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IsoValue-0.18101-0.171727-0.162445-0.153162-0.14388-0.134597-0.125315-0.116032-0.106749-0.0974669-0.0881843-0.0789018-0.0696192-0.0603366-0.0510541-0.0417715-0.032489-0.0232064-0.0139238-0.00464128table:cone20 JEAN-DAVID BENAMOU, BRITTANY D. FROESE, AND ADAM M. OBERMANN∥ ∥ u − u(N) ∞(M1) (M1 ∗ )21 1.4 × 10 −2 2.5 × 10 −341 8.2 × 10 −3 1.5 × 10 −361 5.9 × 10 −3 1.1 × 10 −381 4.6 × 10 −3 8.4 × 10 −4101 3.8 × 10 −3 7.0 × 10 −4121 3.2 × 10 −3 6.0 × 10 −4141 2.8 × 10 −3 5.3 × 10 −4Table 13. Errors for the exact solution u(x, y) = √ x 2 + y 2 on anN × N grid.We monitor the difference between two succesive iterates, called convergencebelow, to check that the iterative process converges. Finally, we monitor the residualdefined as√R[u] = ∆T [u] − u 2 xx + u 2 yy + 2u 2 xy + 2f,which should go to 0 as we reach the solution of Monge-Ampère.In most examples, this finite element implementation is not significantly fasterthan the finite difference implementation. The adaptivity improved performance forthe example of §5.3. This method, initialised on a uniform 40×40 mesh, converges inabout 40 iterations whereas the finite difference implementation requires thousandsof iterations to converge. See Figure 11 and Table 8.(a)fig:nonsmooth_sol(b)fig:nonsmooth_convFigure 11. Results with f = 1 and homogeneous Dirichlet boundaryconditions initialised on a uniform 40 × 40 mesh. (a) Finalmesh and level curves of the solution. (b) Convergence and residualagainst number of iterations.fig:nonsmooth8. ConclusionsTwo simple numerical methods (M1), (M2) were introduced for solving (MA).Computations were performed on a number of solutions whose regularity ranged