Hybridizable discontinuous Galerkin methods for the two-dimensional Monge-Ampère equation. (English) Zbl 07900962
Summary: We introduce two hybridizable discontinuous Galerkin (HDG) methods for numerically solving the two-dimensional Monge-Ampère equation. The first HDG method is devised to solve the nonlinear elliptic Monge-Ampère equation by using Newton’s method. The second HDG method is devised to solve a sequence of the Poisson equation until convergence to a fixed-point solution of the Monge-Ampère equation is reached. Numerical examples are presented to demonstrate the convergence and accuracy of the HDG methods. Furthermore, the HDG methods are applied to \(r\)-adaptive mesh generation by redistributing a given scalar density function via the optimal transport theory. This \(r\)-adaptivity methodology leads to the Monge-Ampère equation with a nonlinear Neumann boundary condition arising from the optimal transport of the density function to conform the resulting high-order mesh to the boundary. Hence, we extend the HDG methods to treat the nonlinear Neumann boundary condition. Numerical experiments are presented to illustrate the generation of \(r\)-adaptive high-order meshes on planar and curved domains.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65H10 | Numerical computation of solutions to systems of equations |
| 47H10 | Fixed-point theorems |
| 35J96 | Monge-Ampère equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
Keywords:
Monge-Ampère equation; hybridizable discontinuous Galerkin methods; grid adaptivity; \(r\)-adaptivity; elliptic equationsSoftware:
ExasimReferences:
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