u)uI = QI (u) + BI (u, u0 )u0,I − KID (u)ud .
(3.4b)
By the identity (2.5), any solution of (3.4b) is a solution of (3.4a). Further, KII (u) ∈ Z+ r (I), and we have a splitting KII (u) = K 0 (u) + K D (u) + K M (u) with K 0 (u) ∈ Z0r (I), K D (u) ∈ D + (I) being the negative row-sum of KID (u) and K M (u) ∈ D + (I) being the—nonnegative by assumption—diagonal matrix established from the difference between the main diagonal entries of KII (u) and the negative sum of the off-diagonal row entries. Let M(u) = BI (u, u0 ) + K D (u) + K M (u),
N(u) = −KID (u)BI (u, u0 ) ,
v0 =
ud u0,I
.
We have M(u) ∈ D ++ (I), N(u) : Rn → RD with nonnegative entries and can rewrite Eq. (3.4b) by M(u)uI + K 0 (u)uI = QI (u) + N(u)v0
(3.4c)
and
uI + M(u)−1 K 0 (u)uI = M(u)−1 QI (u) + N(u)v0 .
(3.4d)
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By Theorem A.1, M(u)−1 K 0 (u) ∈ Z+ r (I). Further, it continuously depends on u. We can estimate Q+ , B− M(u)−1 N(u)v0 max u0,I ∞ , ud ∞ ∞ M(u)−1 QI (u)
∞
because
v0 ∞ = max u0,I ∞ , ud ∞
and the row-sums of M(u)−1 N(u) are less or equal then one by construction. Hence, by Theorem 2.2, Eq. (3.4d) has a solution which can be estimated by (3.3). It is a solution of (3.4a) as well, and by definition, it is a solution of the Dirichlet problem (3.2). If D = ∅, we can everywhere omit ud , KID and K D , thus obtaining the result also for this case. ✷ Remark 3.1. Please note, that we did not demand any non-degeneracy condition for K(u). This allows to apply our result also in the case of degenerating diffusion equations, where rows of K(u) can become zero. The case of degeneracy under the time derivative which occurs, e.g., in saturated/unsaturated flow calculations [10] is covered by the next theorem. Theorem 3.2. Let K ∈ C(Rn , M(n)). For any u ∈ Rn assume K(u) ∈ Z+ r (n).
(3.5a)
Further, let B ∈ C 1 (Rn , Rn ) be a diagonal operator with B (u) B − 0.
(3.5b)
Then for any u0 ∈ Rn , for any (possibly empty) subset D and any ud ∈ RD , the Dirichlet problem
B(u) + K(u)u = B(u0 ), u|D = ud
has a solution u with
uI ∞ max u0,I ∞ , ud ∞
(3.6)
(3.7)
assuming ud ∞ = 0 for D = ∅. Proof. For ε > 0 let Bε (u) = B(u) + εu. Then after Theorem 3.1, the Dirichlet problem
Bε (u) + K(u)u = Bε (u0 ), u|D = ud
has a solution uε which can be estimated by
uε,I ∞ max u0,I ∞ , ud ∞ . Hence, we can find an element uI admitting the estimate (3.7) and a sequence εn with lim εn = 0,
n→∞
lim uεn = u.
n→∞
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As all operators involved continuously depend on u,
u =
ud uI
necessarily is a solution of (3.6).
✷
Remark 3.2. Unfortunately, using the type of estimates of this section, we are not able to prove uniqueness. At the other hand, in the case of Theorem 3.2, for both degenerating B and K, we have to expect multiple solutions. 4. Nonnegative column-sums and L1 -estimates In this section, we apply Theorem 2.2 to the case of nonnegative column-sums and L1 -estimates. The nonlinear operators corresponding to this case come from stable finite volume discretizations of nonlinear convection–diffusion problems. Under certain conditions, the example from Section 5 is valid for this case, as well. For our notation of the Dirichlet problem, please consult Definition 2.1. Theorem 4.1. Let A : Rn → Rn be continuously differentiable with derivative A (u) ∈ Z+ c (n). Further, let B ∈ C 1 (Rn , Rn ) be an isotone diagonal homeomorphism. Let Q ∈ C(Rn , Rn ) be bounded with Q(u)1 Q+ . Then for u0 ∈ Rn , for any (possibly empty) subset D ⊂ {1, . . . , n}, and ud ∈ RD , the Dirichlet problem
B(u) + A(u) = Q(u) + B(u0 ), u|D = ud
(4.1)
has a solution u which in the case D = ∅ admits the estimate B(u) B(u0 ) + Q+ + A B −1 (0) . 1 1 1
(4.2)
Proof. First, regard the case D = ∅. As B is a homeomorphism, we can perform a variable substitution v = B(u). Let AB (v) = A(B −1 (v)). We have the identity
A B (v) = A B −1 (v) B −1 (v) thus because of Theorem A.2, the isotonicity and the continuity of B and hence of its inverse, AB fulfills the conditions of the theorem. Thus we restrict ourselves to the problem v + AB (v) = QB (v) + v0
(4.3a)
with QB (v) = Q(B −1 (v)) and v0 = B(u0 ). Let AB0 (v) = AB (v) − AB (0) and QB0 (v) = QB (v) − AB (0). We get the equivalent problem v + AB0 (v) = QB0 (v) + v0 ,
(4.3b)
where AB0 besides of the condition of the theorem fulfills AB0 (0) = 0 and QB0 is bounded by QB0 (v) Q+ = Q+ + AB (0) = Q+ + A B −1 (0) . B0 1 1 1
(4.3c)
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Let AB0 (v, w) ∈ Z+ c (n) be defined as in Theorem 2.1. Then we can rewrite Eq. (4.3b) as v + AB0 (v, 0)v = QB0 (v) + v0 ,
(4.3d)
which after Theorem 2.2 has a solution v with the estimate v Q+ + v0 1 . B0 1
= B −1 (v) is a solution of (4.1) admitting the It is a solution of (4.3b), too and consequently, u estimate (4.2). For the Dirichlet problem, we regard
BI (u) + AI (u) = QI (u) + BI (u0 ). We have a splitting A (u) =
ADD AID
ADI AII
(4.3e)
.
