FINITE DIFFERENCE METHODS FOR THE INFINITY

LAPLACE AND p-LAPLACE EQUATIONS

ADAM M. OBERMAN

Abstract. We build convergent discretizations and semi-implicit solvers for

the Infinity Laplacian and the game theoretical p-Laplacian. The discretiza-tions simplify and generalize earlier ones. We prove convergence of the solution

of the Wide Stencil finite difference schemes to the unique viscosity solution

of the underlying equation. We build a semi-implicit solver, which solves theLaplace equation as each step. It is fast in the sense that the number of itera-

tions is independent of the problem size. This is an improvement over previous

explicit solvers, which are slow due to the CFL-condition.

1. Introduction

The Infinity Laplacian equation is at the interface of the fields of analysis, non-linear elliptic Partial Differential Equations (PDEs), and probabilistic games. Itwas first studied in the late 1960s by the Swedish mathematician Gunnar Aronsson[Aro67, Aro68, Aro84], motivated by classical analysis problem of building Lips-chitz extensions of a given function. Aronsson found non-classical solutions, but arigorous theory of weak solutions was not yet available. It took a few decades untilanalytical tools were developed to study the equation rigorously, and computationaltools were developed which made numerical solution of the equation possible.

In the last decade, PDE theorists established existence and uniqueness, andregularity results. The theory of viscosity solutions [CIL92] is the appropriateone for studying weak solutions to the PDE. But the general uniqueness theorydid not apply to this very degenerate equation, so proving uniqueness required anew approach. The first uniqueness result was due to Jensen [Jen93], followed bya different proof by Barles and Busca [BB01]. Later, the connection with finitedifference equation was exploited by Armstrong and Smart, and they were to givea short uniqueness proof for the PDE [AS10] . The differentiability of solutionsremained an open question for some time. The first result was obtained by [Sav05] intwo dimensions, followed by [ES08], and [ES11b] and [ES11a] in general dimensions.

The first convergent difference scheme was presented in [Obe05]. A numericalscheme using the extension property can be found in LeGruyer [LG07]. Two differ-ent numerical methods were derived in [ES11a], one adapted the monotone schemein [Obe05] to the standard Infinity Laplacian (which is homogeneous degree twoin ∇u), the second, quite different, used the variational structure of a regularizedPDE.

Date: December 6, 2012.Key words and phrases. Nonlinear Partial Differential Equations, Infinity Laplace, Semi-

Implicit Solver, Viscosity Solutions, p-Lapalcian, Random Turn Games.The author is grateful to Selim Esedoglu for suggesting applying Smereka’s work on the semi-

implicit iteration to this equation.

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2 ADAM M. OBERMAN

Earlier work by LeGruyer [LGA98] proved uniqueness for a related finite dif-ference equation. The proof is a generalization of the uniqueness proof for linearelliptic finite difference schemes from [MW53].

A group of probabilists, Peres-Shramm-Sheffield-Wilson [PSSW09] studying arandomized version of a marble game called Hex found a connection with the InfinityLaplacian equation. This connection gives an interpretation of the equation as atwo player random game. This is related to work by Kohn-Serfaty [KS06] who foundan interpretation of the equation for motion of level sets by mean curvature, [OS88][ES91] as a deterministic two player game. This equation was also interpreted as anonlinear average [Obe04] for the purpose of finite difference schemes. Deterministicgame interpretations for more general PDEs followed in [KS10]. The connectionbetween these various game interpretations was further studied in [Eva07].

The rich connection between games, finite difference schemes, and nonlinearelliptic PDEs is now much better understood. There have been a number of worksin this area, in particular on the game theoretical p-Laplacian. The probabilisticgames interpretation can be found in [PS08] (see also [MPR10]). Related worksinclude biased games which corresponds to a gradient term [ASS11]

This article will further exploit the connection between games, finite differ-ence schemes, and nonlinear elliptic PDE, by building convergent finite differenceschemes which are consistent with the game interpretation. The existence anduniqueness results are now established, efficient numerical solution of the equationremains a challenge. The original convergent scheme proposed in [Obe05] con-verged, but is not efficient: as the grid size grows, so does the number of iterationsrequired to find the solution. This article improves and simplifies the original dis-cretization, and also finds fast solution methods. It also generalizes the scheme andthe solvers to the game-theoretical p-Laplacian, which is a convex combination ofthe Laplacian and the Infinity Laplacian.

1.1. Introduction to numerical methods for degenerate elliptic PDEs.There are two major challenges in building numerical solvers for nonlinear anddegenerate elliptic Partial Differential Equations (PDEs). The first challenge is tobuild convergent approximations, usually by finite difference schemes. The secondchallenge is to build efficient solvers.

The approximation theory developed by Barles and Souganidis [BS91] providescriteria for the convergence of approximation schemes: monotone, consistent, andstable schemes converge to the unique viscosity solution of a degenerate ellipticequation. But this work does not indicate how to build such schemes, or howto produce fast solvers for the schemes. It is not obvious how to ensure thatschemes satisfy the comparison principle. The class of schemes which for whichthis property holds was identified in [Obe06], and were called elliptic, by analogywith the structure condition for the PDE.

