RELAXED KA CANOV SCHEME FOR THE p-LAPLACIAN WITH LARGEp ANNA KH. BALCI LARS DIENING AND JOHANNES STORN

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RELAXED KA ˇ

CANOV SCHEME FOR THE p-LAPLACIAN WITH

LARGE p

ANNA KH. BALCI, LARS DIENING, AND JOHANNES STORN

Abstract. We introduce a globally convergent relaxed Kaˇcanov scheme for

the computation of the discrete minimizer to the p-Laplace problem with 2 ≤

p < ∞. The iterative scheme is easy to implement since each iterate results

only from the solve of a weighted, linear Poisson problem. It neither requires

an additional line search nor involves unknown constants for the step length.

The rate of convergence is independent of the underlying mesh.

1. Introduction

The p-Laplace problem is a prototype of many non-linear problems occurring in

simulations of non-Newtonian ﬂuids, turbulent ﬂows of gases, glaciology, and plastic

modeling. Given a right-hand side f∈Lq(Ω) with 1/p + 1/q = 1 and p∈(1,∞),

the p-Laplace problem seeks the unique minimizer

u= arg min

v∈W1,p

0(Ω)

J(v) with J(v):=1

pZΩ

|∇v|pdx−ZΩ

fv dx.(1)

The functions may be scalar or vector-valued, but for better readability we use

the notation of the scalar-valued case. One diﬃculty that arises in its numerical

approximation is the computation of the discretized minimizer. In fact, established

iterative schemes like Newton or gradient descent methods experience huge insta-

bilities or even fail for values of pthat are not close to two, see Section 7.3. A

numerical scheme that overcomes these diﬃculties for small values of 1 < p ≤2 is

the relaxed Kaˇcanov scheme in [DFTW20] (see also [BDN18] for an alternative ap-

proach using the p-Laplace gradient ﬂow covering the range 1 ≤p < 2). However,

this scheme does not converge for p > 3, see [DFTW20, Rem. 21]. We modify this

approach and suggest a novel algorithm to compute the discrete minimizer of the

p-Laplacian for large values of p≥2, e.g. p= 100 in Section 7.2. The resulting

iterative scheme

•converges globally for all p≥2,

•is cheap and easy to implement, since each iteration computes solely the

Galerkin approximation to a weighted Poisson model problem and avoids

any additional line search,

•includes a regularization that ensures well-posedness of the iterative com-

putations,

•is robust with respect to (adaptive) mesh reﬁnements,

2020 Mathematics Subject Classiﬁcation. 35J70, 65N22, 65N30.

Key words and phrases. p-Laplacian, Kacanov iteration, adaptive FEM, energy relaxation.

The work of the authors was supported by the Deutsche Forschungsgemeinschaft (DFG, Ger-

man Research Foundation) – SFB 1283/2 2021 – 317210226.

arXiv:2210.06402v1 [math.NA] 12 Oct 2022

2 A. KH. BALCI, L. DIENING, AND J. STORN

•allows for a fully adaptive scheme including adaptive mesh reﬁnements and

adaptions of the regularization parameters,

•applies to the scalar and vector-valued case.

These advantages make our Kaˇcanov scheme very attractive for the computation of

p-Laplace problems with large values of pas for example needed in [BBD03;BP00].

We design the Kaˇcanov scheme as follows. Section 2introduces the dual problem

seeking the minimizer σ∈Wq(div=9f, Ω) :={τ∈Lq(Ω; Rd)|div τ=−f}with

σ= arg min

τ∈Wq(div=9f,Ω)

J∗(τ) where J∗(τ):=1

qZΩ

|τ|qdx.(2)

The dual problem allows for a similar relaxation of the energy functional as in

[DFTW20]. We prove the convergence of the relaxed dual functional with respect

in the relaxation parameter in Section 3, provided the solution has the additional

smoothness σ∈L2(Ω; Rd) (see Theorem 2for a discussion on that regularity as-

sumption). We show the convergence of the Kaˇcanov iterations for ﬁxed relaxation

parameters in Section 4and combine the two convergence results to conclude an

algebraic rate of convergence towards the exact discrete minimizer in Section 5.

