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Approximate Homogenization of Fully Nonlinear Elliptic PDEs: Estimates and Numerical Results for Pucci Type Equations

Authors:

Adam M. ObermanAuthors Info & Claims

Journal of Scientific Computing, Volume 77, Issue 2

Pages 936 - 949

https://doi.org/10.1007/s10915-018-0730-x

Published: 01 November 2018 Publication History

Abstract

We are interested in the shape of the homogenized operator $$\overline{F}(Q)$$F (Q) for PDEs which have the structure of a nonlinear Pucci operator. A typical operator is $$H^{a_1,a_2}(Q,x) = a_1(x) \lambda _{\min }(Q) + a_2(x)\lambda _{\max }(Q)$$Ha1,a2(Q,x)=a1(x)ýmin(Q)+a2(x)ýmax(Q). Linearization of the operator leads to a non-divergence form homogenization problem, which can be solved by averaging against the invariant measure. We estimate the error obtained by linearization based on semi-concavity estimates on the nonlinear operator. These estimates show that away from high curvature regions, the linearization can be accurate. Numerical results show that for many values of Q, the linearization is highly accurate, and that even near corners, the error can be small (a few percent) even for relatively wide ranges of the coefficients.

References

[1]

Armstrong, S.N., Smart, C.K.: Quantitative stochastic homogenization of elliptic equations in nondivergence form. Arch. Ration. Mech. Anal. 214(3), 867---911 (2014)

[2]

Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. American Mathematical Society, Providence (2011)

[3]

Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society, Providence (1995)

[4]

Caffarelli, L.A., Glowinski, R.: Numerical solution of the Dirichlet problem for a Pucci equation in dimension two. Application to homogenization. J. Numer. Math. 16(3), 185---216 (2008)

[5]

Engquist, B., Souganidis, P.E.: Asymptotic and numerical homogenization. Acta Numer. 17, 147---190 (2008)

[6]

Evans, L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35(3), 333---363 (1982)

[7]

Evans, L.C.: The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. R. Soc. Edinb. Sect. A Math. 111(3---4), 359---375 (1989)

[8]

Froese, B.D., Oberman, A.M.: Numerical averaging of non-divergence structure elliptic operators. Commun. Math. Sci. 7(4), 785---804 (2009)

[9]

Froese, B.D., Oberman, A.M.: Convergent filtered schemes for the Monge---Ampère partial differential equation. SIAM J. Numer. Anal. 51(1), 423---444 (2013)

[10]

Finlay, C., Oberman, A.M.: Approximate homogenization of convex nonlinear elliptic PDEs (2017). arXiv preprint arXiv:1710.10309

[11]

Gomes, D.A., Oberman, A.M.: Computing the effective Hamiltonian using a variational approach. SIAM J. Control Optim. 43(3), 792---812 (2004)

Digital Library

[12]

Jensen, R.: The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Ration. Mech. Anal. 101(1), 1---27 (1988)

[13]

Krylov, N.V.: Boundedly nonhomogeneous elliptic and parabolic equations in a domain. Izvestiya: Mathematics 22(1), 67---97 (1984)

[14]

Luo, S., Yifeng, Y., Zhao, H.: A new approximation for effective hamiltonians for homogenization of a class of Hamilton---Jacobi equations. Multiscale Model. Simul. 9(2), 711---734 (2011)

[15]

Oberman, A.M.: Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton---Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 44(2), 879---895 (2006). (electronic)

Digital Library

[16]

Oberman, A.M., Takei, R., Vladimirsky, A.: Homogenization of metric Hamilton---Jacobi equations. Multiscale Model. Simul. 8(1), 269---295 (2009)

Cited By

Zhu DFu YZhao XWang XYi H(2022)Facial Emotion Recognition Using a Novel Fusion of Convolutional Neural Network and Local Binary Pattern in Crime InvestigationComputational Intelligence and Neuroscience10.1155/2022/22494172022Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1155/2022/2249417

Approximate Homogenization of Fully Nonlinear Elliptic PDEs: Estimates and Numerical Results for Pucci Type Equations
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations

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Published In

cover image Journal of Scientific Computing

Journal of Scientific Computing Volume 77, Issue 2

November 2018

614 pages

ISSN:0885-7474

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Copyright © Copyright © 2018 Springer Science+Business Media, LLC, part of Springer Nature.

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Plenum Press

United States

Publication History

Published: 01 November 2018

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Cited By

Zhu DFu YZhao XWang XYi H(2022)Facial Emotion Recognition Using a Novel Fusion of Convolutional Neural Network and Local Binary Pattern in Crime InvestigationComputational Intelligence and Neuroscience10.1155/2022/22494172022Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1155/2022/2249417

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