Hybridizable Discontinuous Galerkin Methods for the Two-Dimensional Monge–Ampère Equation
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10915_2024_2604_ReferencePDF.pdf
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Author(s)
Nguyen, Ngoc C.
Peraire, Jaime
Date Issued
June 27, 2024
Journal
Journal of Scientific Computing
Publisher
Springer US
Citation
Nguyen, N.C., Peraire, J. Hybridizable Discontinuous Galerkin Methods for the Two-Dimensional Monge–Ampère Equation. J Sci Comput 100, 44 (2024).
Version
Author's final manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
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Creative Commons Attribution-Noncommercial-ShareAlike
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DOI of Published Version
https://doi.org/10.1007/s10915-024-02604-3