Section 6.1 introduces a discretization of the Kaˇcanov scheme that leads due to

a duality relation on the discrete level to a numerical scheme that computes in

each iterate solely the Galerkin approximation to a weighted Poisson model prob-

lem with lowest-order Lagrange elements. Section 6.2 suggests an adaptive scheme

that estimates the regularization error, the error in the Kaˇcanov iterations, and

the discretization error. Depending on these estimates it either causes an adaption

of the regularization parameter, computes a further Kaˇcanov iteration, or applies

an adaptive mesh reﬁnement. The numerical experiments in Section 7.1 and 7.2

indicate even for large values p= 100 an exponential rate of convergence for that

adaptive approach. The numerical experiment in Section 7.3 compares our scheme

with the steepest descent method from [HLL07]. It illustrates that our scheme does,

unlike the steepest descent method, not depend on the underlying triangulation and

is thus superior on ﬁne meshes.

2. Relaxed dual functional

The equivalence of the primal problem in (1) and its dual formulation in (2) is

a classical result in the calculus of variations, see for example [ET76]. It follows

for a wide class of convex functionals by properties of the convex conjugate. The

Euler–Lagrange equation of (2) involves the spaces

Wq(div,Ω) :={τ∈Lq(Ω; Rd)|div τ∈Lq(Ω)},

(3)

The minimizer σ∈Wq(div,Ω) in (2) solves, with a Lagrange multiplier u∈Lp(Ω)

the saddle point problem

ZΩ

|σ|q−2σ·ξdx+ZΩ

udiv ξdx= 0 for all ξ∈Wq(div,Ω),

ZΩ

vdiv σdx=−ZΩ

fv dxfor all v∈Lp(Ω).

(4)

The Lagrange multiplier u∈Lp(Ω) equals the minimizer in (1) and satisﬁes σ=

|∇u|p−2∇uand ∇u=|σ|q−2σ. Moreover, we have the identity

−J ∗(σ) = J(u).(5)

RELAXED KAˇ

CANOV SCHEME FOR THE p-LAPLACIAN WITH LARGE p3

The problem in (4) has similarities to the mixed formulation of a weighted Pois-

son model problem and so motivates an iterative calculation of functions σn+1 ∈

Wq(div,Ω) and un+1 ∈Lp(Ω) by solving, for all ξ∈Wq(div,Ω) and v∈Lp(Ω),

ZΩ

|σn|q−2σn+1 ·ξdx+ZΩ

un+1 div ξdx= 0,

ZΩ

vdiv σn+1 dx=−ZΩ

fv dx.

(6)

This approach (known as Kaˇcanov iteration, Picard iteration, or method of suc-

cessive substitutions) has already been suggested in [CFP07]. Unfortunately, the

problem in (6) degenerates at points where |σ|= 0 and |σ|=∞. We remedy

this diﬃculty with an idea from [DFTW20]. This idea bases on a modiﬁed energy

functional which reads for all a: Ω →[0,∞) and τ∈Wq(div,Ω)

J∗(τ, a):=ZΩ

2aq−2|τ|2+1

q−1

2aqdx.(7)

Notice that for a=|τ|the energy equals the one in (2). Moreover, the relaxed

energy is convex with respect to τand a, which we show as follows. We set the

function β(t, a):=aq−2t2/2 for all a≥0 and t∈R. The function satisﬁes

(∇2β(t, a)) = aq−2(q−2)aq−3t

(q−2)aq−3t(q−2)(q−3)aq−4t2/2.

Since 0 ≤aq−2and 0 ≤det((∇2β)(t, a) = a2q−6t2(2 −q)(q−1) for all a≥0 and

t∈R, this matrix is non-negative deﬁnite for all 1 < q < 2 and so the relaxed

energy J∗is convex in τand a.

Remark 1 (Opposite case).The restriction 1< q < 2is equivalent to 2<p<∞.

Therefore, this approach covers the range of pthat is excluded in [DFTW20].

Given a: Ω →[0,∞), the Euler-Lagrange equation corresponding to the mini-

mization of (7) over Wq(div=9f, Ω) seeks σ∈Wq(div,Ω) and u∈Lp(Ω) with

aq−2σ− ∇u= 0 and −div σ=f.

This saddle point system degenerates as ess inf a→0 and ess sup a→ ∞. We over-

come this diﬃculty by restricting the minimization of (7) for ﬁxed τ∈Wq(div,Ω)

to functions a: Ω →[ε−, ε+] within a relaxation interval ε= [ε−, ε+]⊂(0,∞).

