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A strongly nonlinear problem arising in glaciology
COLINGE, Jacques, RAPPAZ, Jacques
Abstract
The computation of glacier movements leads to a system of nonlinear partial differential equations. The existence and uniqueness of a weak solution is established by using the calculus of variations. A discretization by the finite element method is done. The solution of the discrete problem is proved to be convergent to the exact solution. A first simple numerical algorithm is proposed and its convergence numerically studied.
COLINGE, Jacques, RAPPAZ, Jacques. A strongly nonlinear problem arising in glaciology.
Mathematical Modelling and Numerical Analysis , 1999, vol. 33, no. 2, p. 395 - 406
DOI : 10.1051/m2an:1999122
Available at:
http://archive-ouverte.unige.ch/unige:12584
Disclaimer: layout of this document may differ from the published version.
1 / 1
Mod´elisation Math´ematique et Analyse Num´erique
A STRONGLY NONLINEAR PROBLEM ARISING IN GLACIOLOGY
Jacques Colinge
1and Jacques Rappaz
2Abstract. The computation of glacier movements leads to a system of nonlinear partial differential equations. The existence and uniqueness of a weak solution is established by using the calculus of variations. A discretization by the finite element method is done. The solution of the discrete problem is proved to be convergent to the exact solution. A first simple numerical algorithm is proposed and its convergence numerically studied.
R´esum´e. La simulation des mouvements d’un glacier conduit `a la r´esolution num´erique d’un syst`eme d’´equations aux d´eriv´ees partielles fortement non-lin´eaire. Dans cet article nous prouvons l’existence et l’unicit´e d’une solution faible de ce syst`eme en utilisant des ´el´ements de calcul des variations. Apr`es avoir ´etabli une discr´etisation du probl`eme par la m´ethode des ´el´ements finis, nous montrons que le probl`eme discret a une solution unique qui converge vers la solution exacte. Nous proposons un algorithme de r´esolution dont nous ´etudions num´eriquement la convergence.
AMS Subject Classification. 35J60, 65N30, 26B25, 35J20, 49J45, 86A40.
Presented June 10, 1998 by J. Rappaz, member of the Editorial Board.
1. The model
In order to conduct theoretical studies, glaciologists often consider the so-calledinfinite parallel sided slab, see [1]. This idealised glacier is made of an infinite ice mass situated between two parallel planes with a slight inclination; ice is treated as an incompressible viscous fluid. In this paper we limit our study to the 2-dimensional first order model (Blatter [1]) in a finite domain,i.e. a rectangle.
Let Λ = (0, L)×(0,1)3(x, z),L >0, be the domain, the boundary of which is denoted by∂Λ. The system of equations to solve in Λ is
∂τ
∂z = −2∂σ
∂x −1, (1)
∂w
∂z = −F(σ, τ)σ, (2)
∂u
∂z = 2F(σ, τ)τ, (3)
∂u
∂x = F(σ, τ)σ. (4)
Keywords and phrases. Nonlinear problem, finite elements, convex analysis, calculus of variations, glaciology.
1 Section of Mathematics, Universit´e de Gen`eve, Rue du Li`evre 2-4, 1227 Acacias, Suisse.
2 Department of Mathematics, ´Ecole Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Suisse.
e-mail:[email protected]
c EDP Sciences, SMAI 1999
The variables have the following meaning: x and z are the Cartesian coordinates, u and w stand for the horizontal and vertical velocities respectively, τ stands for the shear stress andσ stands for the x-component of the deviatoric stress tensor. The function F(σ, τ) represents a flow law; it is related to the inverse of the viscosity. We consider an often used law in glaciology,i.e. Glen’s flow law
F(σ, τ) = T02+τ2+σ2n−12
, (5)
whereT06= 0 is a real constant andnis a positive integer.