+ As A (u) ∈ Z+ c (n), AII ∈ Zc (I), thus for fixed uD , (4.3e) as an equation in uI fulfills the conditions of the first part of the proof. ✷
Remark 4.1. Again, degeneracy of the “main part” A is allowed. But unlike in Section 3, the existence proof for degenerating B does not work here. Further, as L1 -estimates in a certain sense correspond to mass conservation in a closed system, decent estimates for the Dirichlet not necessarily have to be expected. Theorem 4.2. Assume that the conditions of Theorem 4.2 are fulfilled, that Q is diagonal and continuously differentiable with Q 0, and that for any u ∈ Rn , B (u) ∈ D++ (n). Then the solution of (4.1) is unique. Proof. For u, v being two solutions of (4.1) after Theorem 2.1 we can write B(u, v)(u − v) + A(u, v)(u − v) − Q(u, v)(u − v) = 0. v) + A(u, v) − Q(u, v) is an M-matrix, thus necessarily, u = v. By Theorem A.2, B(u,
✷
5. Example: nonlinear diffusion in inhomogeneous domains To motivate our considerations, we give a simple example. In the fields of nonlinear heat transfer and of porous media flow simulation, we have to solve numerically initial boundary value problems of the form
∂t b x, u(x, t) − ∇ · d x, u(x, t) ∇u(x, t) = 0
(5.1)
with b monotonically increasing and continuously differentiable in u and d continuous and nonnegative. For simplicity, we assume here a one-dimensional domain Ω = (0, 2) with two subdomains Ω1 = (0, 1) and Ω2 = (1, 2) with homogeneous Neumann boundary conditions, and an initial value u0 . Assume that b(x, u) = bm (u) and d(x, u) = d m (u) on Ωm , respectively.
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If we introduce an implicit Euler time discretization with time-step τ , in the νth time-step we have to solve a problem of the type
b(x, uν ) − τ ∇ · d(x, uν )∇uν = b(x, uν−1 ).
(5.2)
To be able to do this, we introduce the following control volume scheme which is formulated using the flux functions g 1 , g 2 which are introduced below. Let N = 2n, h = 1/n and for i = 0, . . . , 2n, let xi = ih, and ui ≈ u(xi ). Let [0, h/2], . . . , [(i − 12 )h, (i + 12 )h], . . . , [2 − h/2, 2] be the control volumes. Let 1 1 2 b (ui ), 1 b (ui ), B(u)i = h 12 b1 (ui ) + b2 (ui ) , b2 (ui ), 1 2 2
and
b (ui ),
i = 0, 0 < i < n, i = n, n < i < 2n, i = 2n,
g 1 (ui , ui+1 ), g 1 (ui , ui−1 ) + g 1 (ui , ui+1 ), τ 1 A(u)i = g (ui , ui−1 ) + g 2 (ui , ui+1 ), h g 2 (ui , ui−1 ) + g 2 (ui , ui+1 ),
g 2 (ui , ui−1 ),
i = 0, 0 < i < n, i = n, n < i < 2n, i = 2n.
Then the system of equations B(uν ) + A(uν ) = B(uν−1 ) can be regarded as an approximation to the continuous system yielded by integrating our initial equation over each of the control volumes and approximating the flux between two discretization nodes by the flux functions. We can imagine that in the ideal case, the discrete solution is glued together from solutions of the boundary value problems ∇ · d(x, u)∇u = 0,
u|xi = ui , u|xi+1 = ui+1
(5.3)
in [xi , xi+1 ]. In our example, for d m = D m , this approach yields the flux functions D m (ui ) − D m (ui+1 ) = ∇D m (u)[xi ,xi+1 ] . h The so defined operator A(u) fulfills the conditions of Theorem 4.1, this proves existence and uniqueness of solutions of the discrete problem and further yields the L1 -estimate which implicates L1 -stability of the discrete system independent on the size of the time-step. Defining
g m (ui , ui+1 ) =
m m D (ui ) − D (ui+1 ) 1 , ui = ui+1 , k m (ui , ui+1 ) = ui − ui+1 h d m (u ), ui = ui+1 i
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for D ∈ C 1 (R, R) allows to rewrite A(u) = K(u)u with K ∈ Z0r (n) continuously depending on u: −k 1 (ui , uj ), −k 1 (ui , uj ), 2 τ −k 2 (ui , uj ), K(u)ij = −k (ui , uj ), h −kil (u) l=0...2n,l = i
0,
0 i < n, j = i + 1, 0 < i n, j = i − 1, n i < 2n, j = i + 1, n < i 2n, j = i − 1, i = j, else.
