On the discrete maximum principle for parabolic difference operators

Kuo, Hung-Ju; Trudinger, Neil S.

doi:10.1051/m2an/1993270607191

Cited by 17 publications

(15 citation statements)

References 10 publications

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“…As an application of the technique used in a proof of the AleksandrovBakelman-Pucci (ABP for short) maximum principle, Cabré pointed out in [8] (and the original paper [7] in Catalan) that the ABP method gives a simple proof of the classical isoperimetric inequality (1.1). We refer the reader, if interested in the ABP maximum principle, to [12, [16,21] for linear equations, [17] for nonlinear operators, [18,20] for parabolic cases and [19,20] for general meshes.…”

Section: ) Holds If and Only If E[ω] Is A Cube Ie E[ω]mentioning

confidence: 99%

A Discrete Isoperimetric Inequality on Lattices

Hamamuki

2014

Discrete Comput Geom

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Abstract. We establish an isoperimetric inequality with constraint by ndimensional lattices. We prove that, among all domains which consist of rectangular parallelepipeds with the common side-lengths, a cube is the best shape to minimize the ratio involving its perimeter and volume as long as the cube is realizable by the lattice. For its proof a solvability of finite difference PoissonNeumann problems is verified. Our approach to the isoperimetric inequality is based on the technique used in a proof of the Aleksandrov-Bakelman-Pucci maximum principle, which was originally proposed by Cabré in 2000 to prove the classical isoperimetric inequality.

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Section: ) Holds If and Only If E[ω] Is A Cube Ie E[ω]mentioning

confidence: 99%

A Discrete Isoperimetric Inequality on Lattices

Hamamuki

2014

Discrete Comput Geom

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“…Gonvergence rates for second order elliptic and parabolic equations, without any regularity assumptions, are obtained in example Krylov [8,11], Kuo and Trudinger [12], Baríes and Jacobsen [1], and Gaifarelli and Souganidis [5] and the references therein. The methods used come from regularity theory for nonlinear elliptic PDEs and are substantially more technical than the methods herein.…”

Section: Related Resultsmentioning

confidence: 99%

Convergence Rate for Difference Schemes for Polyhedral Nonlinear Parabolic Equations

Oberman¹

2010

JCM

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We build finite difference schemes for a class of fully nonlinear parabolic equations. The schemes are polyhedral and grid aligned. While this is a restrictive class of schemes, a wide class of equations are well approximated by equations from this class. For regular (C^'") solutions of uniformly parabolic equations, we also establish of convergence rate of O{a). A case study along with supporting numerical results is included.Mathematics subject classification: 65M06, 65M12. Contents 'The remainder of this section recalls the setting for our nonlinear parabolic equations and the necessary regularity results. Section 2 is a case study with a simple example equation. Error estimates are obtained directly in this simpler setting, and supporting numerical results are presented.The first part of section 3 recalls general results on nonlinear elliptic schemes. The second part presents new material on error estimates in terms of the residual for perturbed equations, the methods of lines, and finally for fully discrete difference schemes. |The main results are in the section 4. Here the class of schemes is established. The schemes are shown to be elliptic, and consistent, which is enough to prove convergence. Then the error estimates of the previous section are used to obtain a convergence rate. Nonlinear parabolic equationsOur results concern the fully nonlinear parabolic Partial Differential Equation (PDE)where fi is a domain in R", along with initial and boundary conditionsuix, t) = hix, t), for (x, t) on dU x (0, T).The fully nonlinear elliptic partial differential operator F[u] is given by [] M(1.1)Here Du and D^u denote the gradient and Hessian of u, respectively. The function FiX, p, r, x) is defined on S" x R" x R x fî, and S" is the space of symmetric n x n matrices. The natural setting for equations of this type is viscosity solutions [7]. Definition 1.1. The differential operator (1.1) is nonhnear or degenerate elliptic if FiN,p,r,x) < F(M,p,s,x) whenever r < s and M < N. (1.2) Here M < N means that M -N is a nonnegative definite symmetric matrix. The corresponding parabolic operator (PDE) is called nonlinear or degenerate parabolic.

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“…Also observe that on the discrete parabolic boundary of Q (h) ∩ Q ε 0 we have |v h | ≤ N ε 0 and |u (ε) K (· + t 0 , · + x 0 )| ≤ N ε 0 owing to Lemmas 2.2 and 6.1. It follows by the discrete maximum principle of Kuo and Trudinger [14] applied to v h − u (ε)…”

Section: Proof Of Theorem 26mentioning

confidence: 99%

To the theory of viscosity solutions for uniformly parabolic Isaacs equations

Крылов

2015

Methods and Applications of Analysis

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We show how a theorem about the solvability in W 1,2 ∞ of special parabolic Isaacs equations can be used to obtain the existence and uniqueness of viscosity solutions of general uniformly nondegenerate parabolic Isaacs equations. We apply it also to establish the C 1+χ regularity of viscosity solutions and show that finite-difference approximations have an algebraic rate of convergence. The main coefficients of the Isaacs equations are supposed to be in C γ with respect to the spatial variables with γ slightly less than 1/2. 2010 Mathematics Subject Classification. 35K55, 35B65, 65N15.

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On the discrete maximum principle for parabolic difference operators

Cited by 17 publications

References 10 publications

A Discrete Isoperimetric Inequality on Lattices

A Discrete Isoperimetric Inequality on Lattices

Convergence Rate for Difference Schemes for Polyhedral Nonlinear Parabolic Equations

To the theory of viscosity solutions for uniformly parabolic Isaacs equations

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