The boundary conditions are
u(x, z) = g(x, z), (x, z)∈Γ0, (6)
τ(x, z) = 0, (x, z)∈Γ1, (7)
w(x,0) = 0. (8)
The boundary subset Γ1 contains the ice mass surface,i.e. (0, L)× {1} ⊂Γ1, but it does not contain the left and right sides,i.e. {0} ×[0,1]∩Γ1=∅and{L} ×[0,1]∩Γ1=∅; we define Γ0=∂Λ\Γ1. The places where the ice can move freely are represented by Γ1. The function g is given; it represents the imposed velocity on the left and right sides and at some places of the ice mass base.
2. Analysis
Equation (2) for wcan be left aside for the moment, becausewcan be computed whenτ andσ are known.
We transform the equations in a more convenient form. We have F(σ, τ)>0 and from equations (3) and (4) we obtain
τ = 1
2F(σ, τ)
∂u
∂z,
σ = 1
F(σ, τ)
∂u
∂x· With equation (1) we find
∂
∂z 1
2F(σ, τ)
∂u
∂z
+ 2 ∂
∂x 1
F(σ, τ)
∂u
∂x
=−1. (9)
After the coordinate transformationy= 2z,v(x, y) =u(x, y/2), we have to solve the system
τ = 1
F(σ, τ)
∂v
∂y, (10)
σ = 1
F(σ, τ)
∂v
∂x, (11)
1
2 = −∂
∂y 1
F(σ, τ)
∂v
∂y
− ∂
∂x 1
F(σ, τ)
∂v
∂x
, (12)
in the new domain Ω = (0, L)×(0,2). The boundary conditions are transformed in a straightforward manner for obtainingv(x, y) =g(x, y/2),∀(x, y)∈Γ, ∂u∂ν = 0 on∂Ω\Γ, where Γ ={(x,2z); (x, z)∈Γ0}.
If n = 1 then F(σ, τ) ≡ 1 and the problem is equivalent to −∆v = 1/2, with mixed Neumann-Dirichlet boundary conditions. This is a well-posed linear problem with solution in W1,2(Ω) =H1(Ω). In the sequel we supposen≥2.
We have the relation
F(σ, τ) = T02+σ2+τ2n−12
=
T02+ 1
F2(σ, τ)|∇v|2 n−12
· (13)
We show thatF(σ, τ) can be expressed as a function of|∇v|2. We definef =F(σ, τ) ands=|∇v|2 and have fn2−1 −T02= s
f2· (14)
Lemma 1. For alls∈R+there exists a uniquef ∈R+satisfying (14), which we denote byf =f(s). Moreover there exist two positive constants β andγ (depending onn) such that
sn−12n ≤f(s)≤γ(β+s)n−12n , s≥0. (15) Proof. Let sbe a non-negative number and setR(f) =fn−12 −T02 andTs(f) = fs2. Clearly (14) is equivalent to R(f) = Ts(f). Ifs= 0 we obtainf =f(0) =|T0|n−1. Assume now that s is positive. The function R is strictly increasing with R(0) =−T02<0 and limf→∞R(f) = +∞. At the opposite, the functionTs is strictly decreasing withTs(+0) = +∞andlimf→∞Ts(f) = 0. By using Bolzano’s Theorem, we easily conclude for the existence and uniqueness off(s).
Now we show the estimate (15). By using (14) we have
fn−12n −T02f2=s. (16)
By derivating (16) we obtain, sincef is strictly positive, 2n
n−1fn−12n (s)−2T02f2(s) f0(s)
f(s) = 1
or equivalently
2
n−1fn−12n (s) + 2s f0(s)
f(s) = 1. (17)
Relation (17) proves thatf0(s) is strictly positive (f is strictly increasing) and with (16) we obtain lims→∞f(s) = +∞. Since n2n−1 >2, it is easy to see from (16) that there exist two positive constantsγ andS such that
f(s)≤γsn−12n , ∀s≥S.