Thus, the conditions of Theorem 3.2 are fulfilled, and we get the L∞ -estimate implying unconditional L∞ -stability. A common way in the case that we are not able to obtain the function D is to use the arithmetical mean of the solutions of the neighboring nodes:
g (ui , ui+1 ) = d m
m
ui + ui+1 (ui − ui+1 ) . 2 h
In this case, for A and B, the conditions of Theorem 3.2 are fulfilled. Thus for arbitrarily large timesteps, the time-step problems are solvable, and, furthermore, the estimate (3.7) yields unconditional L∞ -stability. However, in this case the L1 -theory and thus the uniqueness result is not available. The practical experience however does not indicate that multiple solutions are a problem in this case [4,6]. We remark that in the case of homogeneous materials, i.e., b1 = b2 and d 1 = d 2 we could use the variable transformation v = D(u). In this case, the results for the case of diagonal nonlinearity [9] would be sufficient. In [5], using a Voroni box based control volume ansatz, this approach is described for simplicial meshes in higher space dimensions. For a real application, it will be necessary to solve the discrete nonlinear systems. In [4–6] we successfully use Newton’s method with feedback to the time-step size control.
Appendix A. Linear algebra A good reference for the following results still is [14]. Let M(n) be the set of all real n × n-matrices. For u = (ui )i=1...n ∈ Rn , let u1 =
n
|ui |,
i=1
n
u∞ = max |ui |. i=1
(A.1)
Let A ∈ M(n). Then n Au1 n = max |aij | = AT ∞ , j =1 0=u∈Rn u1 i=1
A1 = sup
n Au∞ n = max |aij | = AT 1 . A∞ = sup i=1 0=u∈Rn u∞ j =1
(A.2)
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Define the following subsets of M(n): • Z(n) ⊂ M(n)—nonnegative main diagonal and nonpositive off-diagonal elements (often denoted as Z n×n ); we call them Z-matrices here, • Z+ r (n) ⊂ Z(n)—rowwise weakly diagonally dominant, • Z0r (n) ⊂ Z+ r (n)—row-sum zero, • Z++ (n) ⊂ Z(n)—rowwise strictly diagonally dominant, r • D(n) ⊂ M(n)—diagonal, • D+ (n) ⊂ D(n)—nonnegative diagonal, • D++ (n) ⊂ D+ (n)—positive diagonal. To all rowwise defined subsets, the corresponding columnwise defined subsets are denoted using the subscript c . For a subset X ⊂ {1, . . . , n} we write M(X), Z(X) etc. to denote the sets of matrices corresponding to this subset. A = (aij )i,j =1...n ∈ M(n) is called positive (A > 0) if aij > 0, i, j = 1, . . . , n, and nonnegative (A 0) if aij 0, i, j = 1, . . . , n. A matrix A ∈ Z(n) is called (nonsingular) M-matrix if A−1 0. ++ (n). Then Theorem A.1. Let A ∈ Z+ r (n), D ∈ D + (i) DA ∈ Zr (n), (ii) A + D ∈ Z++ r (n) is an M-matrix, (iii) (I + A)−1 ∞ 1.
Proof. The first part is obvious, for the second, see [1, lemma 6.2]. For the third part, let first A ∈ Z0r (n). Assume that (I + A)−1 ∞ > 1. We know that (I + A)−1 has positive entries. Then for αij being the entries of (I + A)−1 , n
max i=1
n
αij > 1.
j =1
Let k be a row where the maximum is reached. Let e = (1, . . . , 1)T . Then for v = (I + A)−1 e we have that v > 0, vk > 1 and vk vj for all j = k. The kth equation of e = (I + A)v then looks like 1 = vk + vk
j =k
|akj | −
|akj |vj vk + vk
j =k
j =k
|akj | −
|akj |vk = vk > 1.
j =k
This contradiction enforces (I + A)−1 ∞ 1. + 0 Let now A ∈ Z+ r (n). The we can write A = D + A0 with D ∈ D (n) and A ∈ Zc (n). We have I + A = I + D + A0 = (I + D)(I + D)−1 (I + D + A0 ) = (I + D)(I + AD0 ) with AD0 = (I + D)−1 A0 ∈ Z0c (n). Thus
(I + A)−1 = (I + AD0 )−1 (I + D)−1 I + D)−1 1, ∞ ∞ ∞
because all main diagonal entries of I + D are greater or equal to 1.
✷
Remark A.1. The third part of the theorem comes from the M-criterion of [8,11], both refer to [2].
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++ Theorem A.2. Let A ∈ Z+ (n). Then c (n), D ∈ D + (i) AD ∈ Zc (n), (ii) A + D ∈ Z++ c (n) is an M-matrix, −1 (iii) (I + A) 1 1.
Proof. Apply Theorem A.1 to AT .
