A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs

Abstract

We obtain an algebraic rate of convergence for monotone and consistent finite difference approximations to Lipschitz-continuous viscosity solutions of uniformly elliptic partial differential equations. c 2007 Wiley Periodicals, Inc. IntroductionWe obtain an algebraic rate of convergence for monotone and consistent finite difference approximations to Lipschitz-continuous viscosity solutions of fully nonlinear, uniformly elliptic partial differential equations of the formwhere U is an open subset of R n with regular boundary, (0.3) F is uniformly elliptic with ellipticity constants λ and such that λ ≤ , (0.4) f ∈ C 0,1 (Ū ), and (0.5)In a forthcoming paper, we obtain algebraic error estimates for monotone and consistent approximations to equations that also depend on the gradient. The arguments are more complicated and require some additional ideas.A key step in our analysis is a regularity result for Lipschitz-continuous viscosity solutions of (0.1). Roughly speaking, it asserts that, outside sets of arbitrarily small measure, Lipschitz-continuous solutions of (0.1) have pointwise secondorder expansions with an error that is controlled by the size of the exceptional set

A deterministic‐control‐based approach to fully nonlinear parabolic and elliptic equations

“…Recently Caffarelli and Souganidis have shown how refined consistency results can sometimes be used to prove convergence with a rate, though the viscosity solution may not be differentiable [12,13]. Their method requires an improved estimate for the consistency error when the test function is a quadratic polynomial in x.…”

“…We omit the proof, since this assertion will not be used in what follows. (It remains an open question whether the method of [12,13] can be used to prove convergence with a rate in our setting. )…”

Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature

“…The seminal papers [9,11] have drawn an increasing attention to the geometry of nonlocal minimal surfaces, i.e., boundaries of sets R n which are stationary for the s-perimeter functional corresponding to some value of s 2 .0; 1=2/ (see for instance [4,5,12,[18][19][20][21]24] for all X 2 C 1 c .AI R n /:…”

A review of numerical methods for nonlinear partial differential equations

“…In this case, one has L 1 -stability against perturbations which in turn implies the bounded variation (BV)-stability [90,100]. The theory of viscosity solutions for nonlinear Hamilton-Jacobi and fully nonlinear elliptic equations was developed in the L ∞ -framework [57,30], as are the corresponding convergence theories [188,158,31]. To overcome the lack of smoothness, one can use negative Sobolev norms | ·| W −m (L p ) [151,196,85].…”