4.7 General Algorithm and Numerical Experiments
4.7.3 Examples with data involving a Dirac function
The first problem with a Dirac function on the right hand side reads as ( detD2u=πδ(1/2,1/2) inΩ,
u= q
(x−0.5)2+¡
y−0.5¢2
on∂Ω, (4.41)
The exact solutionuof problem (4.41) is defined by u(x,y)=
q
(x−0.5)2+¡
y−0.5¢2
∀(x,y)∈Ω¯,
As in subsection 4.5.4 for both solvers we approximate the Dirac functionδ(α,β) by f(α,β)
ε ¡
x,y¢
= ε2
π³
ε2+(x−α)2+¡
y−β¢2´2 ,
whereεis a small positive value. Therefore, we approximateπδ(1/2,1/2)byfε(1/2,1/2)(x,y). To estimate the error norms, we assume that lim
h→0lim
ε→0uηε=u. Moreover, as in subsection 4.5.4, we setε=10−3/2.
In this experiment, we expect the adaptive algorithm to refine the mesh around (0.5, 0.5), i.e., where the Dirac function occurs. Figure 4.12 shows indeed the refined meshes for both solvers and differentT OL.
Table 4.16 shows numerical results obtained by PMA and LSMA solvers. For both solvers, when T OLdecreases,hmi n,hmax,ηIK,L2error norm decrease, and the number of elements and nodes increase. For PMA, the number of elements and nodes is larger than LSMA. — Figure 4.13 displays the refined meshes for LSMA with fixedT OL=0.25 and different values ofε. In this figure, we observe that the smaller the parameterεis, the larger the number of elements.
More precisely, Table 4.17 shows that asεdecreases, thehmi n,hmax,L2error norm decreases and the number of elements and nodes increases. Asεgets smaller, the problem becomes more challenging, and therefore,ηIK increases.
In this problem we consider two Dirac function on the right hand side ( detD2u=π2δ(1/4,1/2)+π2δ(3/4,1/2) inΩ,
u=g on∂Ω, (4.42)
T OL=0.5 T OL=0.25 T OL=0.125
T OL=0.5 T OL=0.25 T OL=0.125
Figure 4.12 – Examples with Dirac function for data f(x,y) = πδ(1/2,1/2) and g(x,y) = q
(x−0.5)2+¡
y−0.5¢2
. Graphs of the final adapted mesh for various values ofT OL and fixedε=10−3/2. Top row: Graphs obtained with PMA solver after 800 timesteps. Bottom row:
Graphs obtained with LSMA solver after 500 iterations.
ε=10−3/2 ε=10−2 ε=10−5/2
Figure 4.13 – Examples with Dirac function for data f(x,y) = πδ(1/2,1/2) and g(x,y) = q
(x−0.5)2+¡
y−0.5¢2
. Graphs of the final adapted mesh for various values ofεand fixed T OL=0.25. The graphs obtained with LSMA solver after 500 iterations.
4.7. General Algorithm and Numerical Experiments
Table 4.16 – Examples with Dirac function for data f(x,y) = πδ(1/2,1/2) and g(x,y) = q
(x−0.5)2+¡
y−0.5¢2
. Convergence behavior of the algorithm for various values of parame-terT OLand fixedε=10−3/2. The columns contain the final minimal and maximal mesh size, the final numbers of elements and nodes, the value of the estimator, andL2error norms. Top table: Results obtained with PMA solver after 800 timesteps. Bottom table: Results obtained with LSMA solver after 500 iterations.
PMA
T OL hmi n hmax # elem # nodes ηI ||u−uh||L2
0.5 1.21e-02 1.90e-01 627 328 7.24e-01 3.71e-02 0.25 7.85e-03 1.23e-01 1331 739 5.23e-01 1.37e-02 0.125 3.58e-03 5.57e-02 3425 1748 3.50e-01 6.68e-03
LSMA
T OL hmi n hmax # elem # nodes ηI ||u−uh||L2
0.5 1.36e-02 1.77e-01 509 269 8.71e-01 7.73e-02 0.25 8.41e-03 1.24e-01 1128 582 5.50e-01 3.84e-02 0.125 4.89e-03 7.78e-02 2920 1489 3.70e-01 1.92e-02
Table 4.17 – Examples with Dirac function for data f(x,y) = πδ(1/2,1/2) and g(x,y) = q
(x−0.5)2+¡
y−0.5¢2
. Convergence behavior of the algorithm for various values of param-eterε=10−3/2and fixedT OL=0.25. The columns contain the final minimal and maximal mesh size, the final numbers of elements and nodes, the value of the estimator, andL2error norms. The results obtained with LSMA solver after 500 iterations.
