Recent Developments in Numerical Methods for Fully Nonlinear Second Order Partial Differential Equations

Feng, Xiaobing; Glowinski, Roland; Neilan, Michael

doi:10.1137/110825960

Cited by 129 publications

(131 citation statements)

References 110 publications

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“…In analysis, nonlinear PDEs are classified as semilinear, quasi-linear, and fully nonlinear three categories based on the degree of the nonlinearity [76]. A semilinear PDE is a differential equation that is nonlinear in the unknown function but linear in all of its partial derivatives.…”

Section: Definition 2 (Canonical Function and Canonical Transformatiomentioning

confidence: 99%

RETRACTED: Double well potential function and its optimization in the n-dimensional real space: part II

Xia

Sheu

Fang

et al. 2016

Mathematics and Mechanics of Solids

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The following article has been included in a multiple retraction: Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part II; Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. http://mms.sagepub.com/content/early/2015/02/09/1081286514566723.abstract In 2015 SAGE were made aware of concerns regarding the Special Issue of Mathematics & Mechanics of Solids on Advances in Canonical Duality Theory, guest-edited by Professor David Gao. At the request of the Guest Editor, the Special Issue has been retracted, due to conflict of interest regarding Professor Gao’s role as Guest Editor and co-author on a number of submitted papers. In addition the peer review process was less rigorous than the journal requires. The Guest Editor takes full responsibility for the retraction. The following articles that were due to appear in the Special Issue have therefore been retracted: Canonical Duality-Triality: Bridge between Nonconvex Analysis/Mechanics and Global Optimization in complex systems; David Y Gao, Ning Ruan, and Vittorio Latorre. http://mms.sagepub.com/content/early/2015/02/24/1081286514566533.abstract Canonical Dual Approach for Contact Mechanics Problems with Friction; Vittorio Latorre, Simone Sagratella, David Y Gao. http://mms.sagepub.com/content/early/2015/01/20/1081286514566534.abstract Canonical Duality Theory for Solving Non-Monotone Variational Inequality Problems; Guoshan Liu, David Y Gao, Shouyang Wang. http://mms.sagepub.com/content/early/2015/02/04/1081286514566535.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part I; Shu-Cherng Fang, David Y Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wen-Xun Xing. http://mms.sagepub.com/content/early/2015/02/24/1081286514566704.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part II; Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. http://mms.sagepub.com/content/early/2015/02/09/1081286514566723.abstract Analytic Solutions to 3-D Finite Deformation Problems Governed by St Venant-Kirchhoff Material; David Y Gao and E. Hajilarov. http://mms.sagepub.com/content/early/2015/07/06/1081286515591084.abstract Triality Theory and Complete Post-buckling Solutions of Large Deformed Beam by Canonical Dual Finite Element Method; Kun Cai, David Y Gao, Qinghua Qin. http://mms.sagepub.com/content/early/2015/06/28/1081286515591085.abstract Global Solutions to Spherically Constrained Quadratic Minimization via Canonical Duality Theory; Yi Chen, David Y Gao. http://mms.sagepub.com/content/early/2015/04/08/1081286515577122.abstract Unified Canonical Duality Methodology for Global Optimization; Vittorio Latorre, David Y Gao and N. Ruan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591305.abstract A Framework of Canonical Dual Algorithms for Global Optimization; Xiaojun Zhou, David Y Gao, Chunhua Yang. http://mms.sagepub.com/content/early/2015/07/22/1081286515592190.abstract Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Globa...

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Section: Definition 2 (Canonical Function and Canonical Transformatiomentioning

confidence: 99%

RETRACTED: Double well potential function and its optimization in the n-dimensional real space: part II

Xia

Sheu

Fang

et al. 2016

Mathematics and Mechanics of Solids

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“…The method is known as the vanishing moment method. We refer the reader to [12,8] for a detailed exposition.…”

