A LeastSquares/Relaxation Method for the Numerical Solution of the ThreeDimensional Elliptic Monge–Ampère Equation
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Abstract
In this article, we address the numerical solution of the Dirichlet problem for the threedimensional elliptic Monge–Ampère equation using a leastsquares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities. Dedicated numerical solvers are derived for the efficient solution of the local optimization problems with cubicly nonlinear equality constraints. The approximation relies on mixed low order finite element methods with regularization techniques. The results of numerical experiments show the convergence of our relaxation method to a convex classical solution if such a solution exists; otherwise they show convergence to a generalized solution in a leastsquares sense. These results show also the robustness of our methodology and its ability at handling curved boundaries and nonconvex domains.
Keywords
Monge–Ampère equation Leastsquares method Nonlinear constrained minimization Newton methods Mixed finite element methodMathematics Subject Classification
65N30 65K10 35J961 Introduction
The Monge–Ampère equation can be considered as the prototypical example of fully nonlinear elliptic equations [11, 21, 29]. Theoretical investigations of fully nonlinear equations started many years ago [6, 39] and have received a lot of attention lately [4, 5, 8, 14, 16, 19, 35, 36, 37, 38], due to the many applications involving this type of equations, for instance in finance [41], in seismic wave propagation [13], in geostrophic flows [15], in differential geometry [2, 18], and in mechanics and physics. However, the Monge–Ampère equation in three space dimensions is a complicated problem, which is still lacking a full theoretical understanding, particularly when the domain of interest is not strictly convex.
From a computational point of view, various approaches have been identified, relying in particular on finite difference [5, 13] or finite element [3, 9, 17, 27] approximations.
Since the Monge–Ampère equation may not have smooth classical solutions, even for smooth data (see [11]), notions of generalized solutions have been introduced, such as Aleksandrov solutions [1], and viscosity solutions [31, 38]. Actually, another notion of generalized solution for fully nonlinear elliptic equations has been introduced relatively recently, namely generalized solutions in a leastsquares sense. These leastsquares generalized solutions have been obtained via augmented Lagrangian or leastsquares/relaxation approaches [9, 12, 24]. The leastsquares approach will be the one used in this article.
The chosen leastsquares formulation we used here consists in minimizing the \(L^2\)distance between \({\mathbf {D}}^2 \psi \) and a matrixvalued function \(\mathbf {p}\in (L^{2}({\varOmega }))^{3\times 3}\), where \(\psi \) satisfies the boundary conditions of the problem (but not the Monge–Ampère equation) and \(\mathbf {p}\) satisfies \(\det \mathbf {p}= f\). Using a relaxation algorithm to minimize such a distance, we obtained a solution method where one solves alternatively, until convergence, a sequence of linear variational problems (to be approximated by mixed finite element methods) and a sequence of cubicly constrained algebraic optimization problems. Using a similar approach, we have been able to compute generalized solutions of the Monge–Ampère equation when this problem has no classical solutions in two dimensions of space [9]. In this article, the methods discussed in [9] have been generalized to threedimensional problems.
In this article, the linear variational problems we mentioned above are solved by a preconditioned conjugate gradient algorithm, whose computer implementation relies on low order (\({\mathbb {P}}_1\) or \({\mathbb {Q}}_1\)) mixed finite element approximations. The local cubicly constrained optimization problems are solved by Newtonlike methods [42] or timestepping methods associated with a dynamical flow (see, e.g., [30]). The main difference between the twodimensional case discussed in [9] , and the present article is the cubic nature (vs quadratic) of the equality constraints.
The structure of this article is as follows. In Sect. 2, we describe the proposed methodology, while the relaxation algorithm is described in Sect. 3. Sections 4 and 5 detail the algebraic and differential solvers respectively. The mixed finite element discretization is discussed in Sect. 6. The method is applied in Sect. 7 to the solution of several numerical examples, including examples without a classical exact solution. Finally, in Sect. 8, some numerical results are presented where \({\mathbb {P}}_1\) finite elements have been replaced with \({\mathbb {Q}}_1\) ones.
