Approximation of viscosity solutions of elliptic partial differential equations on minimal grids

Kocan, Maciej

doi:10.1007/s002110050160

Cited by 39 publications

(45 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Bounds on the size of M in terms of the ellipticity of Θ and N have been established by Kocan [15], where the remarks of this section are refined, extended and generalizedin particular, general, possibly nonlinear, discretizations are discussed. We also note that the existence of schemes in the nondegenerate case is not established constructively, so even in the constant coefficient nondegenerate case our methods have implications.…”

Section: On Monotone Schemesmentioning

confidence: 99%

“…For more general equations, consistent monotone schemes were derived for fully nonlinear first order equations in [8] and Souganidis [22], Barles and Souganidis [1] give difference schemes for "diagonally dominant" cases of (PE) and Kuo and Trudinger's work [16,17] is relevant for nondegenerate equations. Kocan [15] studies the stencil size required for monotone schemes as a function of the ellipticity. Section 2 is devoted to an interesting aside.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Convergent difference schemes for nonlinear parabolic equations and mean curvature motion

Crandall

Lions

1996

Numerische Mathematik

View full text Add to dashboard Cite

Explicit finite difference schemes are given for a collection of parabolic equations which may have all of the following complex features: degeneracy, quasilinearity, full nonlinearity, and singularities. In particular, the equation of "motion by mean curvature" is included. The schemes are monotone and consistent, so that convergence is guaranteed by the general theory of approximation of viscosity solutions of fully nonlinear problems. In addition, an intriguing new type of nonlocal problem is analyzed which is related to the schemes, and another very different sort of approximation is presented as well.Mathematics Subject Classification (1991): 35A40, 35K55, 65M06, 65M12

show abstract

Section: On Monotone Schemesmentioning

confidence: 99%

mentioning

confidence: 99%

Convergent difference schemes for nonlinear parabolic equations and mean curvature motion

Crandall

Lions

1996

Numerische Mathematik

View full text Add to dashboard Cite

show abstract

“…Finding well localized numerical schemes, involving small stencils, is a natural objective [Koc95]. Theorem 1.9 (Minimality).…”

mentioning

confidence: 99%

Monotone and consistent discretization of the Monge-Ampère operator

Benamou¹,

Collino²,

Mirebeau³

2016

Math. Comp.

View full text Add to dashboard Cite

We introduce a novel discretization of the Monge-Ampere operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices, and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency.

show abstract

“…where Fh is a given function on F = E n x R x M. YN [8,12,15] where A, go, So are constants depending on Ao, Pi, (to, respectively, as well as the dimension n. The constant N (the size of the stencil) will depend upon n and /lo/Ao, while ho depends in addition upon H\/\o (see [8,12,13,15]). Now we set up the Dirichlet problem for the nonlinear difference operator of the form (1.7).…”

Section: Introductionmentioning

confidence: 99%

“…We will assume that Fh [u] is independent of u(x + y) for \\y\\x = sup, \yi\ > Nh for some fixed N € N. We may then replace E' with the stencil [8] …”

Section: Introductionmentioning

confidence: 99%

Higher-order estimates for fully nonlinear difference equations. I

Holtby¹

2000

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

The purpose of this work is to establish a priori C 2 ' a estimates for mesh function solutions of nonlinear positive difference equations in fully nonlinear form on a uniform mesh, where the fully nonlinear finite-difference operator Th, is concave in the second-order variables. The estimate is an analogue of the corresponding estimate for solutions of concave fully nonlinear elliptic partial differential equations. We deal here with the special case that the operator does not depend explicitly upon the independent variables. We do this by discretizing the approach of Evans for fully nonlinear elliptic partial differential equations using the discrete linear theory of Kuo and Trudinger. The result in this special case forms the basis for a more general result in part II. We also derive the discrete interpolation inequalities needed to obtain estimates for the interior C 2 ' a semi-norm in terms of the C° norm.

show abstract

Approximation of viscosity solutions of elliptic partial differential equations on minimal grids

Cited by 39 publications

References 10 publications

Convergent difference schemes for nonlinear parabolic equations and mean curvature motion

Convergent difference schemes for nonlinear parabolic equations and mean curvature motion

Monotone and consistent discretization of the Monge-Ampère operator

Higher-order estimates for fully nonlinear difference equations. I

Contact Info

Product

Resources

About