doi: 10.3934/cpaa.2021182
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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, doi: 10.3934/cpaa.2021182
References:
[1]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asym. Anal., 4 (1991), 271-283.   Google Scholar

[2]

P. W. BatesG. W. Wei and S. Zhao, Minimal molecular surfaces and their applications, J. Comp. Chem., 29 (2008), 380-391.   Google Scholar

[3]

J. D. BenamouB. D. Froese and A. M. Oberman, Numerical solution of the optimal transportation problem using the Monge-Ampère equation, J. Comput. Phys., 260 (2014), 107-126.  doi: 10.1016/j.jcp.2013.12.015.  Google Scholar

[4]

J. D. Benamou, A. Oberman and B. Froese, Numerical solution of the second boundary value problem for the elliptic Monge-Ampère equation, Inst. Nation. Recherche Inform. Automat., 2012, 37 pp. Google Scholar

[5]

D. P. Bertsekas, Convex Analysis and Optimization, Athena Scientific, Belmont, MA, 2003.  Google Scholar

[6]

S. Brendle and M. Warren, A boundary value problem for minimal Lagrangian graphs, J. Differ. Geom., 84 (2010), 267-287.   Google Scholar

[7]

S. C. BrennerT. GudiM. Neilan and L. Y. Sung, C0 penalty methods for the fully nonlinear Monge-Ampére equation, Math. Comp., 80 (2011), 1979-1995.  doi: 10.1090/S0025-5718-2011-02487-7.  Google Scholar

[8]

C. Budd and J. Williams, Moving mesh generation using the parabolic Monge-Ampère equation, SIAM J. Sci. Comput., 31 (2009), 3438-3465.  doi: 10.1137/080716773.  Google Scholar

[9]

M. G. CrandallH. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[10]

E. J. Dean and R. Glowinski, Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type, Comput. Meth. Appl. Mech. Engrg., 195 (2006), 1344-1386.  doi: 10.1016/j.cma.2005.05.023.  Google Scholar

[11]

P. Delanoë, Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampere operator, Ann. Inst. Hen. Poin. Non Lin. Anal., 8 (1991), 443-457.  doi: 10.1016/j.anihpc.2007.03.001.  Google Scholar

[12]

B. Engquist and B. D. Froese, Application of the Wasserstein metric to seismic signals, Commun. Math. Sci., 12 (2014), 979-988.  doi: 10.4310/CMS.2014.v12.n5.a7.  Google Scholar

[13]

X. Feng and M. Neilan, Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations, SIAM J. Sci. Comput., 38 (2009), 74-98.  doi: 10.1007/s10915-008-9221-9.  Google Scholar

[14]

B. D. Froese, A numerical method for the elliptic Monge-Ampère equation with transport boundary conditions, SIAM J. Sci. Comput., 34 (2012), A1432-A1459. doi: 10.1137/110822372.  Google Scholar

[15]

B. D. Froese, Meshfree finite difference approximations for functions of the eigenvalues of the Hessian, Numer. Math., 138 (2018), 75-99.  doi: 10.1007/s00211-017-0898-2.  Google Scholar

[16]

S. HakerL. ZhuA. Tannenbaum and S. Angenent, Optimal mass transport for registration and warping, Int. J. Comp. Vis., 60 (2004), 225-240.   Google Scholar

[17]

B. Hamfeldt, Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature, Comm. Pure Appl. Anal., 17 (2018), 671-707.  doi: 10.3934/cpaa.2018036.  Google Scholar

[18]

B. Hamfeldt, Convergence framework for the second boundary value problem for the Monge-Ampère equation, SIAM J. Numer. Anal., 57 (2019), 945-971.  doi: 10.1137/18M1201913.  Google Scholar

[19]

B. F. Hamfeldt and T. Salvador, Higher-order adaptive finite difference methods for fully nonlinear elliptic equations, SIAM J. Sci. Comput., 75 (2018), 1282-1306.  doi: 10.1007/s10915-017-0586-5.  Google Scholar

[20]

R. Harvey and H. B. Lawson, Calibrated geometries, Act. Math., 148 (1982), 47-157.  doi: 10.1007/BF02392726.  Google Scholar

[21]

R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rat. Mech. Anal., 101 (1988), 1-27.  doi: 10.1007/BF00281780.  Google Scholar

