Convergence of approximation schemes for fully nonlinear second order equations

Barles, Guy; Souganidis, Panagiotis E.

doi:10.1109/cdc.1990.204046

Cited by 212 publications

(392 citation statements)

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“…[7,21]) in 2-D by treating the Monge-Ampère equation and Pucci's equation as a constraint and using a variational criterion to select a particular solution. Very recently, Oberman [30] constructed some wide stencil finite difference scheme which fulfill the convergence criterion established by Barles and Souganidis in [3] for finite difference approximations of fully nonlinear second order PDEs. Consequently, the convergence of the proposed wide stencil finite difference scheme immediately follows from the general convergence framework of [3].…”

Section: Introductionmentioning

confidence: 99%

“…Very recently, Oberman [30] constructed some wide stencil finite difference scheme which fulfill the convergence criterion established by Barles and Souganidis in [3] for finite difference approximations of fully nonlinear second order PDEs. Consequently, the convergence of the proposed wide stencil finite difference scheme immediately follows from the general convergence framework of [3]. Numerical experiments results were reported in [31,30,2,13], however, convergence analysis was not addressed except in [30].…”

Section: Introductionmentioning

confidence: 99%

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Analysis of Galerkin Methods for the Fully Nonlinear Monge-Ampère Equation

Feng

Neilan

2010

J Sci Comput

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Abstract. This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Ampère equation det(D 2 u 0 ) = f (> 0) based on the vanishing moment method which was developed by the authors in [17,15]. In this approach, the Monge-Ampère equation is approximated by the fourth order quasilinear equation −ε∆ 2 u ε + det D 2 u ε = f accompanied by appropriate boundary conditions. This new approach allows one to construct convergent Galerkin numerical methods for the fully nonlinear Monge-Ampère equation (and other fully nonlinear second order partial differential equations), a task which has been impracticable before. In this paper, we first develop some finite element and spectral Galerkin methods for approximating the solution u ε of the regularized fourth order problem. We then derive optimal order error estimates for the proposed numerical methods. In particular, we track explicitly the dependence of the error bounds on the parameter ε, for the error u ε − u ε h . Due to the strong nonlinearity of the underlying equation, the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work here. To overcome the difficulty, we employ a fixed point technique which strongly makes use of the stability property of the linearized problem and its finite element approximations. Finally, using the Aygris finite element method as an example, we present a detailed numerical study of the rates of convergence in terms of powers of ε for the error u 0 − u ε h , and numerically examine what is the "best" mesh size h in relation to ε in order to achieve these rates.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analysis of Galerkin Methods for the Fully Nonlinear Monge-Ampère Equation

Feng

Neilan

2010

J Sci Comput

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“…It is interesting to remark that, even in the case of non-negative velocity, our proof is new. However, the idea of our proof is inspired by the paper by Barles and Georgelin [5] on fronts driven by Mean Curvature where they prove convergence for a scheme in the framework of discontinuous viscosity solutions (we also refer to Barles, Souganidis [7] for convergence in the framework of continuous viscosity solutions). Basically, it is sufficient to consider a test function touching the upper semicontinuous envelope of the numerical solution (obtained as ∆x, ∆t go to zero) which violates the subsolution property and to derive from this some properties of the discrete solution for non-zero ∆x, ∆t.…”

Section: (12)mentioning

confidence: 99%

Convergence of a Generalized Fast-Marching Method for an Eikonal Equation with a Velocity-Changing Sign

Carlini¹,

Falcone

Forcadel³

et al. 2008

SIAM J. Numer. Anal.

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We present a new Fast Marching algorithm for an eikonal equation with a velocity changing sign. This first order equation models a front propagation in the normal direction. The algorithm is an extension of the Fast Marching Method in two respects. The first is that the new scheme can deal with a time-dependent velocity and the second is that there is no restriction on its change in sign. We analyze the properties of the algorithm and we prove its convergence in the class of discontinuous viscosity solutions. Finally, we present some numerical simulations of fronts propagating in R 2 .

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“…The simplest approximations are finite difference schemes based on a Cartesian grid. In this class, monotone schemes are provably convergent [BS91], but only first order accurate [Obe06].…”

Section: Introductionmentioning

confidence: 99%

Filtered schemes for Hamilton–Jacobi equations: A simple construction of convergent accurate difference schemes

Oberman¹,

Salvador²

2015

Journal of Computational Physics

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Abstract. We build a simple and general class of finite difference schemes for first order Hamilton-Jacobi (HJ) Partial Differential Equations. These filtered schemes are convergent to the unique viscosity solution of the equation. The schemes are accurate: we implement second, third and fourth order accurate schemes in one dimension and second order accurate schemes in two dimensions, indicating how to build higher order ones. They are also explicit, which means they can be solved using the fast sweeping method. The accuracy of the method is validated with computational results for the eikonal equation and other HJ equations in one and two dimensions, using filtered schemes made from standard centered differences, higher order upwinding and ENO interpolation.

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Convergence of approximation schemes for fully nonlinear second order equations

Cited by 212 publications

References 10 publications

Analysis of Galerkin Methods for the Fully Nonlinear Monge-Ampère Equation

Analysis of Galerkin Methods for the Fully Nonlinear Monge-Ampère Equation

Convergence of a Generalized Fast-Marching Method for an Eikonal Equation with a Velocity-Changing Sign

Filtered schemes for Hamilton–Jacobi equations: A simple construction of convergent accurate difference schemes

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