L ? stability of finite element approximations to elliptic gradient equations

Kerkhoven, Thomas; Jerome, Joseph W.

doi:10.1007/bf01386428

Cited by 30 publications

(26 citation statements)

References 4 publications

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“…2. It can be shown that this construction fulfills assumption (A2) by studying the local stiffness matrix, see [6] and further [11]. To study the energy norm convergence, a reference solution on a mesh with 128 cubes in each coordinate direction and compared with solutions computed on meshes with 4, 8, 16, 32, and 64 cubes in each coordinate direction.…”

Section: Convergence Studymentioning

confidence: 90%

Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions

et al. 2010

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In this paper we present a finite element discretization of the Joule-heating problem. We prove existence of solution to the discrete formulation and strong convergence of the finite element solution to the weak solution, up to a sub-sequence. We also present numerical examples in three spatial dimensions. The first example demonstrates the convergence of the method in the second example we consider an engineering application.

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Section: Convergence Studymentioning

confidence: 90%

Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions

et al. 2010

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“…In [29] it was also shown that L ∞ stability, in terms of satisfying the maximum principle, is a consequence of the following assumption.…”

Section: Maximum and Discrete Maximum Principlesmentioning

confidence: 99%

“…For these reduced bounds to hold for the Galerkin approximation of (36), certain conditions on the mesh must be assumed, as was detailed in [29]. The primary condition requires some terminology and notation.…”

Section: Maximum and Discrete Maximum Principlesmentioning

confidence: 99%

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The Approximation Problem for Drift-Diffusion Systems

Jerome¹

1995

SIAM Rev.

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This review surveys a significant set of recent ideas developed in the study of nonlinear Galerkin approximation. A significant role is played by the Krasnosel'skii Calculus, which represents a generalization of the classical inf-sup linear saddle point theory. A description of a proper extension of this calculus, and the relation to the inf-sup theory are part of this review. The general study is motivated by steady-state, self-consistent, drift-diffusion systems. The mixed boundary value problem for nonlinear elliptic systems is studied with respect to defining a sequence of convergent approximations, satisfying requirements of: (1) optimal convergence rate; (2) computability; and, (3) stability. It is shown how the fixed point and numerical fixed point maps of the system, in conjunction with the Newton-Kantorovich method applied to the numerical fixed point map, permit a solution of this approximation problem. A critical aspect of the study is the identification of the breakdown of the Newton-Kantorovich method, when applied to the differential system in an approximate way. This is now known as the numerical loss of derivatives. As an antidote, a linearized variant of successive approximation, with locally defined subproblems bounded in number at each iteration, is demonstrated. In (2), a distinction is made between the outer analytical iteration, and the inner iteration, governed by numerical linear algebra. The systems studied are broad enough to include important application areas in engineering and science, for which significant computational experience is available.

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“…We note that β r = d r |e r | , r = i, j, k (see the annexe in (Bank & Rose, 1987;Kerkhoven-Jerome, 1990 With the aid of the new expression of β r the system (39) becomes…”

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confidence: 99%

Mixed and Hybrid Finite Element Methods for Convection-Diffusion Problems and Their Relationships with Finite Volume: The Multi-Dimensional Case

Fortin¹,

Mounim²

2017

JMR

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We introduced in (Fortin & Serghini Mounim, 2005) a new method which allows us to extend the connection between the finite volume and dual mixed hybrid (DMH) methods to advection-diffusion problems in the one-dimensional case. In the present work we propose to extend the results of (Fortin & Serghini Mounim, 2005) to multidimensional hyperbolic and parabolic problems. The numerical approximation is achieved using the Raviart-Thomas (Raviart & Thomas, 1977) finite elements of lowest degree on triangular or rectangular partitions. We show the link with numerous finite volume schemes by use of appropriate numerical integrations. This will permit a better understanding of these finite volume schemes and the large number of DMH results available could carry out their analysis in a unified fashion. Furthermore, a stabilized method is proposed. We end with some discussion on possible extensions of our schemes.

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L ? stability of finite element approximations to elliptic gradient equations

Cited by 30 publications

References 4 publications

Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions

Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions

The Approximation Problem for Drift-Diffusion Systems

Mixed and Hybrid Finite Element Methods for Convection-Diffusion Problems and Their Relationships with Finite Volume: The Multi-Dimensional Case

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