Fors∈[0, S], the functionf(s) is bounded and we set β=
maxs∈[0,S]f(s) γ
n−12n . It follows that
f(s)≤γ(β+s)n−12n , ∀s∈R+. Finally, from (16) we obtain
fn2n−1(s)≥s
and Lemma 1 is proved.
If we return to equation (12) we have, by using (13), the formal expression
−div 1
f(|∇v|2)∇v
= 1
2, in Ω. (18)
Moreover if we define k(x, y) =g(x, y/2),∀(x, y)∈Γ, we obtain the following boundary conditions onv
v(x, y) = k(x, y), ∀(x, y)∈Γ, (19)
∂v
∂y(x, y) = 0, ∀(x, y)∈∂Ω\Γ. (20)
From now on, we assume thatk belongs toW1−1p,p(Γ), wherepis defined by p= n+ 1
n · (21)
In order to analyse Problems (18–20), we use the affine space Vk =
u∈W1,p(Ω); u|Γ=k .
The affine spaceVk is well defined because the functions belonging toW1,p(Ω) possess a trace on Γ which is in W1−p1,p(Γ) (see [8] for instance). In a similar way we denote byV0the linear space
V0=
u∈W1,p(Ω); u|Γ= 0 .
Let p0 be the conjugate integer of p, i.e. 1p +p10 = 1. Equation (21) immediately yields p0 =n+ 1. Now if we take a function uin W1,p(Ω), we easily verify, by using Lemma 1, that f(|∇1u|2)|∇u| belongs to Ln+1(Ω).
Consequently, H¨older’s Inequality allows us to write a weak formulation of Problems (18–20) as follows: Find v∈Vk such that
Z
Ω
1
f(|∇v|2)∇v∇wdxdy =1 2 Z
Ω
wdxdy, ∀w∈V0. (22)
Leth(s) =f(s)1 andH(s) =Rs
0 h(t) dt. Lemma 1 impliesH(s)≤n+12n sn+12n and ifu∈W1,p(Ω), thenH(|∇u|2)∈ L1(Ω). Therefore the following minimization problem possesses a meaning: Find v∈Vk such that
J(v)≤J(u), ∀u∈Vk, (23)
whereJ is defined onVk by
J(u) = 1 2 Z
Ω
H(|∇u|2)−u
dxdy. (24)
We show below that the minimization problem (23) has a unique solution which is just the unique solution to the variational problem (22). To aim at this we use the so-calleddirect methodsfrom the calculus of variations, seee.g.[6]. We denote by
ϕ(u, ξ) =H(|ξ|2)−u
the integrand of (24); it takes its source inR×R2. We observe thatϕ(u, ξ) is continuous.
Lemma 2. ϕ(u,·)is strictly convex.
Proof. We show that l(s) =H(s2),s≥0, is strictly convex. We have l0(s) = 2sH0(s2) = 2s
f(s2), (25)
l00(s) = 2f(s2)−4s2f0(s2)
f2(s2) · (26)
We sett=s2and from
f(t) =
T02+ t f2(t)
n−12
(27) we deduce
f0(t) =n−1 2
T02+ t f2(t)
n−32
f(t)−2tf0(t) f3(t)
. (28)
Hence
l00(s) = 4
n−1f(t)f0(t)
T02+ t f2(t)
3−n2
(29) and, since
f0(t) = n−1 2
T02+ t f2(t)
n−32
f(t) f3(t) + 2tn−21
T02+f2t(t)
n−32 >0, (30)
we havel00(s)>0. Therefore we have established the strict convexity ofl(s).
The conclusion follows from the fact thatl(s) is increasing for positives: letα∈(0,1),ξ, η∈R2, then l(|αξ+ (1−α)η|)≤l(α|ξ|+ (1−α)|η|)< αl(|ξ|) + (1−α)l(|η|).
Theorem 1. There is a uniquev∈Vk⊂W1,p(Ω)such that
J(v) = inf{J(u); u∈Vk}<∞. Moreover, v is the unique weak solution to (22).
