Numerical Solution of the 2-Hessian Equation by a Newton’s Algorithm

The elliptic 2-Hessian equation is a fully nonlinear partial differential equation that is related, for example, to intrinsic curvature for three dimensional manifolds. We solve numerically this equation with periodic boundary condition and with Dirichlet boundary condition using a Newton’s algorithm. We verify numerically, by introducing finite difference schemes, the convergence of the algorithm which is obtained in few iterations. Citation: Haj EA, Khalil H, Hossein M (2018) Numerical Solution of the 2-Hessian Equation by a Newton’s Algorithm. J Appl Computat Math 7: 393. doi: 10.4172/2168-9679.1000393

In the periodic setting, in eqn. (1) reads as follows [2]: given a positive periodic function f on T 3=  3 /Z 3 , find a periodic function u: T 3 7→  such that where M 2 is the nonlinear differential operator defined by M 2 :=u↦ σ 2 [λ(I +D 2 u)]. This equation is none other than eqn. (1) with ψ is of the orm 2 1 2 x u + Note that a necessary condition for eqn. (3) to be well posed is that In the next place, we solve numerically the Dirichlet problem which obtained when treating computationally prescribed curvature problems.

Properties of the 2-Hessian Operator
Then where λ 1 , λ 2 and λ 3 are the eigenvalues of M. Therefore, by expanding in eqn. (7) and using in eqn.

Algorithm of Resolution
Using a global convergence Newton method [1], to linearize the eqn. (1), the algorithm we consider reads: Given u 0 , loop over n ∈ N, • Computation of f n= T 2 [u n ].
• Computation of θ n as solution of the linearized 2-Hessian equation with the stabilization factor τ ≥ 1.
Where T 2= M 2 for the periodic problem and T 2= S 2 for the Dirichlet problem. For τ=1, we obtain the classical Newton's method.

Linearization
Let s be a parameter in R . We have By expanding in eqn. (9) we obtain the linearization of S 2 [u] and M 2 [u]. For u ∈ C 2 : Proof. see [3].

Numerical Experiments
We discretize the problem's domain [0,1] 3 by dividing the domain into a uniform grid with grid space h. We denote by D ν1ν2 u the centered second order finite difference discretization of the operator u ν1ν2 for ν 1 ,ν 2 ∈ {x,y,z}, and by D 2,h u the discretization of the Hessian matrix.

Periodic problem
The numerical scheme connected to the problem in eqn. (3), obtained by using the global convergence Newton method and the discrete operators described above, is given, in each iteration, by the following linear system of (m+1) 3 equations with (m+1) 3 unknowns, where m+2 is the number of discretization points given by  The numerical tests are shown in the following two figures.
In Figure 1, we consider the function f=1+sin(2πx)sin(2πy) sin(2πz) and we solve the problem in the tore T 3 . Figure 1 shows the convergence of the error And in all the cases the curves are very close, that means that the convergence of the algorithm is almost independent from the grid space. Then the algorithm is efficient even on very coarse grid.
In Figure 2 shows the convergence of the error ku ex −u n k L 2 in terms of the number of iterations, where u ex is the solution of the eqn. (3) for f=M 2 [u] with u(x,y,z)=0.02sin(2πx)sin(2πy)sin(2πz). Here we fix m=15 and τ=1.

Dirichlet problem
We consider the 2-Hessian equation in R 3 with Dirichlet boundary conditions:   Finally, Figure 4 shows the order of convergence of the algorithm for different values of τ. We remark that the order of convergence decrease very fast in terms of τ. Note that in practice we have taken τ=1, for which value the order of convergence is close to 2.

Conclusion
We presented a Newton's algorithm to solve the fully nonlinear 2-Hessian equation in the case where it is elliptic and the solution is smooth enough. The numerical experiments show that the convergence is very fast. Then we can solve the 2-Hessian in the cost of solving a few number of linear elliptic problems. The sparsity of the matrix of discretization allows us to solve quickly the linear problems. Moreover, the numerical tests show a good stability of this algorithm.