[22]

C. Y. KaoS. Osher and J. Qian, Lax?Friedrichs sweeping scheme for static Hamilton?Jacobi equations, J. Comput. phys., 196 (2004), 367-391.  doi: 10.1016/j.jcp.2003.11.007.  Google Scholar

[23]

R. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (Classics in Applied Mathematics Classics in Applied Mathemat), SIAM, Philadelphia, PA, USA, 2007. doi: 10.1137/1.9780898717839.  Google Scholar

[24]

Y. Lian and K. Zhang, Boundary Lipschitz regularity and the Hopf lemma for fully nonlinear elliptic equations, arXiv: 1812.11357. Google Scholar

[25]

A. Oberman, The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc., 135 (2007), 1689-1694.  doi: 10.1090/S0002-9939-07-08887-9.  Google Scholar

[26]

A. M. Oberman, Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton?Jacobi equations and free boundary problems, SIAM J. Numer. Anal., 44 (2006), 879-895.  doi: 10.1137/S0036142903435235.  Google Scholar

[27]

A. M. Oberman, Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the {H}essian, Disc. Cont. Dynam. Syst. Ser. B, 10 (2008), 221-238.  doi: 10.3934/dcdsb.2008.10.221.  Google Scholar

[28]

C. R. Prins, R. Beltman, J. H. M. ten Thije Boonkkamp, W. L. IJzerman, and T. W. Tukker, A least-squares method for optimal transport using the Monge-Ampère equation, SIAM J. Sci. Comp., 37 (2015), B937?B961. doi: 10.1137/140986414.  Google Scholar

[29]

L. Qi and J. Sun, A nonsmooth version of Newton's method, Math. program., 58 (1993), 353-367.  doi: 10.1007/BF01581275.  Google Scholar

[30]

J. QianY. T. Zhang and H. K. Zhao, A fast sweeping method for static convex Hamilton?Jacobi equations, J. Sci. Comput., 31 (2007), 237-271.  doi: 10.1007/s10915-006-9124-6.  Google Scholar

[31]

K. Smoczyk and M. T. Wang, Mean curvature flows of Lagrangian submanifolds with convex potentials, J. Differ. Geom., 62 (2002), 243-257.   Google Scholar

[32]

E. L. Thomas, D. M. Anderson, C. S. Henkee and D. Hoffman, Periodic area-minimizing surfaces in block copolymers, Nat., 334 (1988): 598. Google Scholar

[33]

R. P. Thomas and S. T. Yau, Special Lagrangians, stable bundles and mean curvature flow, Commun. Anal. Geom., 10 (2002), 1075-1113.  doi: 10.4310/CAG.2002.v10.n5.a8.  Google Scholar

[34]

J. Urbas, On the second boundary value problem for equations of Monge-Ampere type, J. Rein. Angew. Math., 487 (1997), 115-124.  doi: 10.1515/crll.1997.487.115.  Google Scholar

[35]

H. Zhao, A fast sweeping method for eikonal equations, Math. Comput., 74 (2005), 603-627.  doi: 10.1090/S0025-5718-04-01678-3.  Google Scholar

show all references

References:
[1]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asym. Anal., 4 (1991), 271-283.   Google Scholar

[2]

P. W. BatesG. W. Wei and S. Zhao, Minimal molecular surfaces and their applications, J. Comp. Chem., 29 (2008), 380-391.   Google Scholar

[3]

J. D. BenamouB. D. Froese and A. M. Oberman, Numerical solution of the optimal transportation problem using the Monge-Ampère equation, J. Comput. Phys., 260 (2014), 107-126.  doi: 10.1016/j.jcp.2013.12.015.  Google Scholar

[4]

J. D. Benamou, A. Oberman and B. Froese, Numerical solution of the second boundary value problem for the elliptic Monge-Ampère equation, Inst. Nation. Recherche Inform. Automat., 2012, 37 pp. Google Scholar

[5]

D. P. Bertsekas, Convex Analysis and Optimization, Athena Scientific, Belmont, MA, 2003.  Google Scholar

[6]

S. Brendle and M. Warren, A boundary value problem for minimal Lagrangian graphs, J. Differ. Geom., 84 (2010), 267-287.   Google Scholar