An important distinction for this class of equations is between first order (Hamilton-Jacobi) equations, and the second order (nonlinear elliptic) case. The theory ofviscosity solutions [CIL92] covers both cases, but the numerical methods are quitedifferent. In the first order case, where the method of characteristics is available,there are some exact solutions formulas (e.g. Hopf-Lax) and there is a connectionwith one dimensional conservation laws [Eva98]. The second order case has more incommon with divergence-structure elliptic equations, but because of the degeneracy

FINITE DIFFERENCE METHODS FOR IFINITY LAPLACE EQUATION 3

or nonlinearity, many of the tools from the divergence-structure case (e.g. finiteelements, multi grid solvers) have not been successfully applied.

In the first order case, there is much more work on discretizations and fast solvers.For Hamilton-Jacobi equations, which are first order nonlinear PDEs, monotonicityis necessary for convergence. Early numerical papers studied explicit schemes fortime-dependent equations on uniform grids [CL84, Sou85]. These schemes havebeen extended to higher accuracy schemes, which include second order convergentmethods, the central schemes [LT00], as well as higher order interpolation methods,the ENO schemes [OS91]. Semi-Langrangian schemes take advantage of the methodof characteristics to prove convergence [FF02]. These have been extended to thecase of differential games [BFS99]. Two classes of fast solvers have been developed,fast marching [Set99], and fast sweeping [TCOZ03], The fast marching and fastsweeping methods give fast solution method for first order equations: both takeadvantage of the method of characteristics, which is not available in the secondorder case.

There is much less work in the second order degenerate elliptic equations. Theequation for motion by mean curvature [OS88], [ES91] has been extensively stud-ied. There is an enormous literature on this equation, but we just closely relatedreferences. The connection with games was already mentioned above. Numericalschemes include [CFF06] and [CFF10] . In the case of motion by mean curvature,the equation is time-dependent, so a fast solver would allow larger time steps. Forthis equation, a semi-implicit solver has been built by Smereka [Sme03]. The ideafrom the Smereka paper will be adapted in this work to build fast solvers for InfinityLaplace. Another equation in this class is the Hamilton-Jacobi-Bellman equations,for the value function of a stochastic control problem. Applications include port-folio optimization and option pricing in mathematical finance. Numerical worksinclude the early paper [LM80] and [CF95], and a paper on fast solvers [BOZ04].

For uniformly elliptic PDEs, monotone schemes are not necessary for convergence(for example most higher order Finite Element Methods are not monotone). But forfully nonlinear or degenerate elliptic, the only convergence proof currently availablerequires monotone schemes. One way to ensure monotone schemes is to use WideStencil Finite difference schemes, this has been done for the equation for motion bymean curvature, [Obe04], for the Infinity Laplace equation [Obe05], for functionsof the eigenvalues [Obe08b], for Hamilton-Jacobi-Bellman equations [BZ03], andfor the convex envelope [Obe08a]. Even for linear elliptic equations, a wide stencilscheme maybe necessary for to build a monotone scheme [MW53]. In some cases,simple finite difference schemes, with minor medications, can give good results, as isthe case for the Monge-Ampere equation [BFO10]. But we show below that simplefinite difference schemes are not convergent for the Infinity Laplace equations.

The second challenge, which is quite distinct from the first, is to build solversfor the finite difference schemes. For fixed values of h, the finite difference schemeis a finite dimensional nonlinear algebraic equation which must be solved. Buildingsolvers demands very different techniques, and little progress has been made, inpart, due to the fact that the discrete equations can be non-differentiable, whichprecludes the use of the Newton’s method. To date, the only general solver availableis a fixed point iteration, which corresponds to solving the parabolic version of theequation for long time. This method is restricted by a nonlinear version of theCFL condition [Obe06], which means the number of iterations required to solve the

4 ADAM M. OBERMAN

equation increases with the problem size. For the Monge-Ampere equation, fastsolvers have been built using Newton’s method [FO11b] [FO11a], but this equationhas a different structure (convex, differentiable) from the Infinity Laplace (or thep-Laplace) equation.

1.2. Contribution of this work. The first contribution of this work is to build aprovably convergent discretization of the operator. The issue here is to ensure thatthe discretization convergences (in the limit of the discretization parameters goingto zero) to the unique viscosity solution of the PDE. Simply using standard finitedifferences fails to converge, as shown below.

The appropriate notion of weak solutions for the PDE is provided by viscositysolutions [CIL92, ACJ04]. The only schemes which can be proven to converge to vis-cosity solutions are monotone schemes [BS91]; these schemes satisfy the maximumprinciple at the discrete level [Obe06]. For the variational p-Laplacian, GalerkinFinite Element methods could be used. But the game-theoretical version is nota divergence structure operator, so there is not a natural version of weak so-lutions. Monotone schemes can be proven to converge for the game-theoreticalp-Laplacian (pLap). We prove convergence of the solution of the Wide Stencilfinite difference schemes to the unique viscosity solution of the underlying equa-tion (pLap) (D).