The numerical solution of the 3D elliptic Monge–Ampère equation has been discussed in [8] using piecewise \({\mathbb {P}}_3\) continuous finite element approximations. A fast multigrid scheme has been presented in [34], while smooth cases on structured meshes have been considered in [3] (with numerical results that are consistent with [9]). However, to the best of our knowledge, the method discussed in the present article is one of the very few able to solve the 3D elliptic Monge–Ampère equation on domains with curved boundaries, using piecewise \({\mathbb {P}}_1\) continuous finite element approximations associated with unstructured meshes, while preserving optimal, or nearly optimal, orders of convergence for the approximation errors, including situations where the solution does not have the \(C^2(\bar{{\varOmega }})\) regularity.
2 Mathematical Formulation and LeastSquares Approach
3 Relaxation Algorithm
4 Numerical Approximation of the Local Nonlinear Problems
4.1 Explicit Formulation of the Local Nonlinear Problems
4.2 A Reduced Newton Method
The stopping criterion is based on the residual value \(\left \left \nabla G(\mathbf {X}^{k}) \right \right \), and the iterations are stopped if \(\left \left \nabla G(\mathbf {X}^{k}) \right \right < \varepsilon _{\mathrm{Newton}}\), where \(\varepsilon _{\mathrm{Newton}}\) is a given tolerance.
4.3 A Runge–Kutta Method for the Dynamical Flow Problem
5 Numerical Solution of the Linear Variational Problems
 Step 1
 $$\begin{aligned} u^0 \in V_g \text { given. } \end{aligned}$$(18)
 Step 2
 Solve: find \(g^0 \in V_g\) satisfyingand set$$\begin{aligned} \int _{{\varOmega }} {\varDelta }g^0 {\varDelta }v d\mathbf {x}= \int _{{\varOmega }} {\mathbf {D}}^2 u^0: {\mathbf {D}}^2 v d\mathbf {x}L(v), \quad \forall v \in V_0, \end{aligned}$$(19)Then, for \(k\ge 0\), \(u^k, g^k\) and \(w^k\) being known, the last two different from zero, we compute \(u^{k+1}, g^{k+1}\) and, if necessary, \(w^{k+1}\) as follows.$$\begin{aligned} w^0 = g^0. \end{aligned}$$(20)
 Step 3
 Solve: find \(\bar{g}^k \in V_0\) satisfyingand compute$$\begin{aligned} \int _{{\varOmega }} {\varDelta }\bar{g}^k {\varDelta }v d\mathbf {x}= \int _{{\varOmega }} {\mathbf {D}}^2 w^k: {\mathbf {D}}^2 v d\mathbf {x}, \quad \forall v \in V_0, \end{aligned}$$(21)$$\begin{aligned}&\rho _k = \frac{ \int _{{\varOmega }} \left {\varDelta }g^k \right ^2 d\mathbf {x}}{ \int _{{\varOmega }} {\varDelta }\bar{g}^k {\varDelta }w^k d\mathbf {x}}, \end{aligned}$$(22)$$\begin{aligned}&u^{k+1} = u^k  \rho _k w^k, \end{aligned}$$(23)$$\begin{aligned}&g^{k+1} = g^k  \rho _k \bar{g}^k. \end{aligned}$$(24)
 Step 4
 ComputeIf \(\delta _k < \varepsilon \), take \(u = u^{k+1}\); otherwise, compute:$$\begin{aligned} \delta _k = \frac{ \int _{{\varOmega }} \left {\varDelta }g^{k+1} \right ^2 d\mathbf {x}}{ \int _{{\varOmega }} \left {\varDelta }g^{0} \right ^2 d\mathbf {x}}. \end{aligned}$$(25)and$$\begin{aligned} \gamma _k = \frac{ \int _{{\varOmega }} \left {\varDelta }g^{k+1} \right ^2 d\mathbf {x}}{ \int _{{\varOmega }} \left {\varDelta }g^k\right ^2 d\mathbf {x}}; \end{aligned}$$(26)$$\begin{aligned} w^{k+1} = g^{k+1} + \gamma _k w^k. \end{aligned}$$(27)
 Step 5

Do \(k+1 \rightarrow k\) and return to Step 3.
