Optimal Convergence Rates for Elliptic Homogenization Problems in Nondivergence-Form: Analysis and Numerical Illustrations

Sprekeler, Timo; Tran, Hung V.

doi:10.1137/20m137121x

Cited by 7 publications

(2 citation statements)

References 29 publications

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“…Thus, for ≥ 5, we obtain the optimal rate of convergence for the homogenization of the Dirichlet problem, which is generically of scale −1 . This is consistent with the generically optimal rate −1 for the periodic setting (see the classical books [8,36] for the derivation, and [32,45,30] for discussions on the optimality of the rates.) It is not clear to us what the optimal rates are when 2 ≤ ≤ 4, which deserve further analysis.…”

Section: Theorem 7 Assume (A1) (A2) Consider the Dirichlet Problemsupporting

confidence: 81%

“…In what follows we will apply the classical method of two-scale expansions to quantify the rate of the homogenization for the Dirichlet problem (19). The proof is similar to that in the periodic setting (see [32,45,30] for example). The only difference is that we use the approximate corrector here instead of the actual corrector in the periodic setting.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Quantitative homogenization in a balanced random environment

Guo

Peterson

Tran

2022

Electron. J. Probab.

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We consider discrete non-divergence form difference operators in a random environment and the corresponding process -the random walk in a balanced random environment in Z d with a finite range of dependence. We first quantify the ergodicity of the environment from the point of view of the particle. As a consequence, we quantify the quenched central limit theorem of the random walk with an algebraic rate. Furthermore, we prove an algebraic rate of convergence for the homogenization of the Dirichlet problems for both elliptic and parabolic non-divergence form difference operators.

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Section: Theorem 7 Assume (A1) (A2) Consider the Dirichlet Problemsupporting

confidence: 81%

Section: Proof Of Theoremmentioning

confidence: 99%

Quantitative homogenization in a balanced random environment

Guo

Peterson

Tran

2022

Electron. J. Probab.

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Characterizations of diffusion matrices in homogenization of elliptic equations in nondivergence-form

Guo,

Sprekeler,

Tran

2024

Calc. Var.

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Discontinuous Galerkin and C⁰-IP finite element approximation of periodic Hamilton–Jacobi–Bellman–Isaacs problems with application to numerical homogenization

Kawecki¹,

Sprekeler

2022

ESAIM: M2AN

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In the first part of the paper, we study the discontinuous Galerkin (DG) and $C^0$ interior penalty ($C^0$-IP) finite element approximation of the periodic strong solution to the fully nonlinear second-order Hamilton--Jacobi--Bellman--Isaacs (HJBI) equation with coefficients satisfying the Cordes condition. We prove well-posedness and perform abstract a posteriori and a priori analyses which apply to a wide family of numerical schemes. These periodic problems arise as the corrector problems in the homogenization of HJBI equations. The second part of the paper focuses on the numerical approximation to the effective Hamiltonian of ergodic HJBI operators via DG/$C^0$-IP finite element approximations to approximate corrector problems. Finally, we provide numerical experiments demonstrating the performance of the numerical schemes.

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Optimal Convergence Rates for Elliptic Homogenization Problems in Nondivergence-Form: Analysis and Numerical Illustrations

Cited by 7 publications

References 29 publications

Quantitative homogenization in a balanced random environment

Quantitative homogenization in a balanced random environment

Characterizations of diffusion matrices in homogenization of elliptic equations in nondivergence-form

Discontinuous Galerkin and C⁰-IP finite element approximation of periodic Hamilton–Jacobi–Bellman–Isaacs problems with application to numerical homogenization

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Optimal Convergence Rates for Elliptic Homogenization Problems in Nondivergence-Form: Analysis and Numerical Illustrations

Cited by 7 publications

References 29 publications

Quantitative homogenization in a balanced random environment

Quantitative homogenization in a balanced random environment

Characterizations of diffusion matrices in homogenization of elliptic equations in nondivergence-form

Discontinuous Galerkin and C0-IP finite element approximation of periodic Hamilton–Jacobi–Bellman–Isaacs problems with application to numerical homogenization

Contact Info

Product

Resources

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Discontinuous Galerkin and C⁰-IP finite element approximation of periodic Hamilton–Jacobi–Bellman–Isaacs problems with application to numerical homogenization