The second contribution of this work is to build fast solvers for (pLap). Thereare two reasonable ways to quantify the notion of a fast solver. The first notionof speed is absolute: the number of operations to solve the equation the shouldbe proportional to the problem size. The second notion of speed is relative: wecompare the speed of our solvers to the speed of solvers for a related but easierproblem. Here, it is natural to compare with the solution speed of the Laplaceequation.

Explicit solvers are available and simple to implement, but they are not fast. Anymonotone scheme can be solved using an iterative, explicit method [Obe06]. Theexplicit method can be interpreted as a Gauss-Seidel solver, or the forward Eulermethod for the equation ut = ∆p. However the time step for the Euler method isO(h2), where h is the spatial resolution. The explicit method is not fast becausethe number of iterations required for it to converge is O(1/h2), which increaseswith the problem size.

The method we propose is semi-implicit, with the implicit step given by solvingthe Laplace equation. The Laplace equation can be solved in O(N) operations,using Fast Fourier Transforms, or O(N logN), using sparse linear algebra. Thus,to the extent needed for our rather coarse analysis, both notions of speed coincide,provided the solution is obtained in a finite (small) number of iterations.

1.3. The setting for the PDE. This work is concerned with the efficient numer-ical solution of a nonlinear, degenerate elliptic Partial Differential Equation (PDE),the normalized Infinity Laplacian. The PDE operator is given by

(IL) ∆∞u =1

|∇u|2d∑

i,j=1

uxiuxixj

uxj

where u(x) : Rd → R.

FINITE DIFFERENCE METHODS FOR IFINITY LAPLACE EQUATION 5

We also study a closely related PDE, the game theoretical p-Laplacian, whichinterpolates between the 1-Laplacian, ∆1,

(MC) ∆1u = |∇u|div(∇u/|∇u|) = ∆u−∆∞u

and the infinity Laplacian, ∆∞. Expanding the ∆1 operator above leads to theidentity

(1) ∆ = ∆∞ + ∆1,

which we record for future use. The game theoretical p-Laplacian is the p weightedaverage of the 1- and ∞-Laplacians,

∆p =1

p∆1 +

1

q∆∞, p−1 + q−1 = 1.(pLap)

This is consistent with the definitions given in the probabilistic games interpreta-tion [PS08] (see also [MPR10]). The normalized versions of the operators are alsoused in image processing [CMS98], [CEPB04], [MST06].

Special cases occur for p = 1 and∞, as above, and for p = 2 we obtain ∆2 = 12∆.

Using the identity (1) we can also write

(2) ∆p = α∆ + β∆∞, α = 1/p, β = (p− 2)/p

Equation (2) will be used for p ∈ [2,∞]. If we were to consider the case p ∈ [1, 2],the equation above is not a positive combination of the operators. Instead, for thecase p ∈ [1, 2], the corresponding representation would be a convex combination ofthe monotone discretization of ∆1 and the Laplacian. Here we will focus on thecase where the Infinity Laplace operator is active.

We consider the Dirichlet problem for the operator, in a domain Ω ⊂ Rd, with agiven right hand side function g.

∆pu(x) = g(x), for x ∈ Ω,(PDE)

u(x) = h(x), for x ∈ ∂Ω.(D)

Here g represents a running cost for a probabilistic game [PS08]. When g = 0, theoperator coincides with the variational p-Laplacian. The relationship between thegame theoretical p-Laplacian and the variational p-Laplacian is given below.

1.4. Failure of the standard finite difference scheme. Here we motivate theneed for a convergent scheme, by showing that the standard finite difference schemefails to converge.

A natural scheme is given by standard finite differences, along with a smallregularization for the norm of the gradient. For this we use standard centeredfinite differences for uxx, uyy, ux, uy, and the with the symmetric scheme for uxy,

ux(x, y) =1

2h(u(x+ h, y)− u(x− h, y)) +O(h2),

uxx(x, y) =1

h2(u(x+ h, y)− 2u(x, y) + u(x− h, y)) +O(h2),

uxy(x, y) = +1

4h2(u(x+ h, y + h) + u(x− h, y − h))

− 1

4h2(u(x− h, y + h) + u(x+ h, y − h)) +O(h2)

and similarly for the uy, uyy terms. In order to regularize the gradient, we replaced‖∇u‖2 with maxh2, ‖∇u‖2.

6 ADAM M. OBERMAN

We computed the solution with boundary conditions corresponding to the exactAronsson solution [Aro68]. The finite difference scheme presented above fails toconverge, see Figure 1. In this case, the solution has the form |x|− |y| in the centre.In fact, it can be shown using symmetry considerations that |x| − |y| is an exactsolution of the symmetric finite difference scheme. On the flat parts, the operatoris zero, so we only need to check the corners. In fact, u(x, y) = |x| and u(x, y) = |y|are also exact solutions. While other discretizations are possible which break thissymmetry, we tried several other simple consistent finite difference schemes andwere always able to find examples where they failed to converge.