Section: 3mentioning

confidence: 99%

“…Due to their strong nonlinearity, this class of PDEs are most difficult to analyze analytically and to approximate numerically. In the mean time, fully nonlinear PDEs arise in many applications such as antenna design, astrophysics, differential geometry, fluid mechanics, image processing, meteorology, mesh generation, optimal control, optimal mass transport, etc [8], which calls for the development of efficient and reliable numerical methods for solving their underlying fully nonlinear PDE problems. This is the second paper in a series [9] which is devoted to developing finite difference (FD) and discontinuous Galerkin (DG) methods for approximating viscosity solutions of the following general one-dimensional fully nonlinear second order elliptic and parabolic equations:…”

mentioning

confidence: 99%

“…In fact, it will be seen later that even in the one dimensional case, the construction and analysis of the proposed mIP-DG methods are already quite complicated. It is well known [8] that the primary challenges for approximating viscosity solutions of fully nonlinear PDEs are caused by the very notion of viscosity solutions themselves (see section 2 for the definition). Unlike the notion of weak solutions for linear and quasilinear PDEs, the notion of viscosity solutions by design is nonvariational, and, in general, viscosity solutions do not satisfy the underlying PDEs in a tangible sense.…”

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confidence: 99%

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Mixed Interior Penalty Discontinuous Galerkin Methods for One-Dimensional Fully Nonlinear Second Order Elliptic and Parabolic Equations

Lewis¹

2014

JCM

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Abstract. This paper is concerned with developing accurate and efficient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations (PDEs). In the paper we present a general framework for constructing high order interior penalty discontinuous Galerkin (IP-DG) methods for approximating viscosity solutions of these fully nonlinear PDEs. In order to capture discontinuities of the second order derivative uxx of the solution u, three independent functions p 1 , p 2 and p 3 are introduced to represent numerical derivatives using various one-sided limits. The proposed DG framework, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a nonlinear problem into a mostly linear system of equations where the nonlinearity has been modified to include multiple values of the second order derivative uxx. The proposed framework extends a companion finite difference framework developed by the authors in [9] and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes. In addition to the nonstandard mixed formulation setting, another main idea is to replace the fully nonlinear differential operator by a numerical operator, which is consistent with the differential operator and satisfies certain monotonicity (called g-monotonicity) properties. To ensure such a g-monotonicity, the crux of the construction is to introduce the numerical moment, which plays a critical role in the proposed DG framework. The g-monotonicity gives the DG methods the ability to select the mathematically "correct" solution (i.e., the viscosity solution) among all possible solutions. Moreover, the g-monotonicity allows for the possible development of more efficient nonlinear solvers as the special nonlinearity of the algebraic systems can be explored to decouple the equations. This paper also presents and analyzes numerical results for several numerical test problems which are used to guage the accuracy and efficiency of the proposed DG methods.Key words. Fully nonlinear PDEs, viscosity solutions, discontinuous Galerkin methods, AMS subject classifications. 65N30, 65M60, 35J60, 35K55, 1. Introduction. Fully nonlinear partial differential equations (PDEs) refer to a class nonlinear PDEs which is nonlinear in the highest order derivatives of the unknown functions in the equations. Due to their strong nonlinearity, this class of PDEs are most difficult to analyze analytically and to approximate numerically. In the mean time, fully nonlinear PDEs arise in many applications such as antenna design, astrophysics, differential geometry, fluid mechanics, image processing, meteorology, mesh generation, optimal control, optimal mass transport, etc [8], which calls for the development of efficient and reliable numerical methods for solving their underlying fully nonlinear PDE problems. This is the second paper in a series [9] which is devoted to developing finite difference (FD) and discontinuous Galerkin (DG) methods for approximating ...

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“…Their definition and their analysis are however complex, and their application requires to adjust several parameters. For a recent overview of the numerical approaches to solving the Monge-Ampère equation, see Glowinski, Feng and Neilan [FGN13].…”

Section: Introductionmentioning

confidence: 99%

Monotone and consistent discretization of the Monge-Ampère operator

Benamou¹,

Collino²,

Mirebeau³

2016

Math. Comp.

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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.

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Recent Developments in Numerical Methods for Fully Nonlinear Second Order Partial Differential Equations

Cited by 129 publications

References 110 publications

RETRACTED: Double well potential function and its optimization in the n-dimensional real space: part II

RETRACTED: Double well potential function and its optimization in the n-dimensional real space: part II

Mixed Interior Penalty Discontinuous Galerkin Methods for One-Dimensional Fully Nonlinear Second Order Elliptic and Parabolic Equations

Monotone and consistent discretization of the Monge-Ampère operator

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