✷
References [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10] [11] [12] [13] [14]
O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, 1994. E. Bohl, Finite Modelle gewöhnlicher Randwertaufgaben, Teubner, Stuttgart, 1981. J. Fuhrmann, Finite volume solution of nonlinear elliptic PDEs, in preparation, 2000. J. Fuhrmann, D. Hömberg, Numerical simulation of surface heat treatments, J. Numer. Methods Heat Fluid Flow 6 (1999) 705–724. J. Fuhrmann, H. Langmach, Stability of solutions of time-implicit finite volume schemes for viscous nonlinear conservation laws, Appl. Numer. Math., accepted (2000). J. Fuhrmann, On numerical solution methods for nonlinear parabolic problems, in: R. Helmig, W. Jäger, W. Kinzelbach, P. Knabner, G. Wittum (Eds.), Modeling and Computation in Environmental Sciences, Proceedings of the 1st GAMM-Seminar at ICA Stuttgart, October 12–13, 1995, Notes on Numerical Fluid Mechanics, Vol. 59, Vieweg, Braunschweig, 1997, pp. 170–180. H. Gajewski, H. Langmach, G. Telschow, K. Zacharias, Der 2D-Bauelementesimulator TOSCA—Handbuch, Technical documentation, Karl-Weierstraß-Institut für Mathematik, Berlin, 1989, unpublished. Ch. Großmann, H.-G. Roos, Numerik partieller Differentialgleichungen, Teubner Studienbücher Mathematik, Teubner, Stuttgart, 1992. J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. L.A. Richards, Capillary conduction of liquids through porous mediums, Physics 1 (1931) 318–333. H.-G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer Series in Computational Mathematics, Vol. 24, Springer, 1996. R.S. Stepleman, Difference analogues of quasi-linear elliptic Dirichlet problems with mixed derivatives, Math. Comp. 25 (114) (1971) 257–269. S. Selberherr, Analysis and Simulation of Semiconductor Devices, Computational Microelectronics, Springer, Wien, 1984. R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.
Existence and uniqueness of solutions of certain systems of algebraic equations with off-diagonal nonlinearity Jürgen Fuhrmann Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
Abstract Using inverse positivity properties and Brouwer’s fixed point theorem, we derive existence and uniqueness results for certain nonlinear systems of equations with off diagonal nonlinearity. Systems of the type considered arise, e.g., from stable finite volume discretizations of viscous nonlinear conservation laws. Though connected to such an important field of applications, results of this type are seldom used. 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Systems of algebraic equations; Positive matrices; Viscous conservation laws; Finite volume methods; Nonlinear parabolic PDEs
1. Introduction Existence and uniqueness results for nonlinear systems of equations in Rn with arbitrary right-hand sides can be given only under structural restrictions to the system. One well-understood class of systems in this respect has the structure B(u) + Mu = f,
(1.1)
where M is an M-matrix and B : Rn → Rn is an isotone diagonal mapping [9]. This type of systems arises, e.g., from the nonlinear Poisson equation in semiconductor device simulation when discretized with finite volume methods [7,13]. More general nonlinear systems of equations are established when numerically treating nonlinear heat transfer [4] or porous media flow problems [6,10]. When discretized following certain rules, within each time-step of the solution algorithm for these problems we have to solve a system of the form B(u) + A(u) + Q(u) = B(u0 )
(1.2)
with B being an isotone diagonal operator, Q(u) bounded and A(u) being an operator with off-diagonal nonlinearities and certain structural properties. E-mail address: [email protected] (J. Fuhrmann). 0168-9274/01/$ – see front matter 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 2 7 4 ( 0 0 ) 0 0 0 5 2 - 0
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In this paper, under natural assumptions about isotonicity properties for A and B, we prove existence and uniqueness results for (1.2), also for cases with degenerating B or A. The basic idea of the proof is the usage of estimates of inverse matrix norms from the Perron–Frobenius theory in Brouwer’s fixed point theorem. We describe the structural properties of the discrete system which allow to establish these results without referring to the continuous operator. A similar idea has been used in [12] to establish existence results for the finite difference discretization of quasi-linear elliptic problems. In addition to the considerations in [12], we consider discrete problems corresponding to the doubly nonlinear parabolic case with possible degenerations. Further, existence and uniqueness results based on column sum estimates and L1 -norms are provided. As we do not want to overload the present paper, for the motivation from the application point of view we provide just a simple example of a discretization of a doubly nonlinear parabolic problem in one space dimension. The treatment of the higher dimensional unstructured mesh case together with results about stability and maximum principles based on the estimates given in the present paper the reader can find together with numerical examples in [5]. So far, we did not consider solution algorithms from the theoretical point of view. The elliptic case can be tackled in a similar way replacing weak diagonal dominance with irreducible diagonal dominance. It is subject of the forthcoming paper [3]. The paper is organized as follows. In Section 2 after introducing the notation of a Dirichlet problem in a finite-dimensional setting, we prove a basic existence result using Brouwer’s fixed point theorem. Section 3 contains existence results based on L∞ -estimates and weakly rowwise diagonal dominance. Section 4 is devoted to existence and uniqueness considerations in the case of L1 -estimates and weakly columnwise diagonal dominance. To illustrate which application we have in mind, Section 5 contains an example of a finite difference scheme for a nonlinear parabolic problem in a one-dimensional domain. Appendix A explains our notations, and for the convenience of the reader, collects some well-known definitions and implications of the Perron–Frobenius theory.