6 Mixed Finite Element Approximation
Considering the highly variational flavor of the methodology discussed in the preceding sections, it makes sense to look for finite element methods for the solution of (1). We will use a mixed finite element approximation (closely related to those discussed in, e.g., [25] for the solution of linear and nonlinear biharmonic problems) with low order (piecewise linear and globally continuous) finite elements on a partition of \({\varOmega }\) made of tetrahedra. The modification of the numerical approximation method obtained when replacing the \({\mathbb {P}}_1\) based finite element spaces on tetrahedra by the \({\mathbb {Q}}_1\) based finite element spaces associated with partitions of \({\varOmega }\) made of hexahedra is discussed in Sect. 8 with some numerical experiments.
6.1 Finite Element Spaces
For simplicity, let us assume that \({\varOmega }\) is a bounded polyhedral domain of \({\mathbb {R}}^3\), and define \(\mathcal {T}_h\) as a finite element partition of \({\varOmega }\) made out of tetrahedra (see, e.g., [23, Appendix 1]). Let \({\varSigma }_h\) be the set of the vertices of \(\mathcal {T}_h\), \({\varSigma }_{0h} = \left\{ P \in {\varSigma }_h , P \notin {\varGamma }\right\} \), \(N_h = \mathrm {Card}({\varSigma }_h)\), and \(N_{0h} = \mathrm {Card}({\varSigma }_{0h})\). We suppose that \({\varSigma }_{0h} =\left\{ P_j \right\} _{j=1}^{N_{0h}}\) and \({\varSigma }_h = {\varSigma }_{0h} \cup \left\{ P_j \right\} _{j=N_{0h}+ 1}^{N_{h}}\).
6.2 Finite Element Approximation of the Monge–Ampère Equation
When solving (17) by the conjugate gradient algorithm (18)–(27), one has to (i) compute the discrete analogues of the second order derivatives, e.g., \({\mathbf {D}}^2 w^k\) and \({\mathbf {D}}^2 u^0\), and (ii) solve biharmonic problems such as (19) and (21).
6.3 Discrete Formulation of the LeastSquares Method
6.4 A Discrete Relaxation Algorithm
6.5 Finite Element Approximation of the Local Nonlinear Problems
6.6 Finite Element Approximation of the Linear Variational Problems
7 Numerical Results
In the numerical examples presented hereafter, we consider \(C=0\) for the structured mesh (i.e. no Tychonoff regularization) and \(C=1\) for the isotropic mesh. The local nonlinear problems are solved with a stopping criterion of \(\varepsilon _{\mathrm{Newton}}=10^{9}\) on the residual for the Newton method, with a maximal number of iterations equal to 1000. When using the Runge–Kutta method for the dynamical flow approach, the time step is set to \({\varDelta }t = 0.1\), and is reduced only if needed (only for the first 2–3 times steps usually); the maximal number of iterations is 20,000 and the stopping criterion is \(\varepsilon =10^{7}\) on two successive iterates.
Unless otherwise specified, the relaxation parameter is set to \(\omega =1\) at the beginning of the outer iterations, and gradually increased to 2 to speed up convergence. The conjugate gradient algorithm for the solution of the variational problems has a stopping criterion of \(\varepsilon =10^{8}\) on successive iterates, with a maximal number of iterations equal to 100. Actually, numerical experiments show that the number of conjugate gradient iterations is never larger than 35. The outer relaxation algorithm has a stopping criterion of \(\varepsilon =5\times 10^{4}\) on the residual \(\left \left \left {\mathbf {D}}^2_h\psi _h^{n}\mathbf {p}_h^n \right \right \right _{0h}\), or on successive iterates if the problem does not admit a classical solution (see Sect. 7.3), with a maximal number of iterations equal to 5000.
7.1 Polynomial Examples
(i) Variations with respect to h of the \(L^2({\varOmega })\) and \(H^1({\varOmega })\) norms of the computed approximation error \(\psi _h  \psi \), with \(\psi (x,y,z) =\frac{1}{2}\left( x^2+5y^2+15z^2 \right) \) and related convergence orders. (ii) Variations with respect to h of the number of relaxation iterations necessary to achieve convergence
h  \(\psi _h\psi _{L_2}\)  \(\psi _h\psi _{H_1}\)  Iter.  

