Topic
Piecewise
About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.
Papers published on a yearly basis
Papers
More filters
••
TL;DR: A new divergence-free finite element on 3D Powell–Sabin grids is constructed for Stokes equations, where the velocity is approximating by continuous piecewise quadratic polynomials while the pressure is approximated by discontinuous piecewise linear polynmials on the same grid.
Abstract: Given a tetrahedral grid in 3D, a Powell---Sabin grid can be constructed by refining each original tetrahedron into 12 subtetrahedra. A new divergence-free finite element on 3D Powell---Sabin grids is constructed for Stokes equations, where the velocity is approximated by continuous piecewise quadratic polynomials while the pressure is approximated by discontinuous piecewise linear polynomials on the same grid. To be precise, the finite element space for the pressure is exactly the divergence of the corresponding space for the velocity. Therefore, the resulting finite element solution for the velocity is pointwise divergence-free, including the inter-element boundary. By establishing the inf-sup condition, the finite element is stable and of the optimal order. Numerical tests are provided.
93 citations
••
TL;DR: The truncation is shown to stabilize the method against measurement noise and to have a smoothing effect on the reconstructed conductivity, which can be interpreted as regularization of the D‐bar method.
Abstract: The effects of truncating the (approximate) scattering transform in the D‐bar reconstruction method for two‐dimensional electrical impedance tomography are studied. The method is based on the uniqueness proof of Nachman [Ann. of Math. (2), 143 (1996), pp. 71–96] that applies to twice differentiable conductivities. However, the reconstruction algorithm has been successfully applied to experimental data, which can be characterized as piecewise smooth conductivities. The truncation is shown to stabilize the method against measurement noise and to have a smoothing effect on the reconstructed conductivity. Thus the truncation can be interpreted as regularization of the D‐bar method. Numerical reconstructions are presented demonstrating that features of discontinuous high contrast conductivities can be recovered using the D‐bar method. Further, a new connection between Calderon’s linearization method and the D‐bar method is established, and the two methods are compared numerically and analytically.
93 citations
••
TL;DR: It is asserted that the distance of the piecewise constant discrete gradient to any continuous piecewise affine approximation is a reliable upper error bound up to known higher order terms, consistency terms, and a multiplicative constant.
Abstract: The reliability of frequently applied averaging techniques for a posteriori error control has recently been established for a series of finite element methods in the context of second-order partial differential equations. This paper establishes related reliable and efficient a posteriori error estimates for the energy-norm error of an obstacle problem on unstructured grids as a model example for variational inequalities. The surprising main result asserts that the distance of the piecewise constant discrete gradient to any continuous piecewise affine approximation is a reliable upper error bound up to known higher order terms, consistency terms, and a multiplicative constant.
93 citations
••
TL;DR: Based on the newly established stability criteria, sufficient conditions for the existence of delay-independently periodically intermittent state-feedback controllers are derived and two illustrations are presented to show the validity of the obtained results.
93 citations
••
01 Feb 2004
TL;DR: In this paper, a controller design method for fuzzy dynamic systems based on a piecewise smooth Lyapunov function is presented, which can be used to establish the global stability with H/sub /spl infin/ performance of the resulting closed loop fuzzy control systems.
Abstract: This paper presents a controller design method for fuzzy dynamic systems based on a piecewise smooth Lyapunov function. The basic idea of the proposed approach is to construct controllers for the fuzzy dynamic systems in such a way that a piecewise continuous Lyapunov function can be used to establish the global stability with H/sub /spl infin// performance of the resulting closed loop fuzzy control systems. It is shown that the control law can be obtained by solving a set of linear matrix inequalities (LMI) that is numerically feasible with commercially available software. An example is given to illustrate the application of the proposed method.
93 citations