TY - JOUR
T1 - A CONVERGENT FINITE DIFFERENCE METHOD FOR COMPUTING MINIMAL LAGRANGIAN GRAPHS
AU - Hamfeldt, Brittany Froese
AU - Lesniewski, Jacob
N1 - Funding Information:
35J25, 35J60, 35J66. Key words and phrases. finite difference methods; minimal Lagrangian graphs; second boundary value problem; fully nonlinear elliptic equations; eigenvalue problems. The first author was partially supported by NSF DMS-1619807 and NSF DMS-1751996. The second author was partially supported by NSF DMS-1619807. ∗ Corresponding Author.
Publisher Copyright:
© 2022 American Institute of Mathematical Sciences. All rights reserved.
PY - 2022/2
Y1 - 2022/2
N2 - 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.
AB - 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.
KW - Eigenvalue problems
KW - Finite difference methods
KW - Fully nonlinear elliptic equations
KW - Minimal Lagrangian graphs
KW - Second boundary value problem
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U2 - 10.3934/CPAA.2021182
DO - 10.3934/CPAA.2021182
M3 - Article
AN - SCOPUS:85123839747
SN - 1534-0392
VL - 21
SP - 393
EP - 418
JO - Communications on Pure and Applied Analysis
JF - Communications on Pure and Applied Analysis
IS - 2
ER -