Structured mesh  
2.00e−01  7.19e−02  –  1.58e−00  –  71 
1.00e−01  1.80e−02  1.99  7.91e−01  0.99  228 
6.25e−02  7.06e−03  1.99  4.95e−01  1.00  314 
4.00e−02  2.89e−03  1.99  3.16e−01  0.99  375 
Isotropic mesh  
1.57e−01  1.97e−02  –  9.39e−01  –  84 
1.03e−01  1.01e−02  1.75  6.11e−01  1.03  137 
6.58e−02  5.44e−03  1.63  3.84e−01  1.03  220 
4.10e−02  3.35e−03  1.38  2.35e−01  1.03  314 
CPU time results and numbers of degrees of freedom for the smooth test case with \(\psi (x,y,z) =\frac{1}{2}\left( x^2+5y^2+15z^2 \right) \) (the number of DOFs specified corresponds to the number of vertices of the finite element mesh)
h  #DOFs  Algebraic solver (s)  Variational solver (s)  # outer iter.  Max # CG iter.  Total CPU (s) 

Structured mesh  
0.2000  216  0  16  71  12  16 
0.1000  1331  3  655  228  19  658 
0.0625  4913  26  6070  314  16  6096 
0.0400  17,576  163  38,993  375  15  39,156 
The leastsquares approach never enforces directly the solution \(\psi \) to be convex. However, it enforces explicitly the additional matrixvalued variable \(\mathbf {p}\) to be symmetric positive definite. It is thus remarkable that the positive definiteness property of \(\mathbf {p}\) translates automatically to the Hessian of the main variable \(\psi \). More precisely, when the Monge–Ampère problem has a smooth classical solution, the converged iterate satisfy \({\mathbf {D}}^2 \psi = \mathbf {p}\), and thus the convexity of \(\psi \) is automatically verified. For this test problem, and for both types of meshes, \({\mathbf {D}}^2 \psi \) is indeed symmetric positive definite for all grid points.
7.2 A Smooth Exponential Example
This test problem generalizes to three dimensions a twodimensional one commonly used in the community for Monge–Ampère solver benchmarking (see, e.g., [9, 16]). Let us denote \(\sqrt{x^2 + y^2 + z^2 }\) by r. Taking advantage of the fact that, if \(\phi \) is a radial function, one has (with obvious notation) \(\det {\mathbf {D}}^2 \phi = \phi '' (\phi '/r)^2\), the data for the Monge–Ampère–Dirichlet problem (1) associated with the above function \(\psi \) are \(f(x,y,z) = (1+r^2)e^{3r^2/2}\), and \(g(x,y,z) = e^{r^2/2}\). The stopping criterion for the relaxation algorithm is \(\left \left \left {\mathbf {D}}^2_h\psi _h^{n}\mathbf {p}_h^n \right \right \right _{0h} < 5 \times 10^{4}\), and \(C=1\) for the isotropic unstructured mesh (as for the first test problem).
Figure 4 visualizes the \(L^2({\varOmega })\) and \(H^1({\varOmega })\) computed approximation errors for both approaches for the solution of the local nonlinear problems. The conclusions are similar: both Newton and Runge–Kutta methods provide exactly the same results, and the method is globally secondorder convergent for the \(L^2\) norm. Table 3 confirms these convergence results, showing in particular no loss of convergence orders for the unstructured isotropic mesh.
(i) Variations with respect to h of the \(L^2({\varOmega })\) and \(H^1({\varOmega })\) norms of the computed approximation error \(\psi _h  \psi \), with \(\psi (x,y,z) =e^{\frac{1}{2} (x^2+y^2+z^2)}\) and related convergence orders. (ii) Variations with respect to h of the number of relaxation iterations necessary to achieve convergence
h  \(\psi _h\psi _{L_2}\)  \(\psi _h\psi _{H_1}\)  Iter.  