−10

1

−1

0

1−1

0

1

−1 −0.5 0 0.5 10

0.5

1

Figure 1. Standard finite difference solutions of the ∞-Laplacian. Boundary data is a |x|4/3 − |y|4/3, the computed so-lution is incorrect, with a singularity of the form |x| − |y| at theorigin. (a) Surface Plot, (b) plot of u(x, 0).

1.5. Variational p-Laplacian. To clarify any confusion, we also discuss the vari-ational p-Laplacian. This operator arises in the variational problem

J [u] =

∫Ω

|∇u(x)|p dx, u = h on ∂Ω

for 1 < p <∞. The Euler-Lagrange equation for the minimizer is

div(|∇u(x)|p−2∇u

)= 0.

The game theoretical p-Laplacian, (pLap), is related to the variational version bya normalization which makes it homogeneous of order zero in the norm of thegradient.

(3) ∆pu =1

p|∇u|p−2div(|∇u(x)|p−2∇u

)To check consistency with definition (pLap), expand the two terms in the operator

above to give ∆pu = 1p∆u+ (p−2)

p ∆∞u which is the representation (2).

When the right hand side function, g, is zero, the solutions of both versions ofthe p-Laplacian equations coincide.

FINITE DIFFERENCE METHODS FOR IFINITY LAPLACE EQUATION 7

2. Viscosity solutions

In this section we recall the definition of viscosity solutions for the infinity Lapla-cian and the definition of consistency used in the convergence theory.

Definition 1 (Viscosity solutions). (1) A continuous function u defined on the setU , is a viscosity subsolution of −∆∞u = 0 in U , if for every local maximum pointx ∈ U of u− φ, where φ is C2 in some neighborhood of x, we have

−∆∞φ(x) ≤ 0, if Dφ(x) 6= 0,

−ηTD2φ(x)η ≤ 0 for some |η| ≤ 1, if Dφ(x) = 0.

(2) A continuous function u defined on the set U , is a viscosity supersolution of−∆∞u = 0 in U , if for every local maximum point x ∈ U of u− φ, where φ is C2

in some neighborhood of x, we have−∆∞φ(x) ≥ 0, if Dφ(x) 6= 0,

−ηTD2φ(x)η ≥ 0 for some |η| ≤ 1, if Dφ(x) = 0.

(3) Moreover, a continuous function defined on the set U , is a viscosity solution of−∆∞u = 0 in U , if it is both a viscosity subsolution and a viscosity supersolutionin U .

Consistency requires only that we can apply the test function definition in thelimit.

Definition 2 (Consistency). The numerical scheme ∆dx,dθ∞ is consistent if for every

φ ∈ C2(U), and for every x ∈ U ,

limdx,dθ→0

∆dx,dθ∞ (φ)(x) = −∆∞φ(x)

if Dφ(x) 6= 0, and

(4) λ ≤ lim infdx,dθ→0

∆dx,dθ∞ (φ)(x) ≤ lim sup

dx,dθ→0∆dx,dθ∞ (φ)(x) ≤ Λ

where λ,Λ are the least and greatest eigenvalues of D2φ(x), otherwise.

By a theorem of Barles-Souganidis [BS91], consistent, monotone schemes con-verge to the viscosity solution of the PDE, provided this solution is unique.

3. Discretization

In this section we present the discretization of the Infinity Laplace operator,which is needed for convergence to the viscosity solution. The discretization wepresent here is different from the one in [Obe05]. The previous scheme was givenby solving the a discrete version of the Lipschitz extension problem. This schemeis simpler, since the resulting equation is explicit. In addition, this scheme givesthe correct scaling in h which is needed for a non-zero right hand side.

By now it is well known that, for smooth functions with non-vanishing gradient,the operator is approximated by the average below. This result follows from Taylorexpansions, and the fact that the minimum (or maximum) is in the direction ofthe gradient, at least to O(h). We show below that the accuracy is actually O(h2),which is an improvement over previous results.

8 ADAM M. OBERMAN

Lemma 1. Let u(x) be a smooth function with non-vanishing gradient at x. Then

(5) ∆∞u(x) = min|y−x|=ε

u(y)− u(x)

ε2+ max|y−x|=ε

u(y)− u(x)

ε2+O(ε2).

Proof. We prove the result in two dimensions, which is all that is needed here.A longer proof is possible which in higher dimensions, which requires a Lagrangemultiplier, λ for the constraint, and an asymptotic expansion in λ as well.

It is enough to consider u(x) = pTx + 12x

TQx, for nonzero p = ∇u(x), and

symmetric matrix Q = D2u(x). Use the notation p = p|p| , p

⊥ = (−p2, p1), for

p = (p1, p2). With this notation, the operator, (IL) is given by

∆∞u = (p)TQp(6)

Write

F (θ) = u(εx(θ)) = εpTx(θ) +ε2

2x(θ)TQx(θ), x(θ) = (cos(θ), sin(θ))

Then a critical point of F is given by

0 = F ′(θ) = (p+ εQx(θ))Tx′(θ) = (p+ εQx)Tx⊥,

where x′(θ) = x⊥(θ) = (− sin(θ), cos(θ)). Perform an asymptotic expansion thecondition in ε, with x = x0 + εx1, to obtain

(p+ εQx0)T(x0 + εx1)⊥ +O(ε2) = 0.