2. Introductory results
Throughout the paper, by C(Rn , Rn ), respectively C(Rn , M(n)), we denote the spaces of componentwise continuous vector-, respectively matrix-, valued functions depending on vectors. C 1 denotes the corresponding spaces of continuously differentiable functions. Further, I denotes the identity operator in Rn or any of its subspaces. Having in mind the application of our theory in the framework of numerical methods for the solution of initial boundary value problems for partial differential equations, we introduce here the notation of a Dirichlet problem. We do this strictly algebraically, as it can be implemented in numerical codes as well. In this case, a Dirichlet problem is posed just by fixing the values of a subset D of unknowns and regarding only the equations corresponding to the complementary subset of unknowns I = {1, . . . , n} \ D.
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So, assume a partition of the index set {1, . . . , n} = D ∪ I into the sets of “Dirichlet nodes” and “interior nodes”. For a given nonlinear operator A : Rn → Rn , the splitting Rn = RD ⊕ RI generated by the coordinate projections induces a splitting of an operator equation A(u) = f into the system AD (uD , uI ) = fD ,
(2.1)
AI (uD , uI ) = fI ,
where for any vector u ∈ Rn , uD ∈ RD and uI ∈ RI denote the coordinate projections. The following definition we introduce mainly for convenience and better understanding of our notation. Definition 2.1. For an operator A : Rn → Rn , a given right-hand side f ∈ Rn and a “boundary value” ud ∈ RD , we define the Dirichlet problem
A(u) = f, u|D = ud
(2.2)
to be the system of equations uD = ud ,
(2.3)
AI (uD , uI ) = fI .
Here, we state a mean value theorem which several times is applied in this paper. We are indebted to L. Recke for the hint to use this type of mean value problem within our considerations. Theorem 2.1. Let A : Rn → Rn be continuously differentiable with the derivative A . Then for any v) defined by u, v ∈ Rn , the operator A(u, A(u, v) =
1
A v + θ(u − v) dθ
(2.4)
0
satisfies A ∈ C(Rn × Rn , M(n)) and v)(u − v). A(u) − A(v) = A(u,
(2.5)
v) If A (u) independently of u belongs to any of the subsets of M(n) defined in Appendix A, then A(u, does.
Proof. The existence of A satisfying (2.5) is proven in [9]. The nonzero pattern, the sign pattern and the row/column-sum conditions are straightforward. ✷ The following theorem is the central building block of the theory developed in this paper. It basically states that the equation u + K(u)u = Q(u)
(2.6)
has a solution if Q is bounded and continuous, and K is a Z-matrix continuously depending on u which is row or column weakly diagonally dominant.
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Theorem 2.2. Let K ∈ C(Rn , M(n)) and Q ∈ C(Rn , Rn ). For any u ∈ Rn assume either K(u) ∈ Z+ r (n) and
Q(u)
Q+ ∞
(2.7a)
K(u) ∈ Z+ c (n) and
Q(u) Q+ . 1 1
(2.7b)
∞
or
with either Then Eq. (2.6) has a solution u
u∞ Q+ ∞
(2.8a)
u1 Q+ 1,
(2.8b)
or
respectively. Proof. The solution u would be a fixed point of the mapping
L(u) = I + K(u)
−1
Q(u).
For ξ ∈ {1, ∞} Theorems A.1, A.2 give us I + K(u) −1 1, ξ
hence
L(u) Q+ . ξ ξ
(2.9)
The entries of (I + K(u))−1 are rational functions of the entries of I + K(u). Inverse positivity bounds them below by zero, and the norm estimate together with the explicit norm representation (A.2) + bounds them from above. Hence for K ∈ Z+ r (n), respectively K ∈ Zc (n), these rational functions cannot have poles and need to be continuous. Thus the mapping L is continuous as well. The closed ball {u ∈ Rn | uξ Q+ ξ } is convex and compact, and by the estimate (2.9) it is mapped onto itself. So we can apply Brouwer’s fixed point theorem which ensures the existence of the desired fixed point which at the same time admits the estimates (2.8a) and (2.8b), respectively. ✷ 3. Nonnegative row sums and L∞ -estimates In this section, based on row-sum and L∞ -estimates, we generalize Theorem 2.2 to the case corresponding to doubly nonlinear parabolic equations. An introductory example for this case one finds in Section 5. To be applicable to a wider range of problems, we prove our results for the case of Dirichlet boundary conditions. For this notation, see Definition 2.1. Theorem 3.1. Let K ∈ C(Rn , M(n)) and Q ∈ C(Rn , Rn ). For any u ∈ Rn assume and
K(u) ∈ Z+ r (n)
(3.1a)
Q(u)
(3.1b)
∞
Q+ .
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Further, let B ∈ C 1 (Rn , Rn ) be a diagonal operator with B (u)
∞
B − > 0.
(3.1c)
Then for any u0 ∈ Rn , for any (possibly empty) subset D ⊂ {1, . . . , n} and any ud ∈ RD , the Dirichlet problem
B(u) + K(u)u = Q(u) + B(u0 ), u|D = ud
(3.2)
has a solution u with Q+ + max u0,I ∞ , ud ∞ , − B assuming ud ∞ = 0 for D = ∅.
u∞
(3.3)
Proof. According to Definition 2.1, problem (3.2) rewrites as BI (u) + KII a(u)uI = QI (u) + BI (u0 ) − KID (u)ud ,
(3.4a)
where for any K(u) ∈ C(Rn , M(n)),
K=
KDD (u) KID (u)
KDI (u) KII (u)
is the block splitting corresponding to the subdivision of the set of unknowns into the “Dirichlet” and the “interior” parts. Let BI (u, u0 ) be the diagonal matrix defined from BI : RI → RI by the mean value theorem 2.1. It allows us to rewrite Eq. (3.4a) by BI (u, u0 )uI + KII (u)uI = QI (u) + BI (u, u0 )u0,I − KID (u)ud .
