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A convergent finite difference method for computing minimal Lagrangian graphs

  • * Corresponding Author

    * Corresponding Author 

The first author was partially supported by NSF DMS-1619807 and NSF DMS-1751996. The second author was partially supported by NSF DMS-1619807

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  • We consider the numerical construction of minimal Lagrangian graphs, which is related to recent applications in materials science, molecular engineering, and theoretical physics. It is known that this problem can be formulated as an additive eigenvalue problem for a fully nonlinear elliptic partial differential equation. We introduce and implement a two-step generalized finite difference method, which we prove converges to the solution of the eigenvalue problem. Numerical experiments validate this approach in a range of challenging settings. We further discuss the generalization of this new framework to Monge-Ampère type equations arising in optimal transport. This approach holds great promise for applications where the data does not naturally satisfy the mass balance condition, and for the design of numerical methods with improved stability properties.

    Mathematics Subject Classification: Primary: 65N06, 65N12, 65N25; Secondary: 35J15, 35J25, 35J60, 35J66.


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  • Figure 1.  Discrete solution to Poisson's equation when viewed as an eigenvalue problem

    Figure 2.  Examples of quadtree meshes. White squares are inside the domain, while gray squares intersect the boundary [19]

    Figure 3.  Potential neighbors are circled in gray. Examples of selected neighbors are circled in black [19]

    Figure 4.  Examples of neighbors $ x_1, x_2 $ needed to construct a monotone approximation of the directional derivative in the direction $ n $ at the boundary point $ x_0 $

    Figure 5.  Domain and computed target ellipse

    Figure 6.  Computed maps from a square $ X $ to various targets $ Y $

    Figure 7.  Circular domain $ X $ and square target $ Y $

    Figure 8.  Circular domain $ X $ and degenerate target $ Y $

    Table 1.  Error in mapping an ellipse to an ellipse

    $ h $ $ \|u^h- u_{\text{ex}}\|_\infty $ Ratio Observed Order
    $ 2.625\times 10^{-1} $ $ 1.304 \times 10^{-1} $
    $ 1.313\times 10^{-1} $ $ 5.703\times 10^{-2} $ 2.287 1.194
    $ 6.563\times 10^{-2} $ $ 2.691\times 10^{-2} $ 2.119 1.084
    $ 3.281\times 10^{-2} $ $ 1.423\times 10^{-2} $ 1.891 0.919
    $ 1.641\times 10^{-2} $ $ 6.768\times 10^{-3} $ 2.103 1.072
     | Show Table
    DownLoad: CSV

    Table 2.  Error in mapping a circle to a line segment

    $ h $ $ \|u^h- u_{\text{ex}}\|_\infty $ Ratio Observed order
    $ 1.375\times 10^{-1} $ $ 9.132 \times 10^{-2} $
    $ 6.875\times 10^{-2} $ $ 3.812 \times 10^{-2} $ 2.396 1.261
    $ 3.438\times 10^{-2} $ $ 1.936\times 10^{-2} $ 1.969 0.978
    $ 1.719\times 10^{-2} $ $ 1.082\times 10^{-2} $ 1.790 0.840
    $ 8.59\times 10^{-3} $ $ 4.636\times 10^{-3} $ 2.333 1.222
     | Show Table
    DownLoad: CSV
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