Then the O(1) terms give

pTx⊥0 = 0

which yieldsx0 = ±p,

and the O(ε) terms give

pTx⊥1 + (x⊥0 Q)Tx0 = 0

which yields

x1 = − (p⊥)TQp

|p|p⊥

(Note that we are violating the constraint that x be a unit vector, but the constraintis still satisfied to O(ε2)). Write

x+ = arg maxθF (θ), x− = arg min

θF (θ)

Then we have

x± = ±p+ εcp⊥, c = − (p⊥)TQp

|p|.

Inserting the values for x±, into the expression on the right hand side of (5) gives

min|y−x|=ε

u(y)− u(x)

ε2+ max|y−x|=ε

u(y)− u(x)

ε2=F (εx−) + F (εx+)

ε2

= pTQp+ ε2c2((p⊥)TQp⊥

)= ∆∞u+ ε2

((p⊥)TQp

|p|

)2

∆1u

= ∆∞u+ ε2c2∆1u

which gives the desired result.

FINITE DIFFERENCE METHODS FOR IFINITY LAPLACE EQUATION 9

dθdθ

dθ

dx dx

dx

Figure 2. Stencils for the 5, 9, and 17 point schemes

However, for any given grid, it is impossible to sample the values on the entirecircle. Instead, only values in a discrete set of directions are available. While itis certainly possible to interpolate the values onto the ball, quadratic interpolationis not monotone, so it violates the maximum principle, which is needed for theconvergence proof. As we show in an example below, non-monotone schemes donot converge for this equation.

So an additional discretization parameter is needed, which we present in whatfollows.

Definition 3 (Spatial and directional resolution). Given a stencil of neighbouringgrid points v1, . . . , vn on a Cartesian grid, define the local spatial resolution, h, tobe the maximum length of the neighbours

(7) h =n

maxi=1|vi|,

and the local directional resolution, dθ, to be the maximum directional distance toa neighbour

(8) dθ = max|v|=1

nmini=1

∣∣∣∣v − vi|vi|

∣∣∣∣The direction vectors used will be on a grid, arranged as in Figure 2. In practice,

we obtain acceptable accuracy using a relatively narrow stencil.

Definition 4 (Scheme definition). Define the discretization of ∆∞ to be given by

(9) ∆h,dθ∞ u(x) = max

i

u(x+ vi)− u(x)

|vi|2+ min

i

u(x+ vi)− u(x)

|vi|2

where vi are the neighbours of the point x, as in Figure 2.

Next we prove consistency, when the parameters dx, dθ go to 0. For the gridpoints, we will assume

whenever vi is a neighbouring grid point, −vi is as well,(10)

maxi |vi|mini |vi|

= o(1)(11)

Theorem 2. Let u be a smooth function in a neighbourhood of x, then

(12) ∆h,dθ∞ u(x) = ∆∞u(x) +O(dθ + h2)

10 ADAM M. OBERMAN

Proof. It is enough to consider

u(x) = pTx+1

2xTQx,

for p = ∇u(x), and symmetric matrix Q = D2u(x). Use the notation p = p|p| .

First consider the case Du 6= 0. Define

v+ = arg maxi

u(x+ vi)− u(x)

|vi|2

v− = arg mini

u(x+ vi)− u(x)

|vi|2Using a Taylor expansion for u, we have

u(x+ vi)− u(x)

|vi|2=vip+ C|vi|2

|vi|2

so, as h → 0, the maximum occurs at v+ = arg maxi vip which is the closestdirection vector to p, so

(13) v+ = p+O(dθ)

by (8) and similarly

(14) v− = p+O(dθ),

In addition, since according to (10) we assume that the grid points are arrangedsymmetrically, we have

(15) v+ = −v−

as h, dθ → 0.Then using the Taylor expansion for u, insert (13,14) into (9) to obtain

∆h,dθ∞ u(x) =

(v+

|v+|2+

v−

|v−|2

)p+

1

2

v+Qv+

|v+|2+v−Qv−

|v−|2+O(h)

=1

2

v+Qv+

|v+|2+

1

2

v−Qv−

|v−|2+O(h) by (15)

= pQp+O(h+ dθ) by (13,14)

which is consistent, and accurate to O(h+ dθ).In the case Du(x) = 0, we are only required only to verify (4). In this case we

can assume u(x) = 12x

TQx, and compute

maxi

u(x+ vi)− u(x)

|vi|2=

1

2maxivi

TQvi =1

2Λ +O(dθ + h2)

and similarly

mini

u(x+ vi)− u(x)

|vi|2=

1

2minivi

TQvi =1

2λ+O(dθ + h2)

where λ,Λ are the smallest and largest eigenvalues of Q, respectively. So then

∆h,dθ∞ u(x) =

1

2(λ+ Λ) +O(dθ + h2)

which is consistent, according to (4).