Proof. We follow the approach presented in Dacorogna [6]. There exists ˜k ∈ W1,p such that ˜k|Γ = k, see Ne˘cas [8]. We clearly haveJ(˜k)<∞and then
inf{J(u); u∈Vk}=m <∞. (31)
Let{wν}ν∈Nbe a minimizing sequence for (31). There exists an integerN such that, for allν ≥N, we have m+ 1≥J(wν) =
Z
Ω
H(|∇wν|2)−wν
dxdy. (32)
In the last part of the proof we introduce several constants, they are always positive but their values can change from one equation to another. Lemma 1 yields
H(|∇wν|2) =
Z |∇wν|2 0
1
f(s)ds≥c1 c2+|∇wν|2n+12n
−c3. (33)
We put (33) into (32) and obtain (p=n+1n ) m+ 1 ≥ c1
Z
Ω
c2+|∇wν|2n+12n
dxdy−c3meas(Ω)−Z
Ω
wνdxdy
≥ c1
Z
Ω
|∇wν|pdxdy−c3−Z
Ω
wνdxdy
≥ c1kwνkpW1,p−c3−Z
Ω
|wν|dxdy.
By using Young’s inequality we find Z
Ω|wν|dxdy = Z
Ω
1· |wν|dxdy ≤ Z
Ω
1 p0εp0 +εp
p|wν|p
dxdy
= c4
εp0 meas (Ω) +c5εpkwνkpLp, ∀ε >0, (34) with 1p+p10 = 1. ClearlykwνkpLp≤ kwνkpW1,p and we have
m+ 1≥c1kwνkpW1,p−c3− c4
εp0 −εpc5kwνkpW1,p. (35) By takingε small enough to havec1−εpc5 >0 in (35), we prove that the sequence {wν}ν∈Nis bounded. We can extract a subsequence, still noted{wν}ν∈N, which has a weak limitv∈Vk.
Lemma 1 easily yields
−|u| ≤ϕ(u, ξ)≤(c+|u|) (1 +|ξ|p), c >0. (36) From Lemma 2 we deduce the convexity ofϕ(u,·) and, with (36), we can apply a theorem found in [7] (Thm.
1.1) to prove the weak lower semicontinuity ofJ. So we have lim inf
ν→∞ J(wν)≥J(v)
and it turns out thatvis a minimum. Let us admit the existence of another minimumv0and definev= 12(v+v0).
The functionalJ is convex becauseϕis convex; this yields J(v)≤ 1
2(J(v) +J(v0)) = inf{J(u); u∈Vk}=m.
Hence vis also a solution to the minimization problem. It follows thatJ(v) =12(J(v) +J(v0)) and Z
Ω
1
2H(|∇v|2) +1
2H(|∇v0|2)−H(|1 2∇v+1
2∇v0|2)
dxdy= 0. (37)
If we suppose v and v0 to be different, then we obtain a contradiction with (37) since the strict convexity of H(u,·), see Lemma 2, implies
Z
Ω
1
2H(|∇v|2) +1
2H(|∇v0|2)−H(|1 2∇v+1
2∇v0|2)
dxdy >0.
It remains to prove thatv is a weak solution to (22). A new application of Lemma 1 yields
|gradξϕ| ≤c|ξ|p−1,
wherec >0 is a constant. This bound allows us to apply a result found in [6] (Thm. 4.4, Sect. 3) to conclude.
We observe that the techniques used in the proof of Theorem 1 are not restricted to the rectangular domain Ω. In fact, they remain valid for any bounded open subset ofR2with Lipschitz boundary.
3. Discretization
We set discrete equivalent problems to Problem (22) and Problem (23) by using triangle finite elements of degree 1. We denote byTh a regular triangulation of Ω, with trianglesK∈ Th and
Xh=
w∈ C0(Ω); w|K is a polynomial of degree at most 1, ∀K∈ Th .
