Laying out the future of final-form digital documents

It is just over 20 years since Adobe's PostScript opened a new era in digital documents. PostScript allows most details of rendering to be hidden within the imaging device itself, while providing a rich set of primitives enabling document engineers to think of final-form rendering as being just a sophisticated exercise in computer graphics. The refinement of the PostScript model into PDF has been amazingly successful in creating a near-universal interchange format for complex and graphically rich digital documents but the PDF format itself is neither easy to create nor to amend. In the meantime a whole new world of digital documents has sprung up centred around XML-based technologies. The most widespread example is XHTML (with optional CSS styling) but more recently we have seen Scalable Vector Graphics (SVG) emerge as an XML-based, low-level, rendering language with PostScript-compatible rendering semantics. This paper surveys graphically-rich final-form rendering technologies and asks how flexible they can be in allowing adjustments to be made to final appearance without the need for regenerating a whole page or an entire document. Particular attention is focused on the relative merits of SVG and PDF in this regard and on the desirability, in any document layout language, of being able to manipulate the graphic properties of document components parametrically, and at a level of granularity smaller than an entire page.

Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the hydrodynamic stability problem associated with the incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the eigenvalue problem in channel and pipe geometries. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual-Weighted-Residual approach, originally developed for the estimation of target functionals of the solution, to eigenvalue/stability problems. The underlying analysis consists of constructing both a dual eigenvalue problem and a dual problem for the original base solution. In this way, errors stemming from both the numerical approximation of the original nonlinear flow problem, as well as the underlying linear eigenvalue problem are correctly controlled. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.

Adaptivity and a posteriori error control for bifurcation problems II: Incompressible fluid flow in open systems with Z_2 symmetry

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork or Hopf bifurcation occurs when the underlying physical system possesses reflectional or Z_2 symmetry. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual-Weighted-Residual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.