(3.4b)
By the identity (2.5), any solution of (3.4b) is a solution of (3.4a). Further, KII (u) ∈ Z+ r (I), and we have a splitting KII (u) = K 0 (u) + K D (u) + K M (u) with K 0 (u) ∈ Z0r (I), K D (u) ∈ D + (I) being the negative row-sum of KID (u) and K M (u) ∈ D + (I) being the—nonnegative by assumption—diagonal matrix established from the difference between the main diagonal entries of KII (u) and the negative sum of the off-diagonal row entries. Let M(u) = BI (u, u0 ) + K D (u) + K M (u),
N(u) = −KID (u)BI (u, u0 ) ,
v0 =
ud u0,I
.
We have M(u) ∈ D ++ (I), N(u) : Rn → RD with nonnegative entries and can rewrite Eq. (3.4b) by M(u)uI + K 0 (u)uI = QI (u) + N(u)v0
(3.4c)
and
uI + M(u)−1 K 0 (u)uI = M(u)−1 QI (u) + N(u)v0 .
(3.4d)
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By Theorem A.1, M(u)−1 K 0 (u) ∈ Z+ r (I). Further, it continuously depends on u. We can estimate Q+ , B− M(u)−1 N(u)v0 max u0,I ∞ , ud ∞ ∞ M(u)−1 QI (u)
∞
because
v0 ∞ = max u0,I ∞ , ud ∞
and the row-sums of M(u)−1 N(u) are less or equal then one by construction. Hence, by Theorem 2.2, Eq. (3.4d) has a solution which can be estimated by (3.3). It is a solution of (3.4a) as well, and by definition, it is a solution of the Dirichlet problem (3.2). If D = ∅, we can everywhere omit ud , KID and K D , thus obtaining the result also for this case. ✷ Remark 3.1. Please note, that we did not demand any non-degeneracy condition for K(u). This allows to apply our result also in the case of degenerating diffusion equations, where rows of K(u) can become zero. The case of degeneracy under the time derivative which occurs, e.g., in saturated/unsaturated flow calculations [10] is covered by the next theorem. Theorem 3.2. Let K ∈ C(Rn , M(n)). For any u ∈ Rn assume K(u) ∈ Z+ r (n).
(3.5a)
Further, let B ∈ C 1 (Rn , Rn ) be a diagonal operator with B (u) B − 0.
(3.5b)
Then for any u0 ∈ Rn , for any (possibly empty) subset D and any ud ∈ RD , the Dirichlet problem
B(u) + K(u)u = B(u0 ), u|D = ud
has a solution u with
uI ∞ max u0,I ∞ , ud ∞
(3.6)
(3.7)
assuming ud ∞ = 0 for D = ∅. Proof. For ε > 0 let Bε (u) = B(u) + εu. Then after Theorem 3.1, the Dirichlet problem
Bε (u) + K(u)u = Bε (u0 ), u|D = ud
has a solution uε which can be estimated by
uε,I ∞ max u0,I ∞ , ud ∞ . Hence, we can find an element uI admitting the estimate (3.7) and a sequence εn with lim εn = 0,
n→∞
lim uεn = u.
n→∞
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As all operators involved continuously depend on u,
u =
ud uI
necessarily is a solution of (3.6).
✷
Remark 3.2. Unfortunately, using the type of estimates of this section, we are not able to prove uniqueness. At the other hand, in the case of Theorem 3.2, for both degenerating B and K, we have to expect multiple solutions. 4. Nonnegative column-sums and L1 -estimates In this section, we apply Theorem 2.2 to the case of nonnegative column-sums and L1 -estimates. The nonlinear operators corresponding to this case come from stable finite volume discretizations of nonlinear convection–diffusion problems. Under certain conditions, the example from Section 5 is valid for this case, as well. For our notation of the Dirichlet problem, please consult Definition 2.1. Theorem 4.1. Let A : Rn → Rn be continuously differentiable with derivative A (u) ∈ Z+ c (n). Further, let B ∈ C 1 (Rn , Rn ) be an isotone diagonal homeomorphism. Let Q ∈ C(Rn , Rn ) be bounded with Q(u)1 Q+ . Then for u0 ∈ Rn , for any (possibly empty) subset D ⊂ {1, . . . , n}, and ud ∈ RD , the Dirichlet problem
B(u) + A(u) = Q(u) + B(u0 ), u|D = ud
(4.1)
has a solution u which in the case D = ∅ admits the estimate B(u) B(u0 ) + Q+ + A B −1 (0) . 1 1 1
(4.2)
Proof. First, regard the case D = ∅. As B is a homeomorphism, we can perform a variable substitution v = B(u). Let AB (v) = A(B −1 (v)). We have the identity
A B (v) = A B −1 (v) B −1 (v) thus because of Theorem A.2, the isotonicity and the continuity of B and hence of its inverse, AB fulfills the conditions of the theorem. Thus we restrict ourselves to the problem v + AB (v) = QB (v) + v0
(4.3a)
with QB (v) = Q(B −1 (v)) and v0 = B(u0 ). Let AB0 (v) = AB (v) − AB (0) and QB0 (v) = QB (v) − AB (0). We get the equivalent problem v + AB0 (v) = QB0 (v) + v0 ,
(4.3b)
where AB0 besides of the condition of the theorem fulfills AB0 (0) = 0 and QB0 is bounded by QB0 (v) Q+ = Q+ + AB (0) = Q+ + A B −1 (0) . B0 1 1 1
(4.3c)
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Let AB0 (v, w) ∈ Z+ c (n) be defined as in Theorem 2.1. Then we can rewrite Eq. (4.3b) as v + AB0 (v, 0)v = QB0 (v) + v0 ,
(4.3d)
which after Theorem 2.2 has a solution v with the estimate v Q+ + v0 1 . B0 1
= B −1 (v) is a solution of (4.1) admitting the It is a solution of (4.3b), too and consequently, u estimate (4.2). For the Dirichlet problem, we regard
BI (u) + AI (u) = QI (u) + BI (u0 ). We have a splitting A (u) =
ADD AID
ADI AII
(4.3e)
.