Structured mesh  
2.00e−01  2.74e−02  –  5.16e−01  –  12 
1.00e−01  7.52e−03  1.87  2.81e−01  0.87  20 
6.25e−02  3.06e−03  1.91  1.83e−01  0.91  25 
4.00e−02  1.26e−03  1.98  1.20e−01  0.95  28 
Isotropic mesh  
1.57e−01  1.58e−02  –  3.19e−01  –  24 
1.03e−01  8.07e−03  1.61  2.05e−01  1.05  34 
6.58e−02  3.54e−03  1.83  1.22e−01  1.15  41 
4.57e−02  1.64e−03  2.11  7.67e−02  1.28  42 
CPU time results and numbers of degrees of freedom for the smooth test case with \(\psi (x,y,z) =e^{\frac{1}{2} (x^2+y^2+z^2)}\) (the number of DOFs specified corresponds to the number of vertices of the finite element mesh)
h  #DOFs  Algebraic solver (s)  Variational solver (s)  # outer iter.  Max # CG iter.  Total CPU (s) 

Structured mesh  
0.2000  216  0  1  12  5  1 
0.1000  1331  0  19  20  5  19 
0.0625  4913  2  90  25  4  92 
0.0400  17,576  8  423  28  4  431 
Isotropic mesh  
0.1570  1043  1  70  24  19  71 
0.1030  3339  1  525  34  20  526 
0.0658  12,191  5  3568  41  22  3573 
0.0457  42,176  24  20,926  42  22  20,950 
(i) Variations with respect to h of the \(L^2({\varOmega })\) and \(H^1({\varOmega })\) norms of the computed approximation error \(\psi _h  \psi \), with \(\psi (x,y,z) =  \sqrt{R^2  (x^2 +y^2 + z^2)}\) (\(R = \sqrt{6}\)) and related convergence orders. (ii) Variations with respect to h of the number of relaxation iterations necessary to achieve convergence
h  \(\psi _h\psi _{L_2}\)  \(\psi _h\psi _{H_1}\)  Iter.  

Structured mesh  
2.00e−01  4.96e−03  –  8.60e−02  –  4 
1.00e−01  1.28e−03  1.95  4.41e−02  0.96  5 
6.25e−02  5.09e−04  1.96  2.78e−02  0.97  6 
4.00e−02  2.10e−04  1.97  1.79e−02  0.98  7 
Isotropic mesh  
1.57e−01  3.81e−03  –  8.60e−02  –  13 
1.03e−01  1.81e−03  1.78  3.62e−02  2.07  16 
6.58e−02  7.51e−04  1.94  2.12e−02  1.18  19 
4.10e−02  3.35e−04  2.21  1.30e−02  1.33  19 
7.3 Nonsmooth Test Problems
Some of the test problems we are going to consider in this section do not have exact solution with the \(H^2({\varOmega })\)regularity or may have no solution at all (but may have generalized solutions). These nonsmooth problems are therefore ideally suited to test the robustness of our methodology, and its ability at capturing generalized solutions when no exact solution does exist.
(i) Variations with respect to h of the \(L^2({\varOmega })\) and \(H^1({\varOmega })\) norms of the computed approximation error \(\psi _h  \psi \), with \(\psi (x,y,z) =  \sqrt{R^2  (x^2 +y^2 + z^2)}\) (\(R = \sqrt{3}\)) and related convergence orders. (ii) Variations with respect to h of the number of relaxation iterations necessary to achieve convergence
h  \(\psi _h\psi _{L_2}\)  \(\psi _h\psi _{H_1}\)  Iter.  

Structured mesh  
2.00e−01  1.15e−02  –  6.60e−01  –  9 
1.00e−01  3.06e−03  1.91  6.31e−01  –  14 
6.25e−02  1.24e−03  1.92  6.25e−01  –  17 
4.00e−02  5.17e−04  1.96  6.22e−01  –  19 
Isotropic mesh  
1.57e−01  6.76e−03  –  6.31e−01  –  13 
1.03e−01  3.31e−03  1.69  6.25e−01  –  16 
6.58e−02  1.39e−03  1.93  6.22e−01  –  19 
4.10e−02  6.41e−04  1.63  6.21e−01  –  19 
To conclude this section, we will consider the particular problem (1) associated with \({\varOmega }= (0,1)^3\), \(f=1\) and \(g = 0\). For these particular data, problem (1) has no smooth solution (the arguments developed in [9, 23] for the related twodimensional problem still apply here).
