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February  2022, 21(2): 393-418. doi: 10.3934/cpaa.2021182

## A convergent finite difference method for computing minimal Lagrangian graphs

 Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102

* Corresponding Author

Received  February 2021 Revised  June 2021 Published  February 2022 Early access  November 2021

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

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.

Citation: Brittany Froese Hamfeldt, Jacob Lesniewski. A convergent finite difference method for computing minimal Lagrangian graphs. Communications on Pure & Applied Analysis, 2022, 21 (2) : 393-418. doi: 10.3934/cpaa.2021182
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##### References:
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]
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 $x_0$
Domain and computed target ellipse
Computed maps from a square $X$ to various targets $Y$
Circular domain $X$ and square target $Y$
Circular domain $X$ and degenerate target $Y$
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
 $h$ $\|u^h- u_{\text{ex}}\|_\infty$ 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
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
 $h$ $\|u^h- u_{\text{ex}}\|_\infty$ 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
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