Diﬀerentiation shows that the minimizer for a ﬁxed τ∈Wq(div,Ω) reads

arg min

a: Ω→[ε−,ε+]

J∗(τ, a) = ε−∨ |τ| ∧ ε+.(8)

This leads for all τ∈Wq(div,Ω) to the relaxed energy

J∗

ε(τ):=J∗(τ, ε−∨ |τ| ∧ ε+) = min

a:Ω→[ε−,ε+]J∗(τ, a).

This monotonically decreasing functional (with respect to growing intervals ε=

[ε−, ε+]) hides the minimization with respect to a: Ω →[ε−, ε+]. The functional

can be written in terms of the integrand

κ∗

ε(t):=









2εq−2

−t2+1

q−1

2εq

−for t≤ε−,

qtqfor ε−≤t≤ε+,

2εq−2

+t2+1

q−1

2εq

+for ε+≤t.

(9)

4 A. KH. BALCI, L. DIENING, AND J. STORN

More precisely, the functional equals

J∗

ε(τ) = ZΩ

κ∗

ε(|τ|) dxfor all τ∈Wq(div,Ω).

The function κ∗

εhas quadratic growth in the sense that κ∗

ε(t)'εq−2

+t2for t≥ε+.

Therefore, the relaxed energy J∗

ε(v)<∞with ε+<∞is bounded if and only

if τ∈W2(div,Ω). This shows that there exists a function in Wq(div=9f, Ω) with

ﬁnite energy if and only if the right-hand side fallows for the existence of a function

τ∈W2(div,Ω) with −div τ=f. Since the minimizer σin (2) satisﬁes −div σ=f,

suﬃcient conditions are the following.

Theorem 2 (Maximal regularity).Let σbe the minimizer in (2)and let Ω⊂Rd

be a bounded domain.

(a) If Ωhas a C1,α-boundary ∂Ωfor some α∈(0,1], we have for any r≥q

kσkLr(Ω) .kfkW−1,r (Ω).

(b) If Ωis a convex open set with d≥2, we have

kσkL2d

d−2(Ω) .k∇σkL2(Ω) .kfkL2(Ω).

Proof. The statement in (a) is shown for equations in [KZ01, Thm. 1.6] and for

systems in [BCDKS18, Thm. 4.1]. The ﬁrst estimate in (b) follows from the Sobolev

embedding theorem, the second is shown in [CM18, Thm. 2.3] for equations and in

[CM19;BCDM22] for systems. 

If the set W2(div=9f, Ω) is not empty, the direct method in the calculus of

variations leads to the existence of a unique minimizer

σε= arg min

τ∈Wq(div=9f,Ω)

J∗

ε(τ).(10)

3. Convergence in the relaxation parameter

This section shows that the minimizer σε∈W2(div=9f, Ω) of the relaxed energy

J∗

εconverges to the minimizer σ∈W2(div=9f, Ω) of J∗as the interval ε→(0,∞).

In particular, we derive an upper bound for the relaxation error σε−σ.

Theorem 3 (Convergence in ε).Suppose that the minimizers σε∈Wq(div=9f, Ω)

with ε= [ε−, ε+]⊂(0,∞)exist and that σ∈Lr(Ω; Rd)for some r≥2. Then the

energy diﬀerence is bounded by

J∗(σε)− J ∗(σ)≤ J ∗

ε(σε)− J ∗(σ)≤|Ω|

qεq

−+1

qε−(r−q)

+kσkr

Lr(Ω).

(11)

Proof. The ﬁrst inequality in (11) follows by the monotonicity of Jεwith respect to

the relaxation parameter. The minimizing property of σε∈W2(div=9f, Ω) implies

J∗

ε(σε)− J ∗(σ)≤ J ∗

ε(σ)− J ∗(σ) = ZΩκε(|σ|)−1

q|σ|qdx.(12)

It follows from the deﬁnition of κεin (9) that

κε(|σ|)−1

q|σ|q≤









εq

−/q in {|σ| ≤ ε−},

0 in {ε−<|σ| ≤ ε+},

εq−2

+|σ|2/q in {ε+<|σ|}.

(13)

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RELAXEDKACANOVSCHEMEFORTHEp-LAPLACIANWITHLARGEpANNAKH.BALCI,LARSDIENING,ANDJOHANNESSTORNAbstract.WeintroduceagloballyconvergentrelaxedKacanovschemeforthecomputationofthediscreteminimizertothep-Laplaceproblemwith2p<1.Theiterativeschemeiseasytoimplementsinceeachiterateresultsonlyfromthesolveofaweig...

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