Convergence Rate of a Stable, Monotone and Consistent Scheme for the Monge-Ampère Equation

Awanou, Gerard

doi:10.3390/sym8040018

Cited by 5 publications

(3 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Awanou [2] proved a linear rate of convergence for classical solutions for the widestencil method, when applied to a perturbed Monge-Ampère equation with an extra lower order term δu; the parameter δ > 0 is independent of the mesh and appears in reciprocal form in the rate. In contrast, our analysis hinges on the discrete comparison principle and two discrete barrier functions, which are instrumental in proving convergence to the viscosity solution of (1.1).…”

Section: Introductionmentioning

confidence: 99%

Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution

2018

View full text Add to dashboard Cite

We propose a two-scale finite element method for the Monge-Ampère equation with Dirichlet boundary condition in dimension d ≥ 2 and prove that it converges to the viscosity solution uniformly. The method is inspired by a finite difference method of Froese and Oberman, but is defined on unstructured grids and relies on two separate scales: the first one is the mesh size h and the second one is a larger scale that controls appropriate directions and substitutes the need of a wide stencil. The main tools for the analysis are a discrete comparison principle and discrete barrier functions that control the behavior of the discrete solution, which is continuous piecewise linear, both close to the boundary and in the interior of the domain.

show abstract

Section: Introductionmentioning

confidence: 99%

Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution

2018

View full text Add to dashboard Cite

show abstract

“…Error estimates in H 1 (Ω) are established in [3,4] for solutions with H 3 (Ω) regularity or more. Awanou [1] also proved a linear rate of convergence for classical solutions for the wide-stencil method, when applied to a perturbed Monge-Ampère equation with an extra lower order term δu; the parameter δ > 0 is independent of the mesh and appears in reciprocal form in the rate.…”

Section: Introductionmentioning

confidence: 94%

Two-scale method for the Monge–Ampère equation: pointwise error estimates

Nochetto

Ntogkas

Zhang

2018

IMA Journal of Numerical Analysis

View full text Add to dashboard Cite

In this paper we continue the analysis of the two-scale method for the Monge-Ampère equation for dimension d ≥ 2 introduced in [12]. We prove continuous dependence of discrete solutions on data that in turn hinges on a discrete version of the Alexandroff estimate. They are both instrumental to prove pointwise error estimates for classical solutions with Hölder and Sobolev regularity. We also derive convergence rates for viscosity solutions with bounded Hessians which may be piecewise smooth or degenerate.

show abstract

“…Since the operator M + s is pointwise consistent and convergence to viscosity solution is point by point, we do not discuss linear interpolation. It can be shown that for smooth solutions [2], it leads to a convergence rate O(h + dθ) where dθ is called directional resolution and dθ → 0 as h → 0.…”

Section: Monotone Schemes Let Us Denote Bymentioning

confidence: 99%

Isogeometric Method for the Elliptic Monge-Ampère Equation

Awanou

2014

Springer Proceedings in Mathematics &Amp; Statistics

Self Cite

View full text Add to dashboard Cite

We consider mesh functions which are discrete convex in the sense that their central second order directional derivatives are positive. Analogous to the case of a uniformly bounded sequence of convex functions, we prove that the uniform limit on compact subsets of discrete convex mesh functions which are uniformly bounded is a continuous convex function. Furthermore, if the discrete convex mesh functions interpolate boundary data of a continuous convex function and their Monge-Ampère masses are uniformly bounded, the limit function satisfies the boundary condition strongly. The domain of the solution needs not be uniformly convex. The result is applied to the convergence of some numerical methods for the Monge-Ampère equation.

show abstract

Convergence Rate of a Stable, Monotone and Consistent Scheme for the Monge-Ampère Equation

Cited by 5 publications

References 10 publications

Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution

Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution

Two-scale method for the Monge–Ampère equation: pointwise error estimates

Isogeometric Method for the Elliptic Monge-Ampère Equation

Contact Info

Product

Resources

About