DOI:10.1051/cocv/2012033 www.esaim-cocv.org
A LEAST-SQUARES METHOD FOR THE NUMERICAL SOLUTION OF THE DIRICHLET PROBLEM FOR THE ELLIPTIC MONGE − AMP` ERE
EQUATION IN DIMENSION TWO
∗Alexandre Caboussat
1,2, Roland Glowinski
3and Danny C. Sorensen
4Abstract.We address in this article the computation of the convex solutions of the Dirichlet problem for the real elliptic Monge−Amp`ere equation for general convex domains in two dimensions. The method we discuss combines a least-squares formulation with a relaxation method. This approach leads to a sequence of Poisson−Dirichlet problems and another sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite element approximations with a smoothing procedure are used for the computer implementation of our least-squares/relaxation methodology. Domains with curved boundaries are easily accommodated. Numerical experiments show the convergence of the computed solutions to their continuous counterparts when such solutions exist. On the other hand, when classical solutions do not exist, our methodology produces solutions in a least-squares sense.
R´esum´e. Nous ´etudions, dans cet article, une m´ethode num´erique, pour le calcul des solutions convexes du probl`eme de Dirichlet pour l’´equation de Monge−Amp`ere elliptique, dans des domaines bi-dimensionnel convexes. Une m´ethode de moindres carr´es est coupl´ee `a un algorithme de relaxa- tion, conduisant `a la r´esolution d’une suite de probl`emes de Poisson−Dirichlet, et d’une suite de probl`emes de valeurs propres de petite dimension d’un type nouveau. Une approximation par ´el´ements finis mixtes, coupl´ee `a une m´ethode de r´egularisation, est utilis´ee pour impl´ementer la m´ethode de moindres-carr´es/relaxation ci-dessus, de sorte que les domaines avec fronti`ere courbe sont trait´es fa- cilement. Des exp´eriences num´eriques montrent la convergence des solutions calcul´ees vers la solution convexe du probl`eme continu, lorsqu’une telle solution existe. Par ailleurs, si le probl`eme n’a pas de solution classique, notre m´ethodologie fournit des solutions au sens des moindres carr´es.
Mathematics Subject Classification. 65N30, 65K10, 65F30, 49M15, 49K20.
Received January 31, 2011. Revised August 21, 2012 Published online June 3, 2013.
Keywords and phrases. Monge−Amp`ere equation, least-squares method, biharmonic problem, conjugate gradient method, quadratic constraint minimization, mixed finite element methods.
∗This work was partially supported by the National Science Foundation (Grants NSF DMS-0913982 and DMS-0412267).
1 Haute ´Ecole de Gestion/Geneva School of Business Administration, Gen`eve, Switzerland.alexandre.caboussat@hesge.ch
2 University of Houston, Department of Mathematics, 4800 Calhoun Rd, Houston, 77204-3008 Texas, USA.
caboussat@math.uh.edu
3 University of Houston, Department of Mathematics, 4800 Calhoun Rd, Houston, 77204-3008 Texas, USA.roland@math.uh.edu
4 Rice University, Department of Computational and Applied Mathematics, MS 134, Houston, 77251-1892 Texas, USA.
sorensen@rice.edu
Article published by EDP Sciences c EDP Sciences, SMAI 2013
LEAST-SQUARES METHOD FOR THE ELLIPTIC MONGE−AMP `
1. Introduction
Iff ispositive, the canonicalMonge−Amp`ere equation detD2ψ=f,
is considered by many mathematicians as the prototypical fully nonlinear elliptic equation. As such, it has recently received considerable attention from both the analytical and computational standpoints as shown by, e.g., [2,5,6,14,24,37,40,41,43,44,50], with applications in geometry, mechanics and physics.
In particular, augmented Lagrangian algorithms and least-squares techniques have been used for the nu- merical solution of the Dirichlet problem for the Monge−Amp`ere equation in dimension two. These methods are discussed in [13,16–21,32,33]; actually, [32] contains a review of several methods for the solution of the Monge−Amp`ere equation and related fully nonlinear elliptic equations such as Pucci’s.
LetΩbe a bounded, convex domain of R2; we denote by∂Ω the boundary ofΩ. Assuming thatf ∈L1(Ω) andg∈H3/2(∂Ω), it makes sense (since the operatorϕ→detD2ϕis continuous fromH2(Ω) toL1(Ω)) to look in H2(Ω) for the solutions (convex, in particular) of the Dirichlet problem for the Monge−Amp`ere equation, that is
detD2ψ=f inΩ, ψ=g on∂Ω; (1.1)
see [10,11,20,37] for details. Suppose that problem (1.1) has convex (or concave) solutions with theH2-regularity;
if Ω is a square domain, using the augmented Lagrangian and least-squares methods discussed in [13,16–
21,32,33], combined with piecewise linear continuous finite element approximations, one has been able to solve problem (1.1) rather accurately. Indeed, the numerical experiments reported in the above references show that theL2approximation error isO(h2), which is, generically, optimal for second order elliptic problems, using this type of approximations. Using the above methodology, one has been able to compute least-squares solutions of (1.1) when, despite the smoothness of the dataf andg, this problem has no classical solutions, as it is the case for example whenΩ= (0,1)2,f = 1, andg= 0 [11,20,37]. Moreover, our method can be easily generalized to systems, unlike viscosity solutions which are based on maximum principles.