[7]

S. C. BrennerT. GudiM. Neilan and L. Y. Sung, C0 penalty methods for the fully nonlinear Monge-Ampére equation, Math. Comp., 80 (2011), 1979-1995.  doi: 10.1090/S0025-5718-2011-02487-7.  Google Scholar

[8]

C. Budd and J. Williams, Moving mesh generation using the parabolic Monge-Ampère equation, SIAM J. Sci. Comput., 31 (2009), 3438-3465.  doi: 10.1137/080716773.  Google Scholar

[9]

M. G. CrandallH. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[10]

E. J. Dean and R. Glowinski, Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type, Comput. Meth. Appl. Mech. Engrg., 195 (2006), 1344-1386.  doi: 10.1016/j.cma.2005.05.023.  Google Scholar

[11]

P. Delanoë, Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampere operator, Ann. Inst. Hen. Poin. Non Lin. Anal., 8 (1991), 443-457.  doi: 10.1016/j.anihpc.2007.03.001.  Google Scholar

[12]

B. Engquist and B. D. Froese, Application of the Wasserstein metric to seismic signals, Commun. Math. Sci., 12 (2014), 979-988.  doi: 10.4310/CMS.2014.v12.n5.a7.  Google Scholar

[13]

X. Feng and M. Neilan, Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations, SIAM J. Sci. Comput., 38 (2009), 74-98.  doi: 10.1007/s10915-008-9221-9.  Google Scholar

[14]

B. D. Froese, A numerical method for the elliptic Monge-Ampère equation with transport boundary conditions, SIAM J. Sci. Comput., 34 (2012), A1432-A1459. doi: 10.1137/110822372.  Google Scholar

[15]

B. D. Froese, Meshfree finite difference approximations for functions of the eigenvalues of the Hessian, Numer. Math., 138 (2018), 75-99.  doi: 10.1007/s00211-017-0898-2.  Google Scholar

[16]

S. HakerL. ZhuA. Tannenbaum and S. Angenent, Optimal mass transport for registration and warping, Int. J. Comp. Vis., 60 (2004), 225-240.   Google Scholar

[17]

B. Hamfeldt, Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature, Comm. Pure Appl. Anal., 17 (2018), 671-707.  doi: 10.3934/cpaa.2018036.  Google Scholar

[18]

B. Hamfeldt, Convergence framework for the second boundary value problem for the Monge-Ampère equation, SIAM J. Numer. Anal., 57 (2019), 945-971.  doi: 10.1137/18M1201913.  Google Scholar

[19]

B. F. Hamfeldt and T. Salvador, Higher-order adaptive finite difference methods for fully nonlinear elliptic equations, SIAM J. Sci. Comput., 75 (2018), 1282-1306.  doi: 10.1007/s10915-017-0586-5.  Google Scholar

[20]

R. Harvey and H. B. Lawson, Calibrated geometries, Act. Math., 148 (1982), 47-157.  doi: 10.1007/BF02392726.  Google Scholar

[21]

R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rat. Mech. Anal., 101 (1988), 1-27.  doi: 10.1007/BF00281780.  Google Scholar

[22]

C. Y. KaoS. Osher and J. Qian, Lax?Friedrichs sweeping scheme for static Hamilton?Jacobi equations, J. Comput. phys., 196 (2004), 367-391.  doi: 10.1016/j.jcp.2003.11.007.  Google Scholar

[23]

R. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (Classics in Applied Mathematics Classics in Applied Mathemat), SIAM, Philadelphia, PA, USA, 2007. doi: 10.1137/1.9780898717839.  Google Scholar

[24]

Y. Lian and K. Zhang, Boundary Lipschitz regularity and the Hopf lemma for fully nonlinear elliptic equations, arXiv: 1812.11357. Google Scholar

[25]

A. Oberman, The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc., 135 (2007), 1689-1694.  doi: 10.1090/S0002-9939-07-08887-9.  Google Scholar

[26]

A. M. Oberman, Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton?Jacobi equations and free boundary problems, SIAM J. Numer. Anal., 44 (2006), 879-895.  doi: 10.1137/S0036142903435235.  Google Scholar

[27]

