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

Abstract

We are interested in the shape of the homogenized operator 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)λ min (Q)+a 2 (x)λmax(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.

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

“…The numerical homogenization of HJB equations via a mixed finite element approximation of the approximate correctors has been proposed and analyzed in Gallistl, Sprekeler, Süli [25]. A finite difference approach for numerical effective Hamiltonians to HJB operators can be found in Camilli, Marchi [13], and some exact formulas and numerical simulations for effective Hamiltonians to certain types of HJB operators are available in Finlay, Oberman [21,22].…”

Discontinuous Galerkin and $C^0$-IP finite element approximation of periodic Hamilton--Jacobi--Bellman--Isaacs problems with application to numerical homogenization

“…The numerical homogenization of HJB equations via a mixed finite element approximation of the approximate correctors has been proposed and analyzed in Gallistl, Sprekeler, Süli [24]. A finite difference approach for numerical effective Hamiltonians to HJB operators can be found in Camilli, Marchi [12], and some exact formulas and numerical simulations for effective Hamiltonians to certain types of HJB operators are available in Finlay, Oberman [20,21].…”

Mixed finite element approximation of periodic Hamilton--Jacobi--Bellman problems with application to numerical homogenization

“…For the case of second-order HJB equations, a finite difference scheme for the whole-space problem has been proposed in Camilli, Marchi [10]. In Finlay, Oberman [18,19], the effective Hamiltonian is computed exactly for HJB operators of certain types and numerical simulations have been conducted. It seems that finite element schemes for the numerical homogenization of the problem (1.3) have not yet been constructed.…”