FINITE DIFFERENCE METHODS FOR IFINITY LAPLACE EQUATION 11

3.1. The discretization of the p-Laplacian. We wish to build a consistent,monotone scheme for

∆p = α∆ + β∆∞.

Starting with the scheme for ∆∞, we can simply combine this with the standardfinite differences for the Laplacian,

∆hu(·) = 4u− u(·)h2

,(16)

u(x, y) =1

4(u(x+ h, y) + u(x− h, y) + u(x, y + h) + u(x, y − h))

Then, using the characterization of monotone schemes from [Obe06], the com-bined scheme is still monotone.

Theorem 3 (Convergence). The solution of the difference scheme for the p-Laplacian, (2),which is given by a convex combination of the schemes for (IL), (9) and the standardfinite difference scheme for the Laplacian, (16)

∆h,dθp = α∆h + β∆h,dθ

∞

converges (uniformly on compact sets) as h, dθ → 0 to the solution of (3).

Proof. The uniform convergence of solutions of consistent, monotone schemes fol-lows from the main result of [BS91], provided solutions are unique. The uniquenessfollows in our case (although not in the case p = 1). Thus by appealing to thisresult we need only establish consistency and monotonicity of the schemes.

Consistency of the discretization of the ∆h,dθ∞ operator follows from Theorem 2.

Consistency of the discretization for (3) follows since it is a convex combination of∆∞ and the consistent discretization of the Laplacian (16).

The definition (9) expresses the scheme ∆h,dθ∞ as a nondecreasing combination

of differences between u(xi) − u(x), where xi are neighbouring grids points to thereference point x. This characterizes elliptic schemes, which were proven to bemonotone in [Obe06]. Monotonicity of the discretization of the ∆p operator is aconsequence of the fact that it is a convex combination of two elliptic schemes isalso elliptic, which was also shown in [Obe06].

Together these results prove convergence.

4. Solvers

When we discretize (3) we obtain a system of nonlinear equations. These equa-tions inherit contraction properties from (3), namely stability in the maximumnorm. Our first goal is solve the equations, and our second goal is to solve themquickly, ideally in O(N) iterations, where N is the number of variables in the dis-crete system of equations.

The main issues to consider are stability of the iteration, and the convergencerate. For implicit or semi-implicit schemes, we also need to solve equations involvingthe operator. This generally requires that the implicit operator be linear.

4.1. Explicit methods. There are no general methods available for solving non-divergence form nonlinear elliptic equations. For monotone schemes, explicit meth-ods can be used [Obe06]. The explicit method in the case of (3) is

un+1 = un + ρ(∆pun − g)

12 ADAM M. OBERMAN

where, to ensure stability, the artificial time step ρ is restricted to be the inverseof the Lipschitz constant of the scheme, regarded as a grid function. The Lipschitzconstant of the scheme is the inverse of the coefficient of u(x) in the operatorevaluated at x. It is ρ = O(h2), usually h2/2 although if a wide stencil is also usedfor ∆ is can be made slightly larger. Explicit methods are slow because of thisrestriction, which can be regarded as a nonlinear CFL condition. The number ofiterations required for convergence grows with the problem size.

Fully implicit methods require the solution of nonlinear equations. In the case

of (3), the fully implicit scheme un+1−un

ρ = ∆p[un+1]− g is unconditionally stable,

but requires solving the equation

un+1 − ρ∆pun+1 = un − ρg.

which has the same difficulties as solving the time-independent equation.

4.2. Semi-implicit methods. We are motivated by the work [Sme03] which builta semi-implicit solver for the one-Laplacian, ∆1, by treating the linear part of theoperator implicitly.

Write

∆∞ =1

2(∆ + (∆∞ −∆1))

which follows from (1). This leads to the semi-implicit scheme

(17) −∆un+1 = −(∆1 −∆∞)un − 2g.

The general case for ∆p follows.

Remark 1. In practice, we only use the discretization of ∆∞ and the identity (1),so no discretization of ∆1 is needed.

4.3. The semi-implicit scheme for ∆p. Rewrite the equation (pLap) symmet-rically in terms of p, q as follows

∆p =p−1 + q−1

2(∆1 + ∆∞) +

p−1 − q−1

2(∆1 −∆∞),

which gives

∆p =1

2(∆ + β(∆1 −∆∞)) , β = 1− 2

p.

The resulting scheme, which generalizes (17), is given by

(Iteration) −∆un+1 = β(∆∞ −∆1)un − 2g, β = 1− 2

p.

The convergence of the iteration depends on whether the operator

β(−∆)−1(∆∞ −∆1)

is a contraction is some norm. Clearly we can focus on the case |β| = 1 since thecase |β| < 1 is easier.