The least-squares methodology discussed in this article was introduced in [17] and further discussed in [21,32, 33]. Actually, the most detailed account-published so far-of our least-squares approach can be found in [21] (for a detailed description of the augmented Lagrangian based methodology see [19]). The methodology discussed in [17,21,32,33] relies on the following ingredients:
(i) A well-chosen least-squares formulation in appropriate Hilbert spaces [4].
(ii) Associating with the optimality conditions of the above least-squares problem an initial value problem (flow in the dynamical system terminology).
(iii) The time-discretization of the above initial value problem by an operator-splitting scheme decoupling nonlinearity and differential operators.
(iv) The solution of the nonlinear (resp., linear) problems resulting from the splitting by a Newton’s type algorithm (resp., by a preconditioned conjugate gradient algorithm).
(v) A mixed finite element approximation [8] of the Monge−Amp`ere problem (1.1) based on piecewise linear continuous approximations ofψand of its three second order derivatives.
Actually, since (a) the speed of convergence of the operator-splitting based iterative method mentioned in (iii) improves as the time discretization step increases, and (b) the above algorithm reduces to a simpler to implement relaxation algorithm`a lablock Gauss−Seidel when the time discretization step converges to +∞, it was decided that relaxation will be the method of choice to go beyond the methodology discussed in [17,21,32,33].
In [21] and related publications, all the test problems considered were posed in Ω = (0,1)2 and the finite element spaces were associated with uniform triangulations like the one on the left in Figure 2 (see Sect.10).
When applied to problems where Ω has a curved boundary requiring unstructured meshes, or when using uniform meshes like the one on the right in Figure2, we observed a deterioration of the convergence properties
whenh→0, and even divergence for some test problems. This issue is addressed in this article: an obvious way to overcome this difficulty is to proceed as in, e.g., [24,25], that is, use mixed finite element approximations of the convex solutions of problem (1.1), and of their second order derivatives, based on continuous, piecewise polynomial functions of degree≥2. This approach has several drawbacks, the main ones being that: (i) unlike piecewise linear approximations, the higher order ones do not preserve the maximum principle when this principle holds. (ii) Compared to piecewise linear approximations, the higher order ones are not easy to implement for domains Ω with curved boundaries. Instead, in order to “rescue” the piecewise linear approximations, we advocate a Tychonoff-like regularization method[49] when defining the discrete analogues of the second order derivatives. With this approach we recover convergence of optimal (or nearly optimal) order, ash→0, even for unstructured meshes, or for pathological structured ones like the triangulation on the right in Figure2.
To summarize, in this article, we advocate arelaxationalgorithm for the solution of a well-chosenleast-squares variant of problem (1.1). With such an algorithm we are able to decouple the treatment of the differential operators from the treatment of the nonlinearities. Indeed, the treatment of the differential operators leads to the solution of a sequence of elliptic linear biharmonic problems. The nonlinearity requires the solution of an infinite family of low dimensional constrained minimization problems, one for almost every point of Ω (in practice, one for each interior vertex of the finite element triangulation ofΩ).
To solve the above linear biharmonic problems we advocate a conjugate gradient algorithm operating in well- chosen sub-spaces ofH2(Ω). On the other hand, two quite different methods are considered for the solution of the low dimensional constrained minimization problems: the first one based on the Newton’s method combined with an appropriate parametrization of the two-dimensional manifold{z={zi}3i=1, z1>0, z2 >0, z1z2−z23= 1}.
The second method is based on a novel algorithm for quadratically constrained minimization problems (denoted by Qmin and introduced in [48]). Following [16–22,35], mixed finite element approximations are used for the discretization of (1.1). A regularization procedure for the approximation of second derivatives on arbitrary meshes allows obtaining optimal (or nearly optimal) convergence properties.