LSMA ε2 hmi n hmax # elem # nodes ηI ||u−uh||L2
1e-03 8.41e-03 1.24e-01 1128 582 5.50e-01 3.84e-02 1e-04 7.93e-03 1.40e-01 1224 636 0.12e-01 2.29e-02 1e-05 4.19e-03 7.02e-02 2986 1544 0.15e-01 1.29e-02
where error norms, we assume that lim
h→0lim
ε→0uηε=u. Moreover, as before we setε=10−3/2.
In this experiment, we expect the adaptive algorithm to refine the mesh around the
neighbor-hood of the line between (1/4, 1/2) and (3/4, 1/2). Figure 4.12 displays the refined meshes for both solvers and differentT OL, which shows the expected behavior.
Table 4.16 shows numerical results obtained by PMA and LSMA solvers. For the PMA, although theL2error norm is accurate to the order of 10−3, it does not decrease withT OL. On the other hand, for LSMA, theL2error norm is of order 10−2, and it decreases monotonically withT OL.
T OL=0.5 T OL=0.25 T OL=0.125
T OL=0.5 T OL=0.25 T OL=0.125
Figure 4.14 – Examples with Dirac function for data f(x,y)=π2δ(1/4,1/2)+π2δ(3/4,1/2)andg(x,y) is giver in (4.43). Graphs of the final adapted mesh for various values ofT OL and fixed ε=10−3/2. Top row: Graphs obtained with PMA solver after 800 timesteps. Bottom row:
Graphs obtained with LSMA solver after 500 iterations.
4.7. General Algorithm and Numerical Experiments
Table 4.18 – Examples with Dirac function for dataf(x,y)=π2δ(1/4,1/2)+π2δ(3/4,1/2)andg(x,y) is giver in (4.43). Convergence behavior of the algorithm for various values of parameterT OL and fixedε=10−3/2. The columns contain the final minimal and maximal mesh size, the final numbers of elements and nodes, the value of the estimator, andL2error norms. Top table:
Results obtained with PMA solver after 800 timesteps. Bottom table: Results obtained with LSMA solver after 500 iterations.
PMA
T OL hmi n hmax # elem # nodes ηI ||u−uh||L2
0.5 1.11e-02 3.27e-01 722 373 7.72e-01 7.75e-03 0.25 5.96e-03 2.00e-01 1886 969 4.85e-01 7.71e-03 0.125 3.96e-03 1.26e-02 4664 2371 2.85e-01 8.42e-03
LSMA
T OL hmi n hmax # elem # nodes ηI ||u−uh||L2
0.5 1.23e-02 2.17e-01 687 356 5.98e-01 3.84e-02 0.25 8.14e-03 2.25e-01 1617 828 4.82e-01 2.08e-02 0.125 4.89e-03 7.78e-02 2920 1489 3.70e-01 1.92e-02
5 Conclusion
In this thesis, numerical methods for solutions of first and second-order fully nonlinear equations have been presented. We have discussed an operator-splitting/ finite element method for the numerical solution of the Dirichlet problem for orthogonal maps, and we have extended it by introducing an anisotropic space adaptivity method. Then, we have derived least-squares/relaxation methods for the numerical solutions of Jacobian equation/inequality, and the three dimensional Monge-Ampère equation. We have also developed a second order time integration method for the approximation of a parabolic and elliptic 2D Monge-Ampère equation. Lastly, we have introduced an isotropic adaptive method for the two dimensional elliptic Monge-Ampère equation.
In Chapter 1, we tackled the orthogonal map equation and we proposed an operator-splitting/
finite element method for the numerical solution of the Dirichlet problem. This method is based on a variational principle, the introduction of the associated flow problem, and a time-stepping splitting algorithm. Results show the robustness and the flexibility of this method and its ability to approximate solutions with line singularities on convex domains, with convergence of order one for theL2(Ω) norm of the approximation error. Then the method has been extended by proposing an anisotropic adaptive mesh algorithm that allows to track the singularities of the gradient of the solution more accurately. Numerical experiments have shown the robustness of the derived estimator and the adaptive algorithm.