FINITE DIFFERENCE METHODS FOR IFINITY LAPLACE EQUATION 13

4.4. Proof of contraction for a linear model. It would be desirable to provethat the iteration defined by (Iteration) converges to the solution. But to do sorequires proving it is a contraction in some norm. The operator involved is notmonotone, so we are unable to prove it is a contraction in the uniform norm.The nonlinearity makes proving convergence in other norms difficult. However wepresent a heuristic for why the operator ∆−1(∆∞ −∆1) is a contraction based onan analogy with a linear operator. Instead of the operator above, we’ll perform theanalysis for

(−uxx − uyy)−1

(−uxx + uyy)

which is a linear operator which is also a difference of degenerate elliptic operators.To be concrete, consider the case of the unit square in two dimensions. Use asimultaneous eigenfunction expansion,

u =∑i

aiφi

with

Mφi = λ2iφi, where M ≡ ∂xx

Nφi = ν2i φi, where N ≡ ∂yy

Then

M −NM +N

φi =λ2i − ν2

i

λ2i + ν2

i

φi ≤ φi

So

M −NM +N

u =M −NM +N

∑aiφi =

∑aiλ2i − ν2

i

λ2i + ν2

i

≤∑

aiφi = u

In fact, the contraction rate is given by

maxi,j

|λ2i − ν2

j |λ2i + ν2

j

< 1, on a finite grid

So in this simplified linear setting, the operator is a contraction.

5. Numerical Results

5.1. Plots of solutions of ∆∞ with varying boundary data. In Figure 3 weplot solutions of the ∞-Laplacian, with variable boundary data, given by

|x| − |y|+ ci3x+ 2y

8√

14, for ci = 0, 1, 2, 3.

Displayed are surface plots of the solution and a contour plot of the norm of thegradient of the solutions. Note how the changing boundary data moves the kinks inthe solution from a symmetric arrangement to a perturbed one. The computationaldomain was given by [−1, 1]2 with a 2002 grid.

14 ADAM M. OBERMAN

−0.5 0 0.5

−0.5

0

0.5

−0.5 0 0.5

−0.5

0

0.5

−0.5 0 0.5

−0.5

0

0.5

−0.5 0 0.5

−0.5

0

0.5

Figure 3. Solutions of the ∞-Laplacian. Boundary data is a|x| − |y|+ ci(3x+ 2y)/

√14, for ci = 0, 1/8, 2/8, 3/8, 4/8.

FINITE DIFFERENCE METHODS FOR IFINITY LAPLACE EQUATION 15

5.2. Plots of solutions for varying p. We solved the p-Laplacian on a 2002 grid,with values of β = 0, 1/3, 2/3.1, which corresponds to p = 2, 3, 6,∞. The boundarydata was |x| − |y| on [−1, 1]2 .

In Figure 4 we show the surface plots of the solution and a contour plot of thenorm of the gradient of the solution. The contours plotted are at levels sets equallyspaced between 0 and 1. Note in the surface plot, the sharpening of the saddle toa kink in the gradient as p increases. In the contour plots, the contours get smalleras p increases, showing that the solution is decreasing the size of the gradient. Notealso that the shape of the contours go from circular (for the Laplacian) to diamondshaped (for the Infinity Laplacian). Also note that the gradient in larger along theaxes, which is where (presumably) the singularity in the solution is found.

6. Solver speed

In this section we report on computational experiments which demonstrate theperformance of the semi-implicit method.

The first section compares the semi-implicit method, which convergences at arate which is independent of the problems, to the explicit method, which takesincreasingly longer to converge as the problem size grow.

The next section gives further details on the solution of ∆∞. Engineering preci-sion is obtained in the first few iterations. But numerical precision can still requiremany iterations.

The final section compares the speed of the solution of the p-Laplacian. Settingp <∞ improves the convergence rate of the solver.

6.1. Speed of the explicit method. The explicit method converges exponen-tially, but at a rate which depends on the problem size. For small problem sizes,n = 64, we get an error of .1 after 200 iterations, and .001 after 1000 iterations.But for large problems sizes, n = 512, the error is .7 even after 1000 iterations.This is prohibitively slow for larger problems sizes.

In Figure 5 the error is shown as a function of the number of iterations, fordifferent problem sizes. Increasing the problem size means the error increases aswell, for a fixed number of iterations.

We compared the semi-implicit method with the explicit method. The errordecreases faster for the semi-implicit method. See Figure 6 for a typical example.

6.2. Solver speed for Infinity Laplace. In the section we examine the perfor-mance of the semi-implicit method for solutions of Infinity Laplace.

The advantage of the semi-implicit solver is that the convergence rate does notdepend on the size of the problem. Another advantage is that the semi-implicitsolver reduces the error to engineering tolerances, .1, after one iteration, andachieves close to another order of magnitude after four iterations. See Figure 7.This is useful for applications, for example in imaging, where high accuracy is notrequired.