This article is structured as follows: in Section2, we introduce some fundamental function spaces and sets, and use them to provide a least-squares formulation of problem (1.1). The relaxation algorithm is described in Section 3. In Sections4 and5, we discuss the solutions of the local low dimensional constrained minimiza- tion problems and of the linear variational bi-harmonic problems. The mixed finite element approximation of problem (1.1) is discussed in Section6, while Sections7,8and9 are dedicated to the discrete analogues of the problems discussed in Sections 3, 4 and5. In Section10, the methodology discussed in the preceding sections is applied to the solution of test problems, some of them borrowed from [13,16–21,32,33]; these numerical experiments include test cases whereΩ has a curved boundary and/or when problem (1.1) has no solution in H2(Ω) [11,20,37].
The methodology described in this article owes much toCalculus of VariationsandOptimal Control. Indeed the least-squares criterion that we use is nothing but a multi-dimensional integral defined on the subset of a functional space`a laSobolev. Moreoveradjoint equation techniquesare used to compute some of the derivatives of the discrete cost functional, resulting in substantial memory and computational time savings.
2. Formulation of the Dirichlet problem for the elliptic Monge − Amp` ere equation in two dimensions
LetΩbe a bounded convex domain ofR2; we denote byΓ the boundary ofΩ. The Dirichlet problem for the canonical Monge−Amp`ere equation reads as follows:
detD2ψ=f in Ω, ψ=g onΓ, (2.1)
whereD2ψ is theHessianof the unknown functionψ, that isD2ψ=
∂2ψ
∂xi∂xj
1≤i,j≤2
.
When problem (2.1) has no solution (inH2(Ω)), we were tempted to call generalized solutions the solutions
“captured” by the least-squares/relaxation methodology discussed in the following sections. Actually, in the
LEAST-SQUARES METHOD FOR THE ELLIPTIC MONGE−AMP `
context of Monge−Amp`ere equations,generalized solutionhas a very precise meaning, namely the one introduced by Aleksandrov (see [1,46]) and further discussed in [37], Chapter 1. Following [37], ψis a generalized solution of problem (2.1) ifM ψ=f inΩandψ=gon∂Ω,M ψbeing theMonge−Amp`ere measureassociated withψ, a particular Borel measure whose precise definition can be found in [37], Chapter 1 (ifψis smooth enough and convex, we can identifyM ψ with detD2ψ).
Concerning the existence and uniqueness of generalized solutions, it is proved in the above reference that if Ω ⊂Rd is a bounded strictly convex domain, μis a Borel measure in Ω withμ(Ω)<+∞, andg ∈C0(∂Ω), then there exists a unique ψ ∈ C0(Ω) that is a convex solution to the boundary value problemM ψ =μ in Ω and ψ =g on ∂Ω. It is difficult to be more general, particularly if we compare with the very demanding conditions required from f, g and Ω, so that (2.1) will have a classical solution (see, e.g., [29], Chap. 17 for details). It is also difficult to do better than Aleksandrov generalized solutions as long as problem (2.1) is concerned; unfortunately, the Aleksandrov’s notion of weak solution does not generalize easily to other fully nonlinear second order elliptic equations.
For those more general situations, the right concept seems to be the notion ofviscosity solutionsintroduced in the early eighties by M. Crandall and P.L. Lions. The basic reference concerning the viscosity solution of second order partial differential equations is [15] (for application to a variety of nonlinear elliptic equations, including Monge−Amp`ere’s, see [10–12,26,39,43] or [37] and the references therein). Actually, it is proved in [37], Chapter 1 that ifψis a generalized solution toM ψ=f withf continuous, thenψis a viscosity solution of the Monge−Amp`ere equation. Conversely, it is also proved that iff ∈C0(Ω), f >0, andψ is a viscosity solution of detD2ψ =f, then ψ is a generalized solution ofM ψ =f. Numerical methods for the solution of problem (1.1), based on the Aleksandrov and viscosity solution approaches, can be found in [43,44].
Among the various methods available for the solution of (2.1), we advocate the following one of thenonlinear
least-squarestype:
Find (ψ,p)∈Vg×Qf such that
J(ψ,p)≤J(ϕ,q), ∀(ϕ,q)∈Vg×Qf, (2.2) where:
J(ϕ,q) = 1 2
Ω
D2ϕ−q2dx, (2.3)
|·|being the Fr¨obenius norm, that is|T| =√
T:T, with S:T=2
i,j=1sijtij, for allS= (sij),T = (tij)∈ R2×2. The functional spaces and sets in (2.2) are defined by:
Vg=
ϕ∈H2(Ω), ϕ=g onΓ , (2.4)
Qf ={q∈Q,detq=f, q11>0, q22>0}, Q=
q∈L2(Ω)2×2, q=qt . (2.5) The spaceQin (2.5) is a Hilbert space for the scalar product (q,q)→
Ωq:qdx, and the associated norm. In order to haveVgandQf both non-empty, we assume from now on thatf ∈L1(Ω),f >0 andg∈H3/2(Γ). The introduction of the set Qf allows the decoupling of the differential operators (acting linearly on the unknown functionψ) and of the nonlinearities (acting on the unknown tensor-valued functionp). Indeed, the burden of nonlinearity (algebraic here) has been reported onp, explaining the introduction of the nonlinear manifoldQf. Note that the existence and uniqueness of a convex solution to the least-squares problem (2.2) is still an open problem; however, the numerical experiments reported in Section 10, will show that our least-squares based method never failed finding the convex solution of the test problems considered there, assuming it exists with the right regularity properties (and sometimes even less, as shown in Sect.10.12).