In Chapter 2, we focused on the numerical approximation of the solutions of the prescribed Ja-cobian equation and inequality. The proposed method relies on a least-squares reformulation of the original problem and on a relaxation algorithm. The method is based on variational approaches where the use of relaxation algorithm allows the decoupling of the least-squares problem into local nonlinear and to variational problems. We have presented the implemen-tation in the finite element space and numerical examples to validate the method.
In Chapter 3, a least-squares/relaxation finite element method also has been introduced to solve the Dirichlet problem for the three-dimensional elliptic Monge-Ampère equation.
The method resembles the least squares framework that has been developed for several fully
nonlinear equations, and in Chapter 2. In this case, new solvers have been derived to provide solutions for the local nonlinear and variational problems. Numerical results have been validated by various test cases usingP1andQ1polynomials. The method achieves nearly optimal orders for theL2(Ω) andH1(Ω) error norms when smooth solutions are approximated.
In Chapter 4, we presented a numerical method for the solution of a parabolic 2D Monge-Ampère equation. Moreover, we have shown the efficiency of the method in capturing station-ary solutions, such as the solutions of the elliptic Monge-Ampère equation. In the numerical examples, when tested to smooth solutions, the method had nearly optimal orders for the L2(Ω) andH1(Ω) error norms. The method proved to be also robust at approximating non-smooth solutions. We have derived a heuristic error estimate, and we have designed an adaptive algorithm for the two-dimensional elliptic Monge-Ampère equation. Numerical experiments confirmed the robustness of the estimators and adaptive algorithm by using two different solvers.
It is possible to extent this work in different directions by working further on the following subtopics. For example, the operator-splitting/finite element method that used to solve the orthogonal maps can be broadened to include the rigid maps problems. The prescribed Jacobian equation can be extended to include a transformation map of the formu:Ω⊂R3→ R3. This will allow us to tackle problems in optimal transportation in 3 dimensional setting.
The parabolic and elliptic Monge-Ampère equation can also be extended to solve not only applications in optimal transport, but also financial ones. Last, the derived error estimates and the adaptive algorithms that are essentially designed for orthogonal maps and Monge-Ampère equation can be improved and extended to include other fully nonlinear equations such as Eikonal and Pucci equations.
Bibliography
Abel, Z., Demaine, E. D., Demaine, M. L., Eppstein, D., Lubiw, A., and Uehara, R. (2018). Flat foldings of plane graphs with prescribed angles and edge lengths.Journal of Computational Geometry, 9(1):74–93.
Abugirda, H. and Katzourakis, N. (2018). On the well-posedness of global fully nonlinear first order elliptic systems.Advances in Nonlinear Analysis, 7(2):139–148.
Afkham, B. M., Bhatt, A., Haasdonk, B., and Hesthaven, J. S. (2018). Symplectic model-reduction with a weighted inner product.arXiv preprint arXiv:1803.07799.
Akrivis, G. (2015). Stability of implicit-explicit backward difference formulas for nonlinear parabolic equations.SIAM Journal on Numerical Analysis, 53(1):464–484.
Akrivis, G. and Lubich, C. (2015). Fully implicit, linearly implicit and implicit–explicit back-ward difference formulae for quasi-linear parabolic equations.Numerische Mathematik, 131(4):713–735.
Akrivis, G., Makridakis, C., and Nochetto, R. H. (2009). Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods. Numerische Mathematik, 114(1):133.
Aleksandrov, A. D. (1958). Dirichlet’s problem for the equation Detkz_i jk = ϕ(z_1, . . . ,z_n,z,x_1, . . . ,x_n). I. Vestnik Leningradskogo Universiteta. Matematika, Mekhanika, Astronomiya, 13(1):5–24.
Ambrosio, L., Caffarelli, L., Salsa, S., Brenier, Y., Buttazzo, G., and Villani, C. (2003). Optimal Transportation and Applications. Springer-Verlag Berlin Heidelberg.
Apel, T., Nicaise, S., and Sirch, D. (2011). A posteriori error estimation of residual type for anisotropic diffusion–convection–reaction problems.Journal of Computational and Applied Mathematics, 235(8):2805–2820.
Armstrong, S. N., Sirakov, B., and Smart, C. K. (2012). Singular solutions of fully nonlinear ellip-tic equations and applications.Archive for Rational Mechanics and Analysis, 205(2):345–394.
