Adjoint consistency – in addition to consistency – is the key requirement for discontinuous Galerkin discretisations to be of optimal order in L as well as measured in terms of target functionals. We provide a general framework for analysing adjoint consistency and introduce consistent modifications of target functionals. This framework is then used to derive an adjoint consistent discontinuous...

In this work we present a theoretical analysis for a residual-type error estimator for locally conservative mixed methods. This estimator was first introduced by Braess and Verfürth for the Raviart–Thomas mixed finite element method working in mesh-dependent norms. We improve and extend their results to cover any locally conservative mixed method under minimal assumptions, in particular, avoidi...

[1] We thank Lim [2005] for providing this opportunity to clarify the restrictions and confirm the applicability of the results of LaBolle et al. [2000]. LaBolle et al. [2000] develop generalized stochastic differential equations (SDE) that converge to advection dispersion equations with discontinuous dispersion tensor D. The problem of interest here relates to equation (15a) of LaBolle et al. ...

The hydrodynamic limit for a kinetic model of chemotaxis is investigated. The limit equation is a non local conservation law, for which finite time blow-up occurs, giving rise to measure-valued solutions and discontinuous velocities. An adaptation of the notion of duality solutions, introduced for linear equations with discontinuous coefficients, leads to an existence result. Uniqueness is obta...

THERE is an extensive literature on difference methods of solving equations of the parabolic type. A considerable proportion of the studies concerns equations with constant coefficients. In a number of papers (see, for example, [l]-[6]) the stability and convergence of schemes with continuous and sufficiently smooth coefficients are studied. For example, in [4] the convergence and stability in ...

We define a set of orthogonal functions on the complex projective space CPN−1, and compute their Clebsch-Gordan coefficients as well as a large class of 6–j symbols. We also provide all the needed formulae for the generation of high-temperature expansions for U(N)-invariant spin models defined on CPN−1.

One of the most important issues for Muslim philosophers is the continuous imaginal world and its relationship with the discontinuous imaginative faculty. The continuous imaginal world is a perceptive faculty of the soul known as the faculty of imagination. The discontinuous imaginative faculty is the order of the universe called the imaginal world, which Muslim philosophers have portrayed in a...