Moreover, we assume that for each triangleK ∈ Th,K∩Γ is either void or a side ofK or a vertex ofK. If rh : C0(Ω) → Xh is the interpolation operator and if we still denote byk a continuous extension ofk to Ω, we set
Vk,h = {w∈Xh; w=rhk on Γ}, V0,h = {w∈Xh; w= 0 on Γ}.
We haveVk,h, V0,h⊂H1(Ω)⊂W1,p(Ω). Therefore the two following discrete problems are meaningful:
Variational discrete problem: Findvh∈Vk,h such that Z
Ω
1
f(|∇vh|2)∇vh∇whdxdy=1 2 Z
Ω
whdxdy, ∀wh∈V0,h. (38)
Minimization discrete problem: Findvh ∈Vk,h such that
J(vh)≤J(uh), ∀uh∈Vk,h. (39)
By repeating the proof of Theorem 1 almost unchanged we easily prove the following result:
Theorem 2. Problem (39) possesses a unique solution vh∈Vk,h which is also the unique solution of Problem
(38).
In order to prove the convergence of the discrete solution to the exact solution as Th is refined, we need several preliminary results.
Lemma 3. Let X be a Banach space and{xi}i∈N ⊂ X a sequence such that, for each subsequence {xij}j∈N
extracted from{xi}i∈N, we can extract a new subsequence{xijl}l∈Nwhich always weakly converges to the same weak limit x∈X. Then the original sequence{xi}i∈Nweakly converges to x.
Lemma 4. Forf, g∈LP(Ω)the inequality below holds Z
Ω
| |f|p− |g|p| dxdy≤pk|f|+|g|kpL−p1kf−gkLp.
Proof. Letx, y∈R. By the Mean Value Theorem there exists ξ∈Rsuch that
|y|p− |x|p=p|ξ|p−1(|y| − |x|).
By taking the absolute value of this expression and bounding|ξ|by|x|+|y|we find
||y|p− |x|p| ≤p(|x|+|y|)p−1||y| − |x|| ≤p(|x|+|y|)p−1|y−x|. Now considering the integral we obtain
Z
Ω
| |f|p− |g|p| dxdy≤p Z
Ω
(|f|+|g|)p−1|f−g|dxdy.
The harmonic conjugate ofpisq=p−p1andf, g∈Lp(Ω) implies that|f−g| ∈Lp(Ω) and (|f|+|g|)p−1∈Lq(Ω).
We apply H¨older’s Inequality and obtain p
Z
Ω
(|f|+|g|)p−1|f−g|dxdy≤pk|f|+|g|kpL−p1kf−gkLp.
Lemma 5. The functionalJ is continuous inW1,p(Ω).
Proof. The linear part ofJ is clearly continuous; we consider the nonlinear part only. By using the definition ofH and Lemma 1, we obtain fors, t≥0:
|H(s)−H(t)| = Z s
t
h(r) dr =
Z s t
1 f(r)dr
≤ Z s
t
r1−n2n dr ≤ 2
p
sp2 −tp2. Consequently, we obtain foru, v∈W1,p(Ω):
Z
Ω
H(|∇u|2)−H(|∇v|2) dxdy ≤2
p Z
Ω
| |∇u|p− |∇v|p| dxdy.
Lemma 4 allows us to conclude.
Theorem 3. Let{Th}h∈(0,1]be a sequence of refining regular triangulations. Then the sequence of corresponding discrete solutions {vh}h∈(0,1] ⊂W1,p(Ω) weakly converges to the exact solution v ∈W1,p(Ω) ash tends to 0.
Moreover, {vh}h∈(0,1] strongly converges to v inLp(Ω).