Remark 2.1. As shown in,e.g., [19–21], problem (2.2) may have smooth solutions, even if (2.1) has no such solutions as it is the case ifΩ = (0,1)2,f = 1 and g = 0 [11,37]. Generally speaking, (2.1) admits a smooth solution whenD2Vg∩Qf =∅, as illustrated in Figure 1 (left). On the other hand, when D2Vg∩Qf = ∅, it makes sense to search for a solution, in the sense of (2.2) (see Fig.1(right)).
Q
Qf
D2Vg
p=D2ψ
Q
Qf
D2Vg
p D2ψ
Figure 1.The Monge−Amp`ere problem (2.1) has a solution inVg (left), or no solution inVg
(right).
3. A relaxation algorithm for the solution of problem (2.2)
In order to compute aconvexsolution of problem (2.2) (or at least to force the convexity of the solution) we advocate the followingrelaxation algorithm: solve
−Δψ0=−2
f in Ω, ψ0=g onΓ. (3.1)
Then, forn≥0, assuming thatψn is known, compute pn, ψn+1/2andψn+1 as follows:
pn = arg min
q∈Qf
J(ψn,q), (3.2)
ψn+1/2= arg min
ϕ∈VgJ(ϕ,pn), (3.3)
ψn+1=ψn+ω(ψn+1/2−ψn), (3.4)
withω, 0< ω < ωmax≤2, a relaxation parameter.
Remark 3.1 (initialization strategy).The rationale behind (3.1) is as follows: denote byλ1 andλ2the eigen- values ofD2ψ; we have thenλ1λ2=f. It follows from (λ1+λ2)2−(λ1−λ2)2= 4λ1λ2, that, ifλ1and λ2 are close to each other, thenΔψ=λ1+λ22√
λ1λ2= 2√
f, justifying thus the initialization (3.1).
The relaxation algorithm (3.1)–(3.4) looks simple but the solution of problems (3.2) and (3.3) leads to technical issues that we will address in the following sections.
4. Numerical solution of the sub-problems (3.2)
4.1. Explicit formulation of problem (3.2)
An explicit formulation of problem (3.2) is given bypn = arg min
q∈Qf
1 2
Ω
|q|2dx−
Ω
D2ψn:qdx
. (4.1)
Since neither integrands in (4.1) contains derivatives ofq, the minimization problem (4.1) can be solvedpoint- wise (in practice at the vertices of a finite element or finite difference grid). This leads us, a.e. in Ω, to the solution of the following finite dimensional minimization problem
pn(x) = arg min
q∈Ef(x)
1
2|q|2−Dn(x) :q
, (4.2)
whereDn(x) =D2ψn(x) is a symmetric matrix andEf(x) ={q∈R2×2,q=qt,detq=f(x), q11>0, q22>0}.
LEAST-SQUARES METHOD FOR THE ELLIPTIC MONGE−AMP `
4.2. A Newton-type method for the numerical solution of problem (4.2)
Taking advantage of the symmetry ofqandDn(x), and using the notationz1=q11, z2=q22,z3=q12=q21 andDn(x)ij =dnij(x), the minimization problem in (4.2) can be rewritten as
z∈Zminf(x)
1
2(z12+z22+ 2z23)−dn11(x)z1−dn22(x)z2−2dn12(x)z3
, (4.3)
with Zf(x) =
z∈R3, z1>0, z2>0, z1z2−z32=f(x) . To transform (4.3) into an unconstrained mini- mization problem in R2, we perform the change of variables z1 =
f(x)eρcoshθ, z2 =
f(x)e−ρcoshθ, z3=
f(x) sinhθ, for (ρ, θ)∈R2, so that (4.3) becomes
(ρ,θmin)∈R2j(ρ, θ), with j(ρ, θ) =
√f(x)
2 (cosh 2ρcosh 2θ+ cosh 2ρ+ cosh 2θ−1)−(dn11(x)eρ+dn22(x)e−ρ) coshθ−2dn12(x) sinhθ.
