A robust nonconforming $H^2$-element

Nilssen, Trygve K.; Tai, Xue–Cheng; Winther, Ragnar

doi:10.1090/s0025-5718-00-01230-8

Cited by 101 publications

(119 citation statements)

References 9 publications

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“…We note that finite element approximations of fourth order PDEs, in particular, the biharmonic equation, were carried out extensively in 1970's in the two-dimensional case (see [10] and the references therein),and have attracted renewed interests lately for generalizing the well-know 2-D finite elements to the 3-D case (cf. [34,35,33]) and for developing discontinuous Galerkin methods in all dimensions (cf. [16,28]).…”

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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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%

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

Feng

Neilan

2010

J Sci Comput

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“…For example, the space S h is a three-dimensional analog of the finite element space used in [14] to discretize fourth-order problems which are perturbations of a second-order problem. However, we do not discuss this here.…”

Section: Introductionmentioning

confidence: 99%

A discrete de Rham complex with enhanced smoothness

Tai

Winther

2006

Calcolo

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Abstract. Discrete de Rham complexes are fundamental tools in the construction of stable elements for some finite element methods. The purpose of this paper is to discuss a new discrete de Rham complex in three space dimensions, where the finite element spaces have extra smoothness compared to the standard requirements. The motivation for this construction is to produce discretizations which have uniform stability properties for certain families of singular perturbation problem. In particular, we show how the spaces constructed here lead to discretizations of Stokes type systems which have uniform convergence properties as the Stokes flow approaches a Darcy flow.

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“…We note that finite element approximations of fourth order PDEs, in particular, the biharmonic equation, were carried out extensively in 1970's in the two-dimensional case (see [10] and the references therein), and have attracted renewed interests lately for generalizing the well-know 2-D finite elements to the 3-D case (cf. [35,36,34]) and for developing discontinuous Galerkin methods in all dimensions (cf. [18,27]).…”

mentioning

confidence: 99%

Mixed Finite Element Methods for the Fully Nonlinear Monge–Ampère Equation Based on the Vanishing Moment Method

Feng¹,

Neilan²

2009

SIAM J. Numer. Anal.

120

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Abstract. This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Ampère equation det(D 2 u 0 ) = f (> 0) based on the vanishing moment method which was proposed recently by the authors in [19]. In this approach, the second order fully nonlinear Monge-Ampère equation is approximated by the fourth order quasilinear equation −ε∆ 2 u ε + det D 2 u ε = f . It was proved in [17] that the solution u ε converges to the unique convex viscosity solution u 0 of the Dirichlet problem for the Monge-Ampère equation. This result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations, a task which has been impracticable before. The goal of this paper is threefold. First, we develop a family of Hermann-Miyoshi type mixed finite element methods for approximating the solution u ε of the regularized fourth order problem, which computes simultaneously u ε and the moment tensor σ ε := D 2 u ε . Second, we derive error estimates, which track explicitly the dependence of the error constants on the parameter ε, for the errors u ε − u ε h and σ ε − σ ε h . Finally, we present a detailed numerical study on the rates of convergence in terms of powers of ε for the error u 0 − u ε h and σ ε − σ ε h , and numerically examine what is the "best" mesh size h in relation to ε in order to achieve these rates. 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 for the problem. To overcome the difficulty, we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations.

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A robust nonconforming $H^2$-element

Cited by 101 publications

References 9 publications

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

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

A discrete de Rham complex with enhanced smoothness

Mixed Finite Element Methods for the Fully Nonlinear Monge–Ampère Equation Based on the Vanishing Moment Method

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