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Hybridizable Discontinuous Galerkin Methods for the Two-Dimensional Monge–Ampère Equation

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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.

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Data Availability

The datasets generated in this study are available from the corresponding author on reasonable request.

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Funding

This work was supported by the United States Department of Energy under contract DE-NA0003965, the National Science Foundation under grant number NSF-PHY-2028125, the Air Force Office of Scientific Research under Grant No. FA9550-22-1-0356, and the MIT Portugal program under the seed grant number 6950138.

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  1. Department of Aeronautics and Astronautics, Center for Computational Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Ngoc Cuong Nguyen & Jaime Peraire

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Nguyen, N.C., Peraire, J. Hybridizable Discontinuous Galerkin Methods for the Two-Dimensional Monge–Ampère Equation. J Sci Comput 100, 44 (2024). https://doi.org/10.1007/s10915-024-02604-3

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