This leads us in turn to the solution ofDj(ρ, θ) =0, whereDj(·) is the differential of the functionalj(·). This 2×2 nonlinear system actually reads as follows:
Dj(ρ, θ)1=
f(x)(1 + cosh 2θ) sinh 2ρ−(dn11(x)eρ−dn22(x)e−ρ) coshθ = 0, Dj(ρ, θ)2=
f(x)(1 + cosh 2ρ) sinh 2θ−(dn11(x)eρ+dn22(x)e−ρ) sinhθ−2dn12(x) coshθ= 0.
This system can be solved by using a Newton method. Let (ρ0, θ0) ∈ R2 be given. For k ≥ 0, we compute (ρk+1, θk+1) from (ρk, θk)viathe solution of
D2j(ρk, θk)
ρk+1−ρk θk+1−θk
=−Dj(ρk, θk), whereD2j(ρ, θ) = (D2j(ρ, θ)ij)1≤i,j≤2 is given by:
D2j(ρ, θ)11= 2
f(x) cosh 2ρ(1 + cosh 2θ)−
dn11(x)eρ+dn22(x)e−ρ coshθ, D2j(ρ, θ)12=D2j(ρ, θ)21= 2
f(x) sinh 2ρsinh 2θ−
dn11(x)eρ−dn22(x)e−ρ sinhθ, D2j(ρ, θ)22= 2
f(x) cosh 2θ(1 + cosh 2ρ)−
dn11(x)eρ+dn22(x)e−ρ
coshθ−2d12(x) sinhθ.
Remark 4.1 (choice of the scalar product). Since we are dealing with symmetric matrices, we can equip Q with the following scalar product (q,q)→
Ω(q11q11+q22q22+q12q12)dx. As shown in [13,18,32], this new scalar product has given better results than the one defined by (q,q) →
Ωq : qdx when applied to the numerical solution of the two-dimensional Dirichlet problem for the Pucci’s equation, that is αλ++λ− = f in Ω, together withψ = g on Γ, where λ+ (resp., λ−) denotes the largest (resp., the smallest) eigenvalue of the HessianD2ψof the unknown functionψ, and whereα≥1. Using this new scalar product, (4.3) would be replaced by
z∈Zminf(x)
1 2
z21+z22+z32
−dn11(x)z1−dn22(x)z2−dn12(x)z3
,
withZf(x) defined similarly. The same change of variables and Newton method can be applied to this problem.
4.3. The quadratically constrained minimization method for the numerical solution of problem (4.3)
In [48], a class of quadratically constrained minimization problems has been addressed with a new algorithm denoted byQmin. This algorithm allows the solution of some specific eigenvalue-constrained matrix optimiza- tion problems of dimension N (≥2), its complexity being O(N3). The particular case associated with N = 2 corresponds to (4.2).
This method relies on the following equivalent formulation of problem (4.2):
pn(x) =Sn(x)Λn(x)Sn(x)t, (Λn(x),Sn(x)) = arg min
(Λ,S)∈Ef
1
2(μ21+μ22)−trace
Dn(x)SΛSt
, (4.4) where Ef(x) = {(Λ,S),Λ = diag(μ1, μ2), μ1μ2 = f(x),StS = I}. The algorithm developed in [48] applies beautifully to the solution of (4.4). After scaling ofDn(x) by
f(x), (4.4) is equivalent to arg min
A∈A1trace [AA−2DnA], (4.5)
where A1 ={A ∈ R2×2,A =At, tM = 2,M ≥0}, M = 0 1
1 0
, and = (μ1, μ2)t, {μ1, μ2} being the spectrum of A. The constrainttM = 2 corresponds to λ1+λ2 = 1, while the constraint M ≥ 0 ensures λ1, λ2≥0 to obtain convex solutions. Ultimately, for N = 2 the solution is found by solving a simple rational equation of the form
β21
(1 +μ)2 = 2 + β22 (1−μ)2, where β1 = (λ1+λ2)/√
2 and β22 = (λ21 +λ22)/2 −λ1λ2, {λ1, λ2} being the spectrum of Dn(x)/
f(x).
Remarkably, the same rational equation holds essentially for arbitraryN ≥2. This equation is efficiently solved numerically by first taking reciprocals and then square roots on both sides and applying Newton’s method.
With a starting guessμ0 =−1, the method converges typically in 3 to 5 iterations. This occurs because the reciprocal square root transformation yields a problem that is essentially the intersection of two straight lines.
For more details, see [48], where this algorithm is developed for arbitraryN ≥2.
