Topic
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: This work concerns the development of stabilized finite element methods for the Stokes problem considering nonstable different (or equal) order of velocity and pressure interpolations, and uses a Petrov--Galerkin approach to propose stable variational formulations.
Abstract: This work concerns the development of stabilized finite element methods for the Stokes problem considering nonstable different (or equal) order of velocity and pressure interpolations. The approach is based on the enrichment of the standard polynomial space for the velocity component with multiscale functions which no longer vanish on the element boundary. On the other hand, since the test function space is enriched with bubble-like functions, a Petrov--Galerkin approach is employed. We use such a strategy to propose stable variational formulations for continuous piecewise linear in velocity and pressure and for piecewise linear/piecewise constant interpolation pairs. Optimal order convergence results are derived and numerical tests validate the proposed methods.
70 citations
TL;DR: In this article, a posteriori error analysis of a new fully mixed finite element method for the coupling of fluid flow with porous media flow in 2D is presented, which makes use of the global infsup condition, Helmholtz decompositions in both media, and local approximation properties of the Clement interpolant and Raviart-Thomas operator.
70 citations
TL;DR: An extension of the well-known Filippov's solution concept, that is appropriate for 'open' systems so as to allow interconnections of DDS, is proposed, proven that the existence of a piecewise smooth ISS Lyapunov function for a DDS implies ISS and a (small gain) ISS interconnection theorem is derived.
70 citations
TL;DR: In this paper, the stability of switched linear systems is investigated using piecewise linear Lyapunov functions, and the authors identify classes of switching sequences that result in stable trajectories.
Abstract: In this paper, the stability of switched linear systems is investigated using piecewise linear Lyapunov functions. In particular, we identify classes of switching sequences that result in stable trajectories. Given a switched linear system, we present a systematic methodology for computing switching laws that guarantee stability based on the matrices of the system. In the proposed approach, we assume that each individual subsystem is stable and admits a piecewise linear Lyapunov function. Based on these Lyapunov functions, we compose 'global' Lyapunov functions that guarantee stability of the switched linear system. A large class of stabilizing switching sequences for switched linear systems is characterized by computing conic partitions of the state space. The approach is applied to both discrete-time and continuous-time switched linear systems.
69 citations
TL;DR: This new approach allows the implementation as a classical V-cycle and preserves the usual multigrid efficiency, and gives estimates for the asymptotic convergence rates.
Abstract: We derive globally convergent multigrid methods
for discrete elliptic
variational inequalities of the second kind
as obtained from
the approximation of related continuous
problems by piecewise linear finite elements.
The coarse grid corrections are computed
from certain obstacle problems.
The actual constraints are fixed by the
preceding nonlinear fine grid smoothing.
This new approach allows the implementation
as a classical V-cycle and preserves
the usual multigrid efficiency.
We give
$1-O(j^{-3})$
estimates
for the asymptotic convergence rates.
The numerical results indicate a significant improvement
as compared with previous multigrid approaches.
69 citations