However, the specific rate of convergence varies for different solutions. Compar-ing several solutions, it appears to be slowest for the solution u(x, y) = x4/3− y4/3,see Figure 8

6.3. Solver speed for solutions of p-Laplacian. We study the convergencerate of the iterative method for the p-Laplacian with p < ∞. We also study the

16 ADAM M. OBERMAN

−0.5 0 0.5−0.5

0

0.5

−0.5 0 0.5−0.5

0

0.5

−0.5 0 0.5−0.5

0

0.5

−0.5 0 0.5−0.5

0

0.5

Figure 4. Solution of the p-Laplacian with b = 0, 1/3, 2/3.1,giving p = 2, 3, 6,∞. Boundary data is |x| − |y|. (a) Surface plotof the solutions. (b) Contour plots of the norm of gradient of thesolution.

FINITE DIFFERENCE METHODS FOR IFINITY LAPLACE EQUATION 17

0 200 400 600 800 100010

−3

10−2

10−1

100

Figure 5. Maximum error as a function of the number of it-erations in the solution of ∆∞ using the explicit method. Forn = 64, 128, 256, 512. Errors are increasing with n.

0 20 40 60 80 10010

−3

10−2

10−1

100

Figure 6. Maximum error as a function of iterations for solutionsof Infinity Laplace. Semi-implicit method (better accuracy) andExplicit method (less accurate), with n = 128. Solution is x4/3 −y4/3.

convergence rate in terms of problem size. One notable fact is that the accuracyafter just a few iterations is to engineering tolerances.

First we display the speed of convergence in terms of the number of iterationsfor different values of α, where α = 1/p as given in (2).

The results which study the convergence rate for a particular example solution asa function of α is presented in Figure 9 and Figure 10. The convergence rate is in-dependent of the size of the problem. The convergence increases with α, which is tobe expected, since the operator ∆p is becoming better approximated by the Lapla-cian. The convergence rate is consistent with an exponential error proportional to10µ(α)N , where the convergence rate µ(α) is consistent with an affine expression

µ(α) = −(c1 + c2α)

where c1 is small and and c2 ≈ 1.

18 ADAM M. OBERMAN

1 1.5 2 2.5 3 3.5 410

−2

10−1

100

Figure 7. Maximum error as a function of the number of itera-tions in the solution of ∆∞ using the semi-implicit method. Forn = 64, 128, 256, 512. Errors are nearly independent of n.

0 10 20 30 40 5010

−4

10−3

10−2

10−1

0 10 20 30 40 5010

−3

10−2

10−1

100

0 10 20 30 40 5010

−5

10−4

10−3

10−2

10−1

100

Figure 8. Maximum error as a function of iterations for solutionsof Infinity Laplace. For n = 64, 128, 256. Semi-implicit method.using exact solution of (a) x2 − y2 . (b) x4/3 − y4/3 (c) |x| − |y|

7. Conclusions

We built convergent discretizations and fast solvers for the ∆∞ and ∆p operators.Since solutions of the PDE (3) can be singular, it is important to use a convergent

discretization. The theory of viscosity solutions provides the appropriate notion of

FINITE DIFFERENCE METHODS FOR IFINITY LAPLACE EQUATION 19

0 20 40 60 80 10010

−12

10−10

10−8

10−6

10−4

10−2

100

Convergence

Iterations

Max

imum

err

or

Figure 9. Convergence in the maximum norm as a function ofthe number of iterations, for α = 1/2, 1/22, . . . , 1/218, 0.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035−0.04

−0.035

−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

Figure 10. Convergence rate as a function of α for α =1/2, 1/22, . . . , 1/218, 0. The line of best fit is −.87α− .0085.

weak solutions for the PDE (3). We simplified and generalized the Wide Stencilfinite difference scheme for ∆∞ used in [Obe05] to build a monotone finite differencescheme for ∆p. The discretization took advantage of the fact the the ∆p operatoris a positive combination of ∆ and ∆∞, given by (2).

A convergent discretization provides a finite dimensional equation whose so-lutions approximate the solutions of the PDE. Proving convergence is establishedusing PDE techniques, which does not address efficiency of solution methods. Stan-dard explicit methods are slow, in the sense that the number of iterations requiredto approximately solve the equations to a given level of precision grows with theproblem size. We introduced a semi-implicit solver, which converged at a rate inde-pendent of the problem size. The idea for the solver is to approximate the operator∆p by the Laplacian, and solve a linear equation at each iteration. While each it-eration is more costly than the explicit method, the added cost is more than madeup for by the fact that accurate solutions can be obtained in a few iterations.

Numerical results show that the method converges exponentially, at a rate whichis independent of the problem size, even in the case p = ∞. However, the conver-gence rate increases approximately linearly with 1/p, which is to be expected, since

20 ADAM M. OBERMAN

as p → 2 we recover the Laplacian and the solution is obtained in one step. Forα = 0 or nearby, engineering precision (maximum error of 10−2) is obtained in afew iterations, but the convergence to numerical precision is slower. For α awayfrom 0, the method is very fast, with numerical precision in about 20 iterations.The convergence rate also depends on the particular solution.

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Department of Mathematics, Simon Fraser University

E-mail address: aoberman@sfu.ca