Interior Penalty Discontinuous Galerkin Methods for Second Order Linear Non-divergence Form Elliptic PDEs

This paper develops interior penalty discontinuous Galerkin (IP-DG) methods to approximate \(W^{2,p}\) strong solutions of second order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients. The proposed IP-DG methods are closely related to the IP-DG methods for advection-diffusion equations, and they are easy to implement on existing standard IP-DG software platforms. It is proved that the proposed IP-DG methods have unique solutions and converge with optimal rate to the \(W^{2,p}\) strong solution in a discrete \(W^{2,p}\)-norm. The crux of the analysis is to establish a DG discrete counterpart of the Calderon–Zygmund estimate and to adapt a freezing coefficient technique used for the PDE analysis at the discrete level. To obtain such a crucial estimate, we need to establish broken \(W^{1,p}\)-norm error estimates for IP-DG approximations of constant coefficient elliptic PDEs, which is also of independent interest. Numerical experiments are provided to gauge the performance of the proposed IP-DG methods and to validate the theoretical convergence results.     

Vicarious punishment in a work setting.

Investigated the effects of punishment on the attitudes and behavior of co-workers (CWs) who observed a peer receiving punishment to test the hypothesis that observing a CW receive punishment will have a positive effect on observers' productivity without damaging job satisfaction. 60 students were hired through a university placement office for temporary clerical employment. Ss were exposed to observing either a CW receiving a reduction in pay or a threat of a reduction in pay or to no punishment (controls). Results support the hypothesis that punishment may be used effectively in work settings: Ss who observed a CW receiving a reduction in pay produced significantly more than controls and Ss who observed a threat of a reduction in pay. These effects did not diminish after 1 wk, and Ss across groups did not differ in levels of job satisfaction. Because none of the Ss knew each other, it is suggested that the absence of negative side effects may not have held if CWs had observed someone they knew receiving punishment. (17 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)