A curious property of oscillatory FEM solutions of one-dimensional convection-diffusion problems

summary:Song, Yin and Zhang (Int. J. Numer. Anal. Model. 4: 127-140, 2007) discovered a remarkable property of oscillatory finite element solutions of one-dimensional convection-diffusion problems that leads to a novel numerical method for the solution of such problems. In the present paper this property is described using several figures, then a simple proof of the phenomenon is given which is much more intuitive than the technical analysis of Song et al

A finite difference method for an initial–boundary value problem with a Riemann–Liouville–Caputo spatial fractional derivative

An initial–boundary value problem with a Riemann–Liouville–Caputo space fractional derivative of order a¿(1, 2) is considered, where the boundary conditions are reflecting. A fractional Friedrichs’ inequality is derived and is used to prove that the problem approaches a steady-state solution when the source term is zero. The solution of the general problem is approximated using a finite difference scheme defined on a uniform mesh and the error analysis is given in detail for typical solutions which have a weak singularity near the spatial boundary x=0. It is proved that the scheme converges with first order in the maximum norm. Numerical results are given that corroborate our theoretical results for the order of convergence of the difference scheme, the approach of the solution to steady state, and mass conservation

Walk a mile in my shoes: A case study of the everyday lives and work experiences of a group of Irish primary school Principals

While much has been written about theories and practices of management and leadership in education in recent years, what school Principals actually do on a daily basis is relatively unresearched in Ireland. Moreover, how Principals experience the job personally goes largely unnoticed. To investigate such questions, the researcher adopts a case study approach to gathering data from a group of 31 Irish primary school Principals. Researcher-driven diaries offer opportunities for self-observation by recording Principals’ personal reflections on management and leadership activities as they arise during the day. On completion, all of the diaries are collated into a single bound volume and a copy of the booklet is returned to each participant. Principals report that it is both interesting and worthwhile to read the entries of others and to gain insights into the daily work practices and lived experiences of colleagues. 21 of the 31 Principals are available in the following weeks for a second round of data gathering in recorded interviews. They comment about their own experiences of keeping the diaries and about their impressions of the experiences of other school leaders also. The diary and interview data are then coded and queried using QSR’s NVivo computer application and an organised framework of thematic flowcharts. The results are presented in a case study report with supporting empirical evidence and 12 different aspects of journeying in the Principal’s shoes through a myriad of daily work practices are explored. A narrative account explores Principals’ engagement with the internal and external school environments. It demonstrates evidence of positive work ethic and time management issues. It details Principals’ involvement with Boards of Management. Principals are vocal about their emotional investment in their roles and about the many positives and negatives that they encounter. Conclusions are drawn about career progression and about the sustainability of certain practices within the current system. The collated Principals’ diaries are available in the Appendices. They offer the opportunity to readers in different contexts to draw relevant and meaningful conclusions of their own

Formal consistency versus actual convergence rates of difference schemes for fractional-derivative boundary value problems

Finite difference methods for approximating fractional derivatives are often analyzed by determining their order of consistency when applied to smooth functions, but the relationship between this measure and their actual numerical performance is unclear. Thus in this paper several wellknown difference schemes are tested numerically on simple Riemann-Liouville and Caputo boundary value problems posed on the interval [0, 1] to determine their orders of convergence (in the discrete maximum norm) in two unexceptional cases: (i) when the solution of the boundary-value problem is a polynomial (ii) when the data of the boundary value problem is smooth. In many cases these tests reveal gaps between a method’s theoretical order of consistency and its actual order of convergence. In particular, numerical results show that the popular shifted Gr¨unwald-Letnikov scheme fails to converge for a Riemann-Liouville example with a polynomial solution p(x), and a rigorous proof is given that this scheme (and some other schemes) cannot yield a convergent solution when p(0)¿ 0