Proof. We prove limh→0vh =v by applying Lemma 3. So we consider an arbitrary subsequence{vhi}i∈N of {vh}h∈(0,1], with limi→∞hi= 0. For eachThi we define whi, the element ofVk,hi, which satisfies
kv−whikW1,p= min{kv−wkW1,p; w∈Vk,h}.
Now we show that the sequence {whi}i∈N is a minimizing sequence for J. We know from Lemma 5 that J is continuous. It is then possible for each ε > 0 to find δ > 0 such that, for each ˜v ∈ B(v, δ), we have
|J(v)−J(˜v)| ≤ε. It is known that kv−whikW1,p tends to 0 ashi tends to 0. Hence we have established that the sequence{whi}i∈N is a minimizing sequence.
The discrete solutions {vhi}i∈N satisfy J(vhi) ≤ J(whi), ∀i ∈ N. Therefore {vhi}i∈N is also a minimizing sequence. From the proof of Theorem 1 we know that we can extract a subsequence from{vhi}i∈Nwhich weakly converges to the exact solutionv. Lemma 3 let us conclude for the weak convergence inW1,p(Ω). The compact embedding ofW1,p(Ω) inLp(Ω) implies that the sequence{vh}h∈(0,1] converges tov strongly.
4. Numerical experiment
In this section we present a simple numerical algorithm to compute an approximation of the solution of Problem (22). In fact, this algorithm computes an approximation of the solution of the discrete problem (38).
The above theory would rather suggest to design a minimization algorithm for the equivalent problem (39).
Nevertheless, an algorithm for Problem (38) is certainly easier to design. Therefore in this preliminary numerical investigation we prefer solving the discrete variational problem (38).
Algorithm 1. For solving Problem (38) apply the following steps a) Choose an initial guessvh(0) for the numerical approximation.
b) Fori= 0,1,2, . . . solve the following linear problem: findv(i+1)h ∈Vk,h satisfying Z
Ω
1
f(|∇v(i)h |2)∇v(i+1)h ∇whdxdy =1 2 Z
Ω
whdxdy, ∀wh∈V0,h.
Before considering the numerical convergence, we present a typical solution to Problems (1–4, 6, 7). We select an example with a small central region at the base (z= 0) where we prescribe a Neumann boundary condition;
a homogeneous Dirichlet condition is prescribed elsewhere at the base. We take n = 3, i.e. p = 4/3. The numerical solution is plotted in Figure 1.
We do not provide any theoretical analysis for the convergence of Algorithm 1; we only report observed numerical convergence. A possible numerical experiment consists of applying Algorithm 1 to a modified problem to which we know the exact solution. We do this bya priorichoosing a function ˜v∈W1,43(Ω), with∂˜∂yv(x,2) = 0, and then inserting it in the left-hand side of (18). This defines a new right-hand side and consequently a new problem (with adapted boundary conditions and possibly domain). Access to the exact error is then possible for this new problem. It turns out that the convergence rate is very sensitive to the choice of ˜v.
Hence, in order to obtain an approximation of the convergence rate for the original problem, we use a sequence of refining grids. The exact problem we consider is : Problem (22) with T02= 0.1,L= 8, ((0, L)× {0})∩Γ1= (3.6,4.4), k(x,0) = 0, k(0, y) = g(L, y) = −321y4+ 14y3− T024+3y2+ (T02+ 1)y. (This particular choice for k has a special meaning in glaciology we do not want to explain here.) We estimate the convergence rate by two different means. First, we consider the numerical solution computed on the finest grid as the exact solution and estimate the convergence on much rougher grids. We find:
• Linear convergence for v,i.e.
kv−vhkL43(Ω)=O(h).
