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Piecewise linear function

About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.


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TL;DR: It is shown that, in fact, there exists an elegant proof of this feature independent of the space dimension, and superconvergence for dimensions four and up is proved simultaneously.
Abstract: Superconvergence of the gradient for the linear simplicial finite-element method applied to elliptic equations is a well known feature in one, two, and three space dimensions. In this paper we show that, in fact, there exists an elegant proof of this feature independent of the space dimension. As a result, superconvergence for dimensions four and up is proved simultaneously. The key ingredient will be that we embed the gradients of the continuous piecewise linear functions into a larger space for which we describe an orthonormal basis having some useful symmetry properties. Since gradients and rotations of standard finite-element functions are in fact the rotation-free and divergence-free elements of Raviart-Thomas and Nedelec spaces in three dimensions, we expect our results to have applications also in those contexts.

74 citations

Journal ArticleDOI
TL;DR: The proposed stabilizing controller can be solved directly and effectively, which is applicable to more general situations than those previously covered, and given to illustrate the effectiveness of the proposed method.

74 citations

Posted Content
TL;DR: In this article, the authors studied the dynamics of a point particle in a periodic array of spherical scatterers and constructed a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density.
Abstract: We study the dynamics of a point particle in a periodic array of spherical scatterers, and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (Boltzmann-Grad limit). A generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a Markov process with memory two.

74 citations

Journal ArticleDOI
TL;DR: A new class of piecewise linear methods for the numerical solution of the one-dimensional Euler equations of gas dynamics is presented and it can be shown that they preserve monotonicity.
Abstract: A new class of piecewise linear methods for the numerical solution of the one-dimensional Euler equations of gas dynamics is presented. These methods are uniformly second-order accurate and can be considered as extensions of Godunov’s scheme. With an appropriate definition of monotonicity preservation for the case of linear convection, it can be shown that they preserve monotonicity. Similar to Van Leer’s scheme, they consist of two key steps: a reconstruction step followed by an upwind step. For the reconstruction step, a monotonicity constraint that preserves uniform second-order accuracy is introduced. Computational efficiency is enhanced by devising a criterion that detects the “smooth” part of the data where the constraint is redundant. The concept and coding of the constraint are simplified by the use of the median function. A slope-steepening technique, which has no effect at smooth regions and can resolve a contact discontinuity in four cells, is described. As for the upwind step, existing and new...

74 citations

Journal ArticleDOI
TL;DR: An optimal order multigrid method is developed for the pure displacement problem in two-dimensional linear elasticity, based on a nonconforming mixed formulation, where the displacement is approximated by weakly continuous piecewise linear vector functions, and the pressure is approximating by piecewise constants.
Abstract: An optimal order multigrid method is developed for the pure displacement problem in two-dimensional linear elasticity. It is based on a nonconforming mixed formulation, where the displacement is approximated by weakly continuous piecewise linear vector functions, and the pressure is approximated by piecewise constants. The full multigrid convergence is proved. The performance of the multigrid method does not deteriorate as the material becomes nearly incompressible.

74 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023179
2022377
2021312
2020353
2019329
2018297