Figure 5 shows different features of the approximated solution inside the unit cube. The stopping criterion for this particular case without a classical solution is \(\left \left \psi _h^{n+1}  \psi _h^n \right \right _{0,h} < 10^{5}\). When studying the number of outer iterations of the relaxation algorithm, we observe that the number of iterations is larger for structured meshes than isotropic ones, and that it increases as expected when \(h\rightarrow 0\). Figure 5 (bottom row) visualizes graphs of the computed solutions restricted to the lines \(y=z=1/2\) and \(x=y, z=1/2\) for \(x\in (0,1)\), and shows little influence of the type of partition on the solution.
We can also observe that \({\mathbf {D}}^2 \psi \) is symmetric positive definite for 100% of the grid points, independently of the nature of the discretization when \(h \simeq 0.04\), even though the Monge–Ampère equations does not have a classical solution, that is \({\mathbf {D}}^2 \psi \ne \mathbf {p}\). The (necessary) loss of convexity of the solution is thus located (near the corners) in a region smaller than the mesh size. When arbitrarily refining the mesh in a corner of the domain, we observe that the Hessian \({\mathbf {D}}^2 \psi \) is not symmetric positive definite when evaluated in some grid points in a neighborhood of size \(10^{3}\) around that corner. This effect is highlighted when calculating \(\left \left {\mathbf {D}}^2 \psi _h  \mathbf {p}_h \right \right _{L^2}\), using a structured mesh of the unit cube, both on \({\varOmega }\), but also on \({\varOmega }' \subset {\varOmega }\), as illustrated in Table 7 for \({\varOmega }' = (0.2,0.8)^3\). These results show that the error inside the domain \({\varOmega }' = (0.2 , 0.8)^3\) is significantly smaller than the error on \({\varOmega }\), implying that the error is mainly committed near the boundary.
7.4 Curved Boundaries and Non Convex Domains
(i) Variations with respect to h of the norm of the residuals \(\left \left {\mathbf {D}}^2 \psi _h  \mathbf {p}_h \right \right _{L^2({\varOmega })}\) and \(\left \left {\mathbf {D}}^2 \psi _h  \mathbf {p}_h \right \right _{L^2({\varOmega }')}\) when \({\varOmega }= (0,1)^3\), \({\varOmega }' = (0.2,0.8)^3\), \(f=1\) and \(g = 0\). (ii) Variations with respect to h of the number of relaxation iterations necessary to achieve convergence
h  \(\left \left {\mathbf {D}}^2 \psi _h  \mathbf {p}_h \right \right _{L^2((0,1)^3)}\)  \(\left \left {\mathbf {D}}^2 \psi _h  \mathbf {p}_h \right \right _{L^2((0.2,0.8)^3)}\)  # iter. 

1.00e−01  4.29931e−04  4.20694e−05  467 
6.25e−02  4.32211e−04  8.47222e−06  1857 
4.00e−02  4.32995e−04  2.42009e−06  37,522 
(i) Variations with respect to h of the \(L^2({\varOmega })\) and \(H^1({\varOmega })\) norms of the computed approximation error \(\psi _h  \psi \), with \(\psi (x,y,z) =\frac{1}{2\sqrt{3}} \left( 1 {x}^{2}{y}^{2}{z}^{2}\right) \) and related convergence orders. (ii) Variations with respect to h of the number of relaxation iterations necessary to achieve convergence
h  \(\psi _h\psi _{L_2}\)  \(\psi _h\psi _{H_1}\)  Iter.  