• Sublinear convergence for ∂v∂x or more precisely k∂v
∂x−∂vh
∂xkL43(Ω)=O(h0.7).
u
x z
4 3 2 1 0
60 50 40 30 20 10 0
1.0 0.75
0.5 0.25
0.0
horizontal velocity
w
x z
1.0 0.5 0.0 -0.5 -1.0
60 50 40 30 20 10 0
1.0 0.75
0.5 0.25
0.0
vertical velocity
tau
z
x 1.6
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
50 60 40 20 30
0 10 1.0
0.5 0.0
tau
sig
z
x 1.0
0.5
0.0
-0.5
-1.0
50 60 30 40
10 20
0 1.0
0.5 0.0
sigma
Figure 1. Slab with sliding : u,w,τ andσ.
• Sublinear convergence for ∂v∂y or more precisely
k∂v
∂y−∂vh
∂y kL43(Ω)=O(h0.8).
A second method which allows us to estimate the convergence rate is the following: we observe that, for any three successive grids (4h, 2handh) in the refining grid sequence, we have
kv4h−v2hkL43(Ω)= 2kv2h−vhkL43(Ω). (40) We have established convergence of the sequence of discrete solutions, see Theorem 3. Therefore, for a chosen ε >0, there existsh0∈(0,1] such that
kv−vhkL43(Ω)≤ε, ∀h≤h0.
By applying the triangle inequality, whenhh0, we find:
kv−vhkL43(Ω)≤ kv−vh0kL43(Ω)+kvh0−v2h0kL43(Ω)+· · ·+kvh
2 −vhkL43(Ω).
The observation (40) allows us to write this last inequality as kv−vhkL43(Ω) ≤ ε+
1 +1
2+1 4+· · ·
kvh
2 −vhkL43(Ω)
= ε+ 2kvh
2 −vhkL43(Ω). (41)
By using (40) once more and (41) we find
kv−vhkL43(Ω) ≤ ε+ 2kvh
2 −vhkL43(Ω)
= ε+ 1
2k−1kv2k−1h−v2khkL43(Ω), k∈N. Sinceεis arbitrarily small, we have linear convergence forv.
The same argument can be repeated for the derivatives ofv because we observe k∂v4h
∂x −∂v2h
∂x kL43(Ω) = 1.61k∂v2h
∂x −∂vh
∂xkL43(Ω), k∂v4h
∂y −∂v2h
∂y kL43(Ω) = 1.77k∂v2h
∂y −∂vh
∂y kL43(Ω),
for anyhin the refining sequence. We findk∂v∂x−∂v∂xhkL43(Ω)=O(h0.7) andk∂v∂y −∂v∂yhkL43(Ω)=O(h0.8). This confirms what we already found.
If we estimate the convergence rate for the original variablesτ andσ, instead of the derivatives ofv, we find an almost linear convergence inLp(Λ).
A last aspect we would like to report is the convergence of Algorithm 1 in terms of approximating the exact solution of the discrete problem (38). We observe a linear convergence. At each iteration the difference kv(i)h −vh(i+1)k∞is divided by about 1.75. This number seems not to be problem dependent. We also have that, for a given problem, the number of iterations does not change as the grid is refined.
Finally, the convergence rates reported above only change a little if we take different values forn(we tested n= 1,2,3,4,5), and overrelaxation reduces the number of iterations by about 30% forn= 3.
5. Conclusions and future work
We have proved the existence and uniqueness of the solution to a nonlinear problem coming from the field of glaciology. We have also proved that the discrete solution converges to the exact solution. A first simple numerical scheme has been presented and tested with reasonable performances. The grid size for the proposed algorithm can be made much finer than what was possible with the previously used techniques in glaciology (see [1, 2, 4, 5]).
Several aspects of the work presented in this paper may certainly be improved or completed. A mixed formulation of the discrete problem or finite volume methods should provide more precision in the approximation of∇v, which is of prime importance to the glaciologist. The convergence rate of Algorithm 1 is only linear and an improvement would be desirable. From a theoretical point of view, a convergence theory for the numerical algorithm anda posteriorierror estimates (adaptative grid refinement) are two aspects we would like to develop in order to complete this work.
The authors would like to thank Henri-Michel Maire for Lemma 4.
References
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