+ As A (u) ∈ Z+ c (n), AII ∈ Zc (I), thus for fixed uD , (4.3e) as an equation in uI fulfills the conditions of the first part of the proof. ✷
Remark 4.1. Again, degeneracy of the “main part” A is allowed. But unlike in Section 3, the existence proof for degenerating B does not work here. Further, as L1 -estimates in a certain sense correspond to mass conservation in a closed system, decent estimates for the Dirichlet not necessarily have to be expected. Theorem 4.2. Assume that the conditions of Theorem 4.2 are fulfilled, that Q is diagonal and continuously differentiable with Q 0, and that for any u ∈ Rn , B (u) ∈ D++ (n). Then the solution of (4.1) is unique. Proof. For u, v being two solutions of (4.1) after Theorem 2.1 we can write B(u, v)(u − v) + A(u, v)(u − v) − Q(u, v)(u − v) = 0. v) + A(u, v) − Q(u, v) is an M-matrix, thus necessarily, u = v. By Theorem A.2, B(u,
✷
5. Example: nonlinear diffusion in inhomogeneous domains To motivate our considerations, we give a simple example. In the fields of nonlinear heat transfer and of porous media flow simulation, we have to solve numerically initial boundary value problems of the form
∂t b x, u(x, t) − ∇ · d x, u(x, t) ∇u(x, t) = 0
(5.1)
with b monotonically increasing and continuously differentiable in u and d continuous and nonnegative. For simplicity, we assume here a one-dimensional domain Ω = (0, 2) with two subdomains Ω1 = (0, 1) and Ω2 = (1, 2) with homogeneous Neumann boundary conditions, and an initial value u0 . Assume that b(x, u) = bm (u) and d(x, u) = d m (u) on Ωm , respectively.
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If we introduce an implicit Euler time discretization with time-step τ , in the νth time-step we have to solve a problem of the type
b(x, uν ) − τ ∇ · d(x, uν )∇uν = b(x, uν−1 ).
(5.2)
To be able to do this, we introduce the following control volume scheme which is formulated using the flux functions g 1 , g 2 which are introduced below. Let N = 2n, h = 1/n and for i = 0, . . . , 2n, let xi = ih, and ui ≈ u(xi ). Let [0, h/2], . . . , [(i − 12 )h, (i + 12 )h], . . . , [2 − h/2, 2] be the control volumes. Let 1 1 2 b (ui ), 1 b (ui ), B(u)i = h 12 b1 (ui ) + b2 (ui ) , b2 (ui ), 1 2 2
and
b (ui ),
i = 0, 0 < i < n, i = n, n < i < 2n, i = 2n,
g 1 (ui , ui+1 ), g 1 (ui , ui−1 ) + g 1 (ui , ui+1 ), τ 1 A(u)i = g (ui , ui−1 ) + g 2 (ui , ui+1 ), h g 2 (ui , ui−1 ) + g 2 (ui , ui+1 ),
g 2 (ui , ui−1 ),
i = 0, 0 < i < n, i = n, n < i < 2n, i = 2n.
Then the system of equations B(uν ) + A(uν ) = B(uν−1 ) can be regarded as an approximation to the continuous system yielded by integrating our initial equation over each of the control volumes and approximating the flux between two discretization nodes by the flux functions. We can imagine that in the ideal case, the discrete solution is glued together from solutions of the boundary value problems ∇ · d(x, u)∇u = 0,
u|xi = ui , u|xi+1 = ui+1
(5.3)
in [xi , xi+1 ]. In our example, for d m = D m , this approach yields the flux functions D m (ui ) − D m (ui+1 ) = ∇D m (u)[xi ,xi+1 ] . h The so defined operator A(u) fulfills the conditions of Theorem 4.1, this proves existence and uniqueness of solutions of the discrete problem and further yields the L1 -estimate which implicates L1 -stability of the discrete system independent on the size of the time-step. Defining
g m (ui , ui+1 ) =
m m D (ui ) − D (ui+1 ) 1 , ui = ui+1 , k m (ui , ui+1 ) = ui − ui+1 h d m (u ), ui = ui+1 i
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for D ∈ C 1 (R, R) allows to rewrite A(u) = K(u)u with K ∈ Z0r (n) continuously depending on u: −k 1 (ui , uj ), −k 1 (ui , uj ), 2 τ −k 2 (ui , uj ), K(u)ij = −k (ui , uj ), h −kil (u) l=0...2n,l = i
0,
0 i < n, j = i + 1, 0 < i n, j = i − 1, n i < 2n, j = i + 1, n < i 2n, j = i − 1, i = j, else.