Avinyó, A., Solà-Morales, J., and València, M. (2012). A singular initial value problem to construct density-equalizing maps. Journal of Dynamics and Differential Equations, 24(1):51–59.
Awanou, G. (2014). Standard finite elements for the numerical resolution of the elliptic Monge-Ampère equation: classical solutions.IMA Journal of Numerical Analysis, 35(3):1150–1166.
Awanou, G. (2016). On standard finite difference discretizations of the elliptic Monge-Ampère equation.Journal of Scientific Computing, 69(2):892–904.
Awanou, G. (2020). A two-grid method for theC0interior penalty discretization of the Monge-Ampère equation.Journal of Computational Mathematics, 38(4):547–564.
Awanou, G. and Leopold, M. M. (2015). A variational method for computing numerical solutions of the Monge-Ampère equation.arXiv preprint arXiv:1510.00453.
Babuška, I., Durán, R., and Rodríguez, R. (1992). Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements.SIAM Journal on Numerical Analysis, 29(4):947–964.
Babuška, I. and Rheinboldt, W. C. (1978). A-posteriori error estimates for the finite element method.International Journal for Numerical Methods in Engineering, 12(10):1597–1615.
Ball, J. M. and James, R. D. (1992). Proposed experimental tests of a theory of fine microstruc-ture and the two-well problem.Philosophical Transactions: Physical Sciences and Engineer-ing, 338(1650):389–450.
Barles, G. and Jakobsen, E. R. (2002). On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations.ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, 36(1):33–54.
Barles., G. and Souganidis, P. E. (1991). Convergence of approximation schemes for fully nonlinear equations.Asymptotic Analysis, 4(3):271–283.
Bartels, S., Bonito, A., and Nochetto, R. H. (2017). Bilayer plates: Model reduction, Γ-convergent finite element approximation and discrete gradient flow. Communications on Pure and Applied Mathematics, 70(3):547–589.
Basterrechea, S. and Dacorogna, B. (2014). Existence of solutions for Jacobian and Hessian equations under smallness assumptions.Numerical Functional Analysis and Optimization, 35(7-9):868–892.
Benamou, J., Froese, B. D., and Oberman, A. M. (2014). Numerical solution of the optimal transportation problem using the Monge-Ampère equation. Journal of Computational Physics, 260:107–126.
Benamou, J. D. and Brenier, Y. (2000). A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem.Numerische Mathematik, 84:375–393.
Bibliography
Benamou, J.-D. and Froese, B. D. (2014). A viscosity framework for computing Pogorelov solutions of the Monge-Ampere equation.arXiv preprint arXiv:1407.1300.
Benamou, J. D., Froese, B. D., and Oberman, A. M. (2010). Two numerical methods for the elliptic Monge-Ampère equation.ESAIM: Mathematical Modelling and Numerical Analysis -Modélisation Mathématique et Analyse Numérique, 44(4):737–758.
Bern, M. W. and Hayes, B. (1996). The complexity of flat origami. InSODA ’96.
Birindelli, I. and Galise, G. (2019). The Dirichlet problem for fully nonlinear degenerate elliptic equations with a singular nonlinearity. Calculus of Variations and Partial Differential Equations, 58(5):180.
Boehmer, K. (2008). On finite element methods for fully nonlinear elliptic equations of second order.SIAM Journal on Numerical Analysis, 46(3):1212–1249.
Bonito, A., Guignard, D., Nochetto, R. H., and Yang, S. (2020). LDG approximation of large deformations of prestrained plates.arXiv preprint arXiv:2011.01086.
Bonnans, J. F., Bonnet, G., and Mirebeau, J.-M. (2020). Monotone and second order consistent scheme for the two dimensional pucci equation.hal archives hal-02383521.
Botsaris, C. A. (1978). A class of methods for unconstrained minimization based on stable numerical integration techniques. Journal of Mathematical Analysis and Applications, 63(3):729–749.
Bourgault, Y., Picasso, M., Alauzet, F., and Loseille, A. (2009). On the use of anisotropic a posteriori error estimators for the adaptative solution of 3D inviscid compressible flows.
International Journal For Numerical Methods In Fluids, 59:47–74.
Brenner, S. C., Gudi, T., Neilan, M., and Sung, L.-Y. (2011). C0penalty methods for the fully nonlinear Monge-Ampère equation.Mathematics of Computation, 80(276):1979–1995.