2.98e−01  3.26e−02  2.60e−01  –  14  
1.61e−01  1.11e−02  1.74  1.28e−01  1.14  19 
8.32e−02  3.22e−03  1.88  6.16e−02  1.11  21 
4.34e−02  9.89e−04  1.80  2.86e−02  1.17  20 
CPU time results and numbers of degrees of freedom for the smooth test case with \(\psi (x,y,z) =\frac{1}{2\sqrt{3}} \left( 1 {x}^{2}{y}^{2}{z}^{2}\right) \) on the unit sphere (the number of DOFs specified corresponds to the number of vertices of the finite element mesh)
h  #DOFs  Algebraic solver (s)  Variational solver (s)  # outer iter.  Max # CG iter.  Total CPU (s) 

Structured mesh  
0.2980  631  0  34  14  25  34 
0.1610  3570  0  327  19  19  327 
0.0832  22,640  4  4385  21  22  4399 
0.0434  184,034  23  66,027  20  23  66,050 
The nonconvexity of \({\varOmega }\) may prevent problem (1) to have solutions (see, e.g., [11]). However, it makes sense to assess the capabilities of our methodology at handling problems having smooth solutions despite the nonconvexity of \({\varOmega }\). To do so, we consider the particular problem (1) where: (i) \({\varOmega }\) is the subset of \(B_1\) obtained by removing from this ball a part of angular size \(\theta \), symmetric about Ox and oriented along the Oz axis (as shown on Fig. 7 for \(\theta = \pi /2\) and \(\theta = \pi /9\)), (ii) \(f=1/(3\sqrt{3})\), g being the restriction to \(\partial {\varOmega }\) of the function \(\psi \) defined by (41). The function \(\psi \) defined by (41) is clearly a convex solution of the above problem (1). The solution methodology discussed in Sects. 3–6 still applies for this case where an exact smooth solution does exist, some of the numerical results we obtained being reported in Fig. 7. We observe in particular that the convergence orders are essentially independent of the value of the reentrant angle \(\theta \).
8 An Alternative Discretization Method Based on \({\mathbb {Q}}_1\) Finite Elements
(i) Variations with respect to h of the \(L^2({\varOmega })\) and \(H^1({\varOmega })\) norms of the computed approximation error \(\psi _h  \psi \), with \(\psi (x,y,z) =e^{\frac{1}{2}(x^2 + y^2 + z^2)}\) and \(\psi (x,y,z) =\frac{1}{2}\left( x^2+5y^2+15z^2 \right) \) and related convergence orders. (ii) Variations with respect to h of the number of relaxation iterations necessary to achieve convergence
h  \(\psi _h\psi _{L_2}\)  \(\psi _h\psi _{H_1}\)  Iter.  

Exact solution \(\psi (x,y,z) =e^{\frac{1}{2}(x^2 + y^2 + z^2)}\)  
1/10  8.09e−03  –  1.18e−01  –  56 
1/20  2.28e−03  1.82  5.67e−02  1.06  50 
1/30  1.05e−03  1.90  3.71e−02  1.04  46 
1/40  6.02e−04  1.93  2.75e−02  1.03  44 
1/50  3.90e−04  1.95  2.18e−02  1.03  42 
Exact solution \(\psi (x,y,z) =\frac{1}{2}\left( x^2+5y^2+15z^2 \right) \)  
1/10  1.26e−02  –  1.65e−01  –  713 
1/20  3.62e−03  1.79  7.88e−02  1.06  716 
1/30  1.71e−03  1.85  5.15e−02  1.05  696 
1/40  9.91e−04  1.88  3.82e−02  1.03  681 
1/50  6.48e−04  1.90  3.03e−02  1.03  671 
All the numerical results reported below are related to \({\varOmega }= (0,1)^3\). On Table 10 we have reported the variations with respect to h of the \(L^2({\varOmega })\) and \(H^1({\varOmega })\) norms of the computed approximation error \(\psi _h  \psi \), for \(\psi \) defined by \(\psi (x,y,z) =e^{\frac{1}{2}(x^2 + y^2 + z^2)}\) and \(\psi (x,y,z) =\frac{1}{2}\left( x^2+5y^2+15z^2 \right) \), the related convergence orders, and the number of relaxation iterations necessary to achieve convergence. The local optimization problems are solved using the Newton method described in Sect. 4.2. As expected, nearly optimal orders of convergence are obtained for both the \(L^2({\varOmega })\) and \(H^1({\varOmega })\) norms of the computed approximation error. Both solutions exhibit comparable orders of convergence, however, the larger anisotropy of the second one implies a larger number of iterations for the relaxation algorithm to achieve its convergence. Approximation errors and iteration numbers are consistent with those reported in Sect. 7.2 for the same test problems.
