Numerical solution of the two-dimensional elliptic Monge-Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach Export

@article{citeulike:2854028, abstract = {The main goal of this Note is to discuss a method for the numerical solution of the two-dimensional elliptic Monge-Amp\`{e}re equation with Dirichlet boundary conditions (the E-MAD problem). This method relies on the reformulation of E-MAD as a problem of Calculus of Variation involving the biharmonic operator (or closely related operators), and then to a saddle-point formulation for a well-chosen augmented Lagrangian functional, leading to iterative methods such as Uzawa-Douglas-Rachford. The above methodology applies to problems other than E-MAD (such as the Pucci equation). The results of numerical experiments are presented. They concern the solution of E-MAD on the unit square (0,1)×(0,1); the first test problem has a known smooth closed form solution which is easily computed with optimal order of convergence. The second test problem has also a known closed form solution; the fact that this solution has the H2([Omega])-regularity, but not the one, does not prevent optimal order of convergence. Finally, the third test problem having no smooth solution is more costly to solve and leads to discrete solutions showing negative curvature near the corners. To cite this article: E.J. Dean, R. Glowinski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).}, author = {Dean, Edward J. and Glowinski, Roland}, citeulike-article-id = {2854028}, citeulike-linkout-0 = {http://dx.doi.org/10.1016/S1631-073X(03)00149-3}, citeulike-linkout-1 = {http://www.sciencedirect.com/science/article/B6X1B-48GP3CK-3/1/489befa17ef1ca0ed2c578a4f45fc8b7}, doi = {10.1016/S1631-073X(03)00149-3}, journal = {Comptes Rendus Mathematique}, keywords = {2d, monge-ampere, numerical}, month = {May}, number = {9}, pages = {779--784}, posted-at = {2008-06-01 08:56:59}, priority = {2}, title = {Numerical solution of the two-dimensional elliptic Monge-Amp\`{e}re equation with Dirichlet boundary conditions: an augmented Lagrangian approach}, url = {http://dx.doi.org/10.1016/S1631-073X(03)00149-3}, volume = {336}, year = {2003} }

TY - JOUR ID - citeulike:2854028 L3 - citeulike-article-id:2854028 N2 - The main goal of this Note is to discuss a method for the numerical solution of the two-dimensional elliptic Monge-Ampère equation with Dirichlet boundary conditions (the E-MAD problem). This method relies on the reformulation of E-MAD as a problem of Calculus of Variation involving the biharmonic operator (or closely related operators), and then to a saddle-point formulation for a well-chosen augmented Lagrangian functional, leading to iterative methods such as Uzawa-Douglas-Rachford. The above methodology applies to problems other than E-MAD (such as the Pucci equation). The results of numerical experiments are presented. They concern the solution of E-MAD on the unit square (0,1)×(0,1); the first test problem has a known smooth closed form solution which is easily computed with optimal order of convergence. The second test problem has also a known closed form solution; the fact that this solution has the H2([Omega])-regularity, but not the one, does not prevent optimal order of convergence. Finally, the third test problem having no smooth solution is more costly to solve and leads to discrete solutions showing negative curvature near the corners. To cite this article: E.J. Dean, R. Glowinski, C. R. Acad. Sci. Paris, Ser. I 336 (2003). IS - 9 JF - Comptes Rendus Mathematique EP - 784 TI - Numerical solution of the two-dimensional elliptic Monge-Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach VL - 336 SP - 779 KW - 2d KW - monge-ampere KW - numerical AU - Dean, Edward AU - Glowinski, Roland PY - 2003/05/01/ UR - http://dx.doi.org/10.1016/S1631-073X(03)00149-3 ER -