5. Conjugate gradient solution of the sub-problems (3.3)
Written in variational form, the Euler−Lagrange equation of the sub-problem (3.3) reads as follows:
Find ψn+1/2∈Vg such that
Ω
D2ψn+1/2:D2ϕdx=
Ω
pn:D2ϕdx, ∀ϕ∈V0, (5.1) where V0 =H2(Ω)∩H01(Ω). The linear variational problem (5.1) is well-posed and belongs to the following family of linear variational problems:
u∈Vg :
Ω
D2u:D2vdx=L(v), ∀v∈V0, (5.2) with the functionalL(·) linear and continuous overH2(Ω); problem (5.2) is clearly of the biharmonic type. The conjugate gradient solutionof linear variational problems in Hilbert spaces, such as (5.2), has been addressed in, e.g., [30], Chapter 3. Following the above reference, we are going to solve (5.2) by a conjugate gradient algorithm operating in the spaces V0 and Vg, both spaces being equipped with the scalar product defined by (v, w)→
ΩΔvΔwdx, and the corresponding norm. This conjugate gradient algorithm reads as follows:
Step 1.
u0∈Vg given. (5.3)
Step 2.Solve:
Find g0∈V0 such that
Ω
Δg0Δvdx=
Ω
D2u0:D2vdx−L(v), ∀v∈V0, (5.4) and set the first descent direction:
w0=g0. (5.5)
LEAST-SQUARES METHOD FOR THE ELLIPTIC MONGE−AMP `
Then, fork ≥0,uk, gk, andwk being known, the last two different from zero, we compute uk+1, gk+1 and, if necessary,wk+1 as follows.
Step 3.Solve:
Find ¯gk ∈V0 such that
Ω
Δ¯gkΔvdx=
Ω
D2wk :D2vdx, ∀v∈V0, (5.6) and compute the new iterates as follows:
ρk=
ΩΔgk2dx
ΩΔ¯gkΔwkdx, (5.7)
uk+1=uk−ρkwk, (5.8)
gk+1=gk−ρkg¯k. (5.9)
Step 4.Compute
δk=
ΩΔgk+12dx
Ω|Δg0|2dx . (5.10)
Ifδk< ε(meaning that the residual is small enough), takeu=uk+1; otherwise, compute:
γk =
ΩΔgk+12dx
Ω|Δgk|2dx , (5.11)
and update the descent directionvia
wk+1=gk+1+γkwk. (5.12)
Step 5.Dok+ 1→kand return toStep 3.
The numerical experiments reported in Section10 show that the conjugate gradient algorithm (5.3)–(5.12) enjoys a fast convergence, typically less than 10 iterations for all the meshes and mesh sizes which have been considered. Combined with an appropriate mixed finite element approximation, it requires, at each iteration, the solution of two discrete Poisson problems.
6. On a mixed finite element approximation
6.1. Generalities
Considering the highly variational flavor of the methodology discussed in the preceding sections, it makes sense to look for finite element based methods for the approximation of (2.1). In order to avoid the complications associated with the construction of finite element sub-spaces ofH2(Ω) (see, however, [5,25] for such an approach), we employ here a mixed finite element approximation (closely related to those discussed in, e.g., [22,23,31, 36,47] for the solution of linear and nonlinear bi-harmonic problems). Following this approach, it is possible to solve (2.1) employing approximations commonly used for the solution of second order elliptic problems (piecewise linear and globally continuous over a triangulation of Ω for example). The use of low order finite elements is justified in order to have the flexibility to consider computational domains with arbitrary (convex) shapes. However, since low order finite elements may have difficulties at handling some biharmonic problems, an additional regularization may be required, when approximating the second derivatives.
6.2. Mixed finite element approximation
For simplicity, we assume thatΩis a bounded polygonal domain ofR2. Let us denote byTh a finite element triangulation of Ωas discussed in,e.g., [31], Appendix 1. FromTh, we approximate the spacesL2(Ω), H1(Ω) and H2(Ω) (respectively, H01(Ω) and H2(Ω)∩H01(Ω)) by the finite dimensional spaceVh (respectively, V0h) defined by:
Vh=
v∈C0 Ω
, v|T ∈P1, ∀T∈ Th , V0h=Vh∩H01(Ω) ={v∈Vh, v= 0 onΓ}, (6.1) withP1 the space of the two-variables polynomials of degree≤1.
For a function ϕbeing given in H2(Ω), we denote ∂2ϕ/∂xi∂xj byD2ij(ϕ). It follows fromGreen’s formula
that
Ω
∂2ϕ
∂xi∂xj
vdx=−1 2
Ω
∂ϕ
∂xi
∂v
∂xj
+ ∂ϕ
∂xj
∂v
∂xi
dx, ∀v∈H01(Ω), ∀i, j= 1,2. (6.2) Consider nowϕ∈Vh. Taking advantage of the relations (6.2), we define the discrete analogues of the differential operatorsD2ij by
D2hij(ϕ)∈V0h,
Ω
D2hij(ϕ)vdx=−1 2
Ω
∂ϕ
∂xi
∂v
∂xj
+ ∂ϕ
∂xj
∂v
∂xi
dx, ∀v∈V0h, ∀i, j= 1,2. (6.3) The functionsD2hij(ϕ) are uniquely defined by the relations (6.3). However, in order to simplify the computation of the above discrete second order partial derivatives, it is tempting to consider using the trapezoidal rule to evaluate the integrals in the left hand sides of (6.3). Owing to their practical importance, let us detail these calculations:
(i) First, we introduce the set Σh of the vertices of Th and then Σ0h ={P ∈Σh, P /∈Γ}. Next, we define the integers Nh and N0h by Nh = Card(Σh) and N0h = Card(Σ0h). We have then dimVh = Nh and dimV0h=N0h. We suppose that Σ0h={Pj}Nj=10h andΣh=Σ0h∪ {Pj}Nj=hN0h+1.
