7 research outputs found
Finite Element Approximation of Elliptic Homogenization Problems in Nondivergence-Form
We use uniform estimates to obtain corrector results for periodic
homogenization problems of the form
subject to a homogeneous Dirichlet boundary condition. We propose and
rigorously analyze a numerical scheme based on finite element approximations
for such nondivergence-form homogenization problems. The second part of the
paper focuses on the approximation of the corrector and numerical
homogenization for the case of nonuniformly oscillating coefficients. Numerical
experiments demonstrate the performance of the scheme.Comment: 39 page
Homogenization of nondivergence-form elliptic equations with discontinuous coefficients and finite element approximation of the homogenized problem
We study the homogenization of the equation
posed in a bounded
convex domain subject to a Dirichlet boundary
condition and the numerical approximation of the corresponding homogenized
problem, where the measurable, uniformly elliptic, periodic and symmetric
diffusion matrix is merely assumed to be essentially bounded and (if )
to satisfy the Cordes condition. In the first part, we show existence and
uniqueness of an invariant measure by reducing to a Lax--Milgram-type problem,
we obtain -bounds for periodic problems in double-divergence-form, we
prove homogenization under minimal regularity assumptions, and we generalize
known corrector bounds and results on optimal convergence rates from the
classical case of H\"{o}lder continuous coefficients to the present case. In
the second part, we suggest and rigorously analyze an approximation scheme for
the effective coefficient matrix and the solution to the homogenized problem
based on a finite element method for the approximation of the invariant
measure, and we demonstrate the performance of the scheme through numerical
experiments.Comment: 19 page
Mixed finite element approximation of periodic Hamilton--Jacobi--Bellman problems with application to numerical homogenization
In the first part of the paper, we propose and rigorously analyze a mixed
finite element method for the approximation of the periodic strong solution to
the fully nonlinear second-order Hamilton--Jacobi--Bellman equation with
coefficients satisfying the Cordes condition. These problems arise as the
corrector problems in the homogenization of Hamilton--Jacobi--Bellman
equations. The second part of the paper focuses on the numerical homogenization
of such equations, more precisely on the numerical approximation of the
effective Hamiltonian. Numerical experiments demonstrate the approximation
scheme for the effective Hamiltonian and the numerical solution of the
homogenized problem.Comment: 23 page
Computational multiscale methods for nondivergence-form elliptic partial differential equations
This paper proposes novel computational multiscale methods for linear
second-order elliptic partial differential equations in nondivergence-form with
heterogeneous coefficients satisfying a Cordes condition. The construction
follows the methodology of localized orthogonal decomposition (LOD) and
provides operator-adapted coarse spaces by solving localized cell problems on a
fine scale in the spirit of numerical homogenization. The degrees of freedom of
the coarse spaces are related to nonconforming and mixed finite element methods
for homogeneous problems. The rigorous error analysis of one exemplary approach
shows that the favorable properties of the LOD methodology known from
divergence-form PDEs, i.e., its applicability and accuracy beyond scale
separation and periodicity, remain valid for problems in nondivergence-form.Comment: 21 page
26th Annual Computational Neuroscience Meeting (CNS*2017): Part 3 - Meeting Abstracts - Antwerp, Belgium. 15â20 July 2017
This work was produced as part of the activities of FAPESP Research,\ud
Disseminations and Innovation Center for Neuromathematics (grant\ud
2013/07699-0, S. Paulo Research Foundation). NLK is supported by a\ud
FAPESP postdoctoral fellowship (grant 2016/03855-5). ACR is partially\ud
supported by a CNPq fellowship (grant 306251/2014-0)
Discontinuous Galerkin and
In the first part of the paper, we study the discontinuous Galerkin (DG) and C0 interior penalty (C0-IP) finite element approximation of the periodic strong solution to the fully nonlinear second-order HamiltonâJacobiâBellmanâIsaacs (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/C0-IP finite element approximations to approximate corrector problems. Finally, we provide numerical experiments demonstrating the performance of the numerical schemes