Brenner, S. C. and Neilan, M. (2012). Finite element approximations of the three dimen-sional Monge-Ampère equation.ESAIM: Mathematical Modelling and Numerical Analysis, 46(5):979–1001.
Budd, C., Huang, W., and Russell, R. (2009). Adaptivity with moving grids. Acta Numerica, 18:111–241.
Budd, C. J. and Williams, J. F. (2006). Parabolic Monge–Ampère methods for blow-up prob-lems in several spatial dimensions. Journal of Physics A: Mathematical and General, 39(19):5425–5444.
Budd, C. J. and Williams, J. F. (2009). Moving mesh generation using the parabolic Monge-Ampère equation.SIAM Journal on Numerical Analysis, 31(5):3438–3465.
Burago, D. and Kleiner, B. (1998). Separated nets in Euclidean space and Jacobians of bilips-chitz maps.Geometric and Functional Analysis, 8(2):273–282.
Busca, J., Esteban, M. J., and Quaas, A. (2005). Nonlinear eigenvalues and bifurcation prob-lems for Pucci’s operators. Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 22(2):187–206.
Böhmer, K. and Schaback, R. (2019). A meshfree method for solving the Monge–Ampère equation.Numerical Algorithms, 82(2):539–551.
Caboussat, A. (2014). On the numerical solution of the Dirichlet problem for the elliptic (σ2) equation. In Fitzgibbon, W., Kuznetsov, Y. A., Neittaanmäki, P., and Pironneau, O., editors, Modeling, Simulation and Optimization for Science and Technology, volume 34, page 23–40.
Springer Netherlands.
Caboussat, A. (2019). Least-squares/relaxation method for the numerical solution of a 2D Pucci’s equation.Methods and Applications of Analysis, 26(2):113–132.
Caboussat, A. and Glowinski, R. (2008). A numerical method for a non-smooth advection-diffusion problem arising in sand mechanics.Communications on Pure & Applied Analysis, 8(1):161–178.
Caboussat, A. and Glowinski, R. (2012). Regularization methods for the divergence equation
∇ ·u=f.Journal of Computational Mathematics, 30(4):354–380.
Caboussat, A. and Glowinski, R. (2015a). A numerical algorithm for a fully nonlinear PDE involving the Jacobian determinant. InNumerical Mathematics and advanced applications - ENUMATH 2013, volume 103 ofLecture Notes in Computational Science and Engineering,
page 143–151. Springer.
Caboussat, A. and Glowinski, R. (2015b). A penalty-regularization-operator splitting method for the numerical solution of a scalar Eikonal equation. Chinese Annals of Mathematics Series B, 36(5):659–688.
Caboussat, A. and Glowinski, R. (2018). An alternating direction method of multipliers for the numerical solution of a fully nonlinear partial differential equation involving the Jacobian determinant.SIAM Journal on Scientific Computing, 40:A52–A80.
Caboussat, A., Glowinski, R., and Gourzoulidis, D. (2018). A least-squares/relaxation method for the numerical solution of the three-dimensional elliptic Monge-Ampère equation. Jour-nal of Scientific Computing, 77(1):1–26.
Caboussat, A., Glowinski, R., Gourzoulidis, D., and Picasso, M. (2019). Numerical approxima-tion of orthogonal maps.SIAM Journal on Scientific Computing, 41(6):B1341–B1367.
Caboussat, A., Glowinski, R., and Pan, T. W. (2015). On the numerical solution of some Eikonal equations: An elliptic solver approach. Chinese Annals of Mathematics Series B, 36(5):689–702.
Bibliography
Caboussat, A., Glowinski, R., and Sorensen, D. C. (2013). A least-squares method for the numerical solution of the Dirichlet problem for the elliptic Monge-Ampère equation in dimension two.ESAIM: Control, Optimization and Calculus of Variations, 19(3):780–810.
Caboussat, A. and Gourzoulidis, D. (2019). A second order time integration method for the approximation of a parabolic 2d Monge-Ampère equation. InProceedings of the European numerical mathematics and advanced applications.
Caboussat, A., Gourzoulidis, D., and Picasso, M. (2020). An anisotropic adaptive method for the numerical approximation of orthogonal maps.Journal of Computational and Applied Mathematics, (submitted).
Caffarelli, L., Crandall, M. G., Kocan, M., and ´Swi˛ech, A. (1996). On viscosity solutions of fully nonlinear equations with measurable ingredients.Communications on Pure and Applied Mathematics, 49(4):365–398.