Thus, the conditions of Theorem 3.2 are fulfilled, and we get the L∞ -estimate implying unconditional L∞ -stability. A common way in the case that we are not able to obtain the function D is to use the arithmetical mean of the solutions of the neighboring nodes:
g (ui , ui+1 ) = d m
m
ui + ui+1 (ui − ui+1 ) . 2 h
In this case, for A and B, the conditions of Theorem 3.2 are fulfilled. Thus for arbitrarily large timesteps, the time-step problems are solvable, and, furthermore, the estimate (3.7) yields unconditional L∞ -stability. However, in this case the L1 -theory and thus the uniqueness result is not available. The practical experience however does not indicate that multiple solutions are a problem in this case [4,6]. We remark that in the case of homogeneous materials, i.e., b1 = b2 and d 1 = d 2 we could use the variable transformation v = D(u). In this case, the results for the case of diagonal nonlinearity [9] would be sufficient. In [5], using a Voroni box based control volume ansatz, this approach is described for simplicial meshes in higher space dimensions. For a real application, it will be necessary to solve the discrete nonlinear systems. In [4–6] we successfully use Newton’s method with feedback to the time-step size control.
Appendix A. Linear algebra A good reference for the following results still is [14]. Let M(n) be the set of all real n × n-matrices. For u = (ui )i=1...n ∈ Rn , let u1 =
n
|ui |,
i=1
n
u∞ = max |ui |. i=1
(A.1)
Let A ∈ M(n). Then n Au1 n = max |aij | = AT ∞ , j =1 0=u∈Rn u1 i=1
A1 = sup
n Au∞ n = max |aij | = AT 1 . A∞ = sup i=1 0=u∈Rn u∞ j =1
(A.2)
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Define the following subsets of M(n): • Z(n) ⊂ M(n)—nonnegative main diagonal and nonpositive off-diagonal elements (often denoted as Z n×n ); we call them Z-matrices here, • Z+ r (n) ⊂ Z(n)—rowwise weakly diagonally dominant, • Z0r (n) ⊂ Z+ r (n)—row-sum zero, • Z++ (n) ⊂ Z(n)—rowwise strictly diagonally dominant, r • D(n) ⊂ M(n)—diagonal, • D+ (n) ⊂ D(n)—nonnegative diagonal, • D++ (n) ⊂ D+ (n)—positive diagonal. To all rowwise defined subsets, the corresponding columnwise defined subsets are denoted using the subscript c . For a subset X ⊂ {1, . . . , n} we write M(X), Z(X) etc. to denote the sets of matrices corresponding to this subset. A = (aij )i,j =1...n ∈ M(n) is called positive (A > 0) if aij > 0, i, j = 1, . . . , n, and nonnegative (A 0) if aij 0, i, j = 1, . . . , n. A matrix A ∈ Z(n) is called (nonsingular) M-matrix if A−1 0. ++ (n). Then Theorem A.1. Let A ∈ Z+ r (n), D ∈ D + (i) DA ∈ Zr (n), (ii) A + D ∈ Z++ r (n) is an M-matrix, (iii) (I + A)−1 ∞ 1.
Proof. The first part is obvious, for the second, see [1, lemma 6.2]. For the third part, let first A ∈ Z0r (n). Assume that (I + A)−1 ∞ > 1. We know that (I + A)−1 has positive entries. Then for αij being the entries of (I + A)−1 , n
max i=1
n
αij > 1.
j =1
Let k be a row where the maximum is reached. Let e = (1, . . . , 1)T . Then for v = (I + A)−1 e we have that v > 0, vk > 1 and vk vj for all j = k. The kth equation of e = (I + A)v then looks like 1 = vk + vk
j =k
|akj | −
|akj |vj vk + vk
j =k
j =k
|akj | −
|akj |vk = vk > 1.
j =k
This contradiction enforces (I + A)−1 ∞ 1. + 0 Let now A ∈ Z+ r (n). The we can write A = D + A0 with D ∈ D (n) and A ∈ Zc (n). We have I + A = I + D + A0 = (I + D)(I + D)−1 (I + D + A0 ) = (I + D)(I + AD0 ) with AD0 = (I + D)−1 A0 ∈ Z0c (n). Thus
(I + A)−1 = (I + AD0 )−1 (I + D)−1 I + D)−1 1, ∞ ∞ ∞
because all main diagonal entries of I + D are greater or equal to 1.
✷
Remark A.1. The third part of the theorem comes from the M-criterion of [8,11], both refer to [2].
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++ Theorem A.2. Let A ∈ Z+ (n). Then c (n), D ∈ D + (i) AD ∈ Zc (n), (ii) A + D ∈ Z++ c (n) is an M-matrix, −1 (iii) (I + A) 1 1.
Proof. Apply Theorem A.1 to AT .
✷
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