(ii) With eachPk ∈Σh, we associate the function wk uniquely defined by
wk ∈Vh, wk(Pk) = 1, wk(Pl) = 0, ∀l= 1, . . . , Nh, l=k.
It is well-known (see,e.g., [31], Appendix 1) that the sets Bh ={wk}Nk=1h andB0h ={wk}Nk=10h arevector basesforVh andV0h, respectively.
(iii) Let us denote byAk the area of the polygonal domain which is the union of those triangles of Th which havePk as a common vertex. Applying the trapezoidal rule to the integrals in the left-hand side of the relations (6.3), we obtain
D2hij(ϕ)∈V0h, Dhij2 (ϕ)(Pk) =− 3 2Ak
Ω
∂ϕ
∂xi
∂wk
∂xj
+ ∂ϕ
∂xj
∂wk
∂xi
dx, ∀k= 1, . . . , N0h, ∀i, j= 1,2.
(6.4) Computing the integrals in the right hand side of (6.4) is quite simple since the first order derivatives of ϕ and wk are piecewise constant. Finally, with ϕ∈ Vh, we associateΔhϕ ∈ V0h uniquely defined by Δhϕ(Pk) =D2h11(ϕ)(Pk) +D2h22(ϕ)(Pk), fork= 1, . . . , N0h.
Taking the above relations into account, approximating problem (2.1) is now fairly straightforward. Assuming that the boundary function g is continuous overΓ (which is definitely the case ifg ∈H3/2(Γ)), let us denote bygh the interpolant of g associated with the triangulationTh. We approximate the affine spaceVg byVgh= {ϕ∈Vh, ϕ(P) =g(P),∀P∈Σh∩Γ}and then problem (2.1) by:
Find ψh∈Vgh such that D2h11(ψh)(Pk)Dh222(ψh)(Pk)−Dh212(ψh)(Pk)2=fh(Pk), k= 1, . . . , N0h, (6.5)
LEAST-SQUARES METHOD FOR THE ELLIPTIC MONGE−AMP `
where fh is a continuous approximation of f (we can always assume that fh∈Vh). In addition, we define the discrete equivalent ofQf as follows:
Qf h={q∈Qh, detq(Pk) =fh(Pk), q11(Pk)>0, q22(Pk)>0, k= 1, . . . , N0h},
with Qh = {q ∈ (V0h)2×2, q(Pk) = qt(Pk), k = 1, . . . , N0h}. We associate with V0h and Qh the following discrete scalar products and corresponding Euclidean norms:
(v, w)0h =1 3
Nh
k=1
Akv(Pk)w(Pk), ∀v, w∈V0h, ||v||20h= (v, v)0h, ∀v∈V0h,
((S,T))0h =1 3
N0h
k=1
AkS(Pk) :T(Pk), ∀S,T∈Qh, |||S|||20h= ((S,S))0h, ∀S∈Qh. The solution of problem (6.5) will be discussed in the sequel.
Remark 6.1. Suppose thatΩ= (0,1)2and that the triangulationThis uniform like the one shown in Figure2 (left). Suppose that h = 1/(I+ 1), I being a positive integer greater than one. In this particular case, the setsΣhandΣ0hare given byΣh={Pij = (ih, jh),0≤i, j≤I+ 1}, andΣ0h={Pij= (ih, jh),1≤i, j≤I}, implying thatNh= (I+ 2)2andN0h=I2. It follows then from the relations (6.4) that (with obvious notation):
D2h11(ϕ)(Pij) =ϕi+1,j+ϕi−1,j−2ϕij
h2 , 1≤i, j≤I, D2h22(ϕ)(Pij) =ϕi,j+1+ϕi,j−1−2ϕij
h2 , 1≤i, j≤I, D2h12(ϕ)(Pij) =ϕi+1,j+1+ϕi−1,j−1+ 2ϕij−(ϕi+1,j+ϕi−1,j+ϕi,j+1+ϕi,j−1)
2h2 , 1≤i, j≤I.
The above discrete second order derivatives of finite difference type have the easily verified yet remarkable property that they areexact for polynomial functions of degree ≤2.