Caffarelli, L. and Milman, M. (1999a). Monge Ampere Equation: Applications to Geometry and Optimization: Applications to Geometry and Optimization. American Mathematical Society.
Caffarelli, L., Nirenberg, L., and Spruck, J. (2014). The Dirichlet problem for nonlinear second-order elliptic equations I. Monge-Ampère equation.Communications on Pure and Applied Mathematics, 37:369–402.
Caffarelli, L. A. (1991). Some regularity properties of solutions of Monge-Ampère equation.
Communications on Pure and Applied Mathematics, 44(8-9):965–969.
Caffarelli, L. A. and Cabré, X. (1995).Fully Nonlinear Elliptic Equations. American Mathemati-cal Society.
Caffarelli, L. A. and Glowinski, R. (2008). Numerical solution of the Dirichlet problem for a Pucci equation in dimension two. Application to homogenization. Journal of Numerical Mathematics, 16(3):185–216.
Caffarelli, L. A. and Milman, M., editors (1999b). Monge-Ampère Equation: Application to Geometry and Optimization. American Mathematical Society, Providence, RI.
Caffarelli, L. A. and Stefanelli, U. (2008). A counterexample to C2,1regularity for parabolic fully nonlinear equations.Communications in Partial Differential Equations, 33(7):1216–1234.
Calabi, E. (1958). Improper affine hyperspheres of convex type and a generalization of a theorem by k. Jörgens.Michigan Mathematical Journal, 5:105–126.
Carlier, G. and Dacorogna, B. (2012). Solution of the Dirichlet problem for the prescribed Jacobian equation using the Monge-Ampère equation. Comptes Rendus Mathematique, 350(7):371–374.
Carstensen, C. (2006). Clément Interpolation and Its Role in Adaptive Finite Element Error Control, page 27–43. Birkhäuser Basel, Basel.
Chang, S.-Y. A., Han, Z.-C., and Yang, P. (2011). On the prescribingσ2curvature equation on S4.Calculus of Variations and Partial Differential Equations, 40(3):539–565.
Charbonneau, A., Dossou, K., and Pierre, R. (1997). A residual-based a posteriori error estima-tor for the Ciarlet-Raviart formulation of the first biharmonic problem.Numerical Methods for Partial Differential Equations, 13(1):93–111.
Ciarlet, P. G. (2002). The Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics.
Clément, P. (1975). Approximation by finite element functions using local regularization.
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, 9(R2):77–84.
Cossette, J.-F. and Smolarkiewicz, P. K. (2011). A Monge-Ampère enhancement for semi-Lagrangian methods.Computers & Fluids, 46(1):180–185.
Crandall, M., Evans, L., and Lions, P.-L. (1984). Some properties of viscosity solutions of Hamilton-Jacobi equations. Transactions of the American Mathematical Society, 282(2):487–502.
Csató, G., Dacorogna, B., and Kneuss, O. (2011).The Pullback Equation for Differential Forms.
Birkhäuser Boston.
Cupini, G., Dacorogna, B., and Kneuss, O. (2009). On the equation det∇u=f with no sign hypothesis.Calculus of Variations and Partial Differential Equations, 36:251–283.
Dacorogna, B., Glowinski, R., Kuznetzov, Y., and Pan, T.-W. (2004). On a conjuguate gradien-t/Newton/penalty method for the solution of obstacle problems. application to the solution of an Eikonal system with Dirichlet boundary conditions. In M. Kˇrížek, P. Neittaanmäki, R. G. and Korotov, S., editors,Conjugate Gradient Algorithms and Finite Element Methods, page 263–283. Springer-Verlag, Berlin, Heidelberg.
Dacorogna, B., Glowinski, R., and T.-W.Pan (2003). Numerical methods for the solution of a system of Eikonal equations with Dirichlet boundary conditions.C. R. Acad. Sci. Paris, Sér. I, 336:511–518.
Dacorogna, B. and Marcellini, P. (1999). Implicit Partial Differential Equations. Birkhaüser, Basel.
Dacorogna, B., Marcellini, P., and Paolini, E. (2008a). An explicit solution to a system of implicit differential equations.Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 25:163–171.
Dacorogna, B., Marcellini, P., and Paolini, E. (2008b). Lipschitz-continuous local isomet-ric immersions: rigid maps and origami.Journal de Mathématiques Pures et Appliquées, 90:66–81.