A. M. Oberman, Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the {H}essian, Disc. Cont. Dynam. Syst. Ser. B, 10 (2008), 221-238.  doi: 10.3934/dcdsb.2008.10.221.  Google Scholar

[28]

C. R. Prins, R. Beltman, J. H. M. ten Thije Boonkkamp, W. L. IJzerman, and T. W. Tukker, A least-squares method for optimal transport using the Monge-Ampère equation, SIAM J. Sci. Comp., 37 (2015), B937?B961. doi: 10.1137/140986414.  Google Scholar

[29]

L. Qi and J. Sun, A nonsmooth version of Newton's method, Math. program., 58 (1993), 353-367.  doi: 10.1007/BF01581275.  Google Scholar

[30]

J. QianY. T. Zhang and H. K. Zhao, A fast sweeping method for static convex Hamilton?Jacobi equations, J. Sci. Comput., 31 (2007), 237-271.  doi: 10.1007/s10915-006-9124-6.  Google Scholar

[31]

K. Smoczyk and M. T. Wang, Mean curvature flows of Lagrangian submanifolds with convex potentials, J. Differ. Geom., 62 (2002), 243-257.   Google Scholar

[32]

E. L. Thomas, D. M. Anderson, C. S. Henkee and D. Hoffman, Periodic area-minimizing surfaces in block copolymers, Nat., 334 (1988): 598. Google Scholar

[33]

R. P. Thomas and S. T. Yau, Special Lagrangians, stable bundles and mean curvature flow, Commun. Anal. Geom., 10 (2002), 1075-1113.  doi: 10.4310/CAG.2002.v10.n5.a8.  Google Scholar

[34]

J. Urbas, On the second boundary value problem for equations of Monge-Ampere type, J. Rein. Angew. Math., 487 (1997), 115-124.  doi: 10.1515/crll.1997.487.115.  Google Scholar

[35]

H. Zhao, A fast sweeping method for eikonal equations, Math. Comput., 74 (2005), 603-627.  doi: 10.1090/S0025-5718-04-01678-3.  Google Scholar

Figure 1.  Discrete solution to Poisson's equation when viewed as an eigenvalue problem
19]">Figure 2.  Examples of quadtree meshes. White squares are inside the domain, while gray squares intersect the boundary [19]
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
$ 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
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
$ 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
[1]

Youngmok Jeon, Dongwook Shin. Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions. Electronic Research Archive, 2021, 29 (5) : 3361-3382. doi: 10.3934/era.2021043

[2]

Navnit Jha. Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization. Conference Publications, 2013, 2013 (special) : 355-363. doi: 10.3934/proc.2013.2013.355

[3]

Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061

[4]

Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845

[5]

Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247

[6]

Martino Bardi, Paola Mannucci. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2006, 5 (4) : 709-731. doi: 10.3934/cpaa.2006.5.709

[7]

J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177

[8]

Sofia Giuffrè, Giovanna Idone. On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1347-1363. doi: 10.3934/dcds.2011.31.1347

[9]

Mark I. Vishik, Sergey Zelik. Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2059-2093. doi: 10.3934/cpaa.2014.13.2059

[10]

Pablo Blanc. A lower bound for the principal eigenvalue of fully nonlinear elliptic operators. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3613-3623. doi: 10.3934/cpaa.2020158

[11]

Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187

[12]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003

[13]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763

[14]

Angelo Favini, Rabah Labbas, Stéphane Maingot, Maëlis Meisner. Boundary value problem for elliptic differential equations in non-commutative cases. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4967-4990. doi: 10.3934/dcds.2013.33.4967

[15]

Guoliang Zhang, Shaoqin Zheng, Tao Xiong. A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations. Electronic Research Archive, 2021, 29 (1) : 1819-1839. doi: 10.3934/era.2020093

[16]

Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems & Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795

[17]

Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133

[18]

Johnny Henderson, Rodica Luca. Existence of positive solutions for a system of nonlinear second-order integral boundary value problems. Conference Publications, 2015, 2015 (special) : 596-604. doi: 10.3934/proc.2015.0596

[19]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[20]

Junjie Zhang, Shenzhou Zheng, Chunyan Zuo. $ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3305-3318. doi: 10.3934/dcdss.2021080

2020 Impact Factor: 1.916

Article outline

Figures and Tables

[Back to Top]