6.3. A smoothing procedure for the approximation of the second derivatives
As emphasized in [45], when using piecewise linear mixed finite elements, thea prioriestimates for the error on the second derivatives of the solution ψ are, in general, O(1) in the L2-norm. Therefore the convergence properties of the solution method depend strongly on the type of triangulations one employs. Indeed, assuming that the discrete second order derivatives have been computed via (6.3) and (6.4), numerical experiments performed by the authors showed the triangulation dependence of the convergence; non-convergence cases (in the L2-norm) were also observed. Unfortunately, the approximations of ∂2ϕ
∂xi∂xj
provided by (6.3) and (6.4) converge to the above second derivative, no better than in H−1(Ω) in general. This allows oscillations and explains the growth of the approximation error in L2(Ω) and H1(Ω) as h → 0. Such pathological behavior can be observed in the results presented in Section 10. From that point of view a dramatic confirmation of these non-convergence properties is provided by the numerical results associated with the structured symmetric mesh shown on the right of Figure2 (also called “British flag” mesh or “crisscross” pattern). To cure the non- convergence properties associated with the approximations (6.3) and (6.4) of the second derivatives, we see two options:
(i) Use, as in, e.g., [24,25], mixed finite elements methods based on piecewise polynomial approximations of degree ≥2. This approach has several drawbacks, among them: (a) it is more complicated to implement than the mixed methods described in Section6.2, particularly ifΩhas a curved boundary. (b) These higher order polynomial approximations do not preserve the maximum principle, if this principle takes place for the continuous problem.
(ii) Use a regularization procedure `a la Tychonoff [49], while keeping a piecewise linear approximation based mixed finite element approach.
Focusing on the second approach, a simple and novel (in this context) way to obtain better convergence properties of the discrete second order derivatives is to use the following regularization procedure: with C > 0 and
|K|= meas(K), when computing the discrete second derivativesD2hij(ϕ) replace (6.3) by:
Find D2hij(ϕ)∈V0h such that,∀v∈V0h, i, j= 1,2,
Ω
D2hij(ϕ)vdx+C
K∈Th
|K|
K
∇Dhij2 (ϕ)· ∇vdx=−1 2
Ω
∂ϕ
∂xi
∂v
∂xj
+ ∂ϕ
∂xj
∂v
∂xi
dx, (6.6)
and (6.4) by
FindDhij2 (ϕ)∈V0h such that,∀v∈V0h, i, j= 1,2, (D2hij(ϕ), v)0h+C
K∈Th
|K|
K
∇D2hij(ϕ)· ∇vdx=−1 2
Ω
∂ϕ
∂xi
∂v
∂xj
+ ∂ϕ
∂xj
∂v
∂xi
dx. (6.7)
The above linear systems can be solved by a sparse Cholesky solver (with the Cholesky factorization made once and for all at the beginning of the algorithm). The overhead in computational time appears to be non significant.
Numerical results in Section10 show that the above regularization procedure generally provides a significant improvement to the orders of convergence of the approximations of the solutionψof problem (2.1). On the other hand, in the particular case of triangulations like the one on the left of Figure2, the regularization associated with (6.6) or (6.7), deteriorates significantly theL2(Ω)-approximation error, while preserving optimal orders of convergence.
Remark 6.2. The regularization method we employed in (6.6) and (6.7) is reminiscent of the stabilization one employed by Hughes et al. in [38] to construct convergent approximations of the Stokes problem using, essentially, the same finite element spaces to approximate velocity and pressure (equal-order interpolation), a very popular method indeed.
7. Discrete least-squares formulation and discrete relaxation algorithm
We advocate the followingnonlinear least-squaresmethod for the solution of problem (6.5):
Find (ψh,ph)∈Vgh×Qf h such that Jh(ψh,ph)≤Jh(ϕ,q), ∀(ϕ,q)∈Vgh×Qf h, (7.1) where
Jh(ϕ,q) =1
2D2h(ϕ)−q2
0h.
In order to solve the nonlinear least-squares problem (7.1), we suggest the followingrelaxationalgorithm:
Find ψh0∈Vgh such that
Ω
∇ψ0h· ∇ϕdx=−2(
fh, ϕ)0h, ∀ϕ∈V0h. (7.2) Forn≥0, assuming thatψnh is known, computepnh, ψhn+1/2 andψnh+1 as follows:
pnh = arg min
q∈Qfh
Jh(ψhn,q), (7.3)
ψnh+1/2= arg min
ϕ∈VghJh(ϕ,pnh), (7.4)
ψhn+1=ψnh+ω(ψhn+1/2−ψnh), (7.5) with 0< ω < ωmax≤2. The solution of the finite dimensional problems (7.3) and (7.4) will be addressed in the following sections.