(i) Variations with respect to h of the \(L^2({\varOmega })\) and \(H^1({\varOmega })\) norms of the computed approximation error \(\psi _h  \psi \), with \(\psi (x,y,z) = \sqrt{R^2  (x^2 + y^2 + z^2)}\) with \(R=\sqrt{6}\); related convergence orders. (ii) Variations with respect to h of the number of relaxation iterations necessary to achieve convergence
h  \(\psi _h\psi _{L_2}\)  \(\psi _h\psi _{H_1}\)  \({\mathbf {D}}^2_h\psi _h^{n}{\mathbf {D}}^2\psi _{L_2}\)  Iter.  

1/10  1.63e−03  2.24e−02  –  8.18e−03  –  22  
1/20  4.49e−04  1.85  1.06e−02  1.07  2.95e−03  1.47  18 
1/30  2.06e−04  1.91  6.94e−03  1.05  1.62e−03  1.48  16 
1/40  1.18e−04  1.94  5.14e−03  1.03  1.05e−03  1.48  15 
1/50  7.65e−05  1.94  4.09e−03  1.03  7.55e−04  1.49  15 
(i) Variations with respect to h of the \(L^2({\varOmega })\) and \(H^1({\varOmega })\) norms of the computed approximation error \(\psi _h  \psi \), with \(\psi (x,y,z) = \sqrt{R^2  (x^2 + y^2 + z^2)}\) with \(R=\sqrt{3}\); related convergence orders. (ii) Variations with respect to h of the number of relaxation iterations necessary to achieve convergence
h  \(\psi _h\psi _{L_2}\)  \(\psi _h\psi _{H_1}\)  \({\mathbf {D}}^2_h\psi _h^{n}{\mathbf {D}}^2\psi _{L_2}\)  Iter.  

1/10  3.06e−03  4.70e−02  –  2.71e−01  35  
1/20  8.66e−04  1.82  2.30e−02  1.02  2.59e−01  34 
1/30  4.00e−04  1.90  1.53e−02  1.01  2.55e−01  31 
1/40  2.29e−04  1.93  1.14e−02  1.00  2.55e−01  30 
The numerical results we have just reported show that, as long as accuracy and number of iterations are concerned, \({\mathbb {Q}}_1\) based finite element approximations of problem (1) compared well with \({\mathbb {P}}_1\) based ones if \({\varOmega }\) is a cube and uniform structured partitions of \({\varOmega }\) are used to define the finite element spaces. However the \({\mathbb {P}}_1\) based methods can easily handle domains \({\varOmega }\) of arbitrary shapes and unstructured finite element partitions, properties that the \({\mathbb {Q}}_1\) based methods do not share.
9 Further Comments and Conclusions
In this article, we have discussed a leastsquares/relaxation/mixed finite element methodology for the numerical solution of the Dirichlet problem for the threedimensional elliptic Monge–Ampère equation \(\det {\mathbf {D}}^2 \psi = f (>0)\) in \({\varOmega }\). The results reported in Sects. 7 and 8 show the robustness and flexibility of this methodology and its ability at approximating smooth convex solutions (if such solutions do exist) with nearly optimal orders of accuracy for the \(L^2({\varOmega })\) and \(H^1({\varOmega })\) norms of the approximation error \(\psi _h  \psi \). To the best of our knowledge, the above methodology is one of the very few which can solve, with nearly optimal orders of accuracy, the threedimensional Monge–Ampère equation on domains \({\varOmega }\) with curved boundaries using \({\mathbb {P}}_1\) based finite element approximations.
Notes
Acknowledgements
The authors thank Prof. Marco Picasso (EPFL), the participants of the SCPDE 2017 conference (Hong Kong, June 5–8, 2017) for fruitful discussions, and the anonymous referees for their constructive comments.
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