Discontinuous Galerkin Methods for the Biharmonic Problem

This work is concerned with the design and analysis of hp-version discontinuous Galerkin (DG) finite element methods for boundary-value problems involving the biharmonic operator. The first part extends the unified approach of Arnold, Brezzi, Cockburn & Marini (SIAM J. Numer. Anal. 39, 5 (2001/02), 1749-1779) developed for the Poisson problem, to the design of DG methods via an appropriate choice of numerical flux functions for fourth order problems; as an example we retrieve the interior penalty DG method developed by Suli & Mozolevski (Comput. Methods Appl. Mech. Engrg. 196, 13-16 (2007), 1851-1863). The second part of this work is concerned with a new a-priori error analysis of the hp-version interior penalty DG method, when the error is measured in terms of both the energy-norm and L2-norm, as well certain linear functionals of the solution, for elemental polynomial degrees $p\ge 2$. Also, provided that the solution is piecewise analytic in an open neighbourhood of each element, exponential convergence is also proven for the p-version of the DG method. The sharpness of the theoretical developments is illustrated by numerical experiments.

Social support in cyberspace: A content analysis of communication within a Huntington’s Disease online support group

Objective: Huntington’s Disease (HD) is an inherited disorder, characterised by a progressive degeneration of the brain. Due to the nature of the symptoms, the genetic element of the disease and the fact that there is no cure, HD patients and those in their support network often experience considerable stress and anxiety. With an expansion in Internet access, individuals affected by HD have new opportunities for information retrieval and social support. The aim of this study is to examine the provision of social support in messages posted to a HD online support group bulletin board. Methods: In total, 1313 messages were content analysed using a modified version of the Social Support Behaviour Code developed by Cutrona & Suhr (1992). Results: The analysis indicates that group members most frequently offered informational (56.2%) and emotional support (51.9%) followed by network support (48.4%) with esteem support (21.7%) and tangible assistance (9.8%) least frequently offered. Conclusion: This study suggests that exchanging informational and emotional support represents a key function of this online group. Practice implications: Online support groups provide a unique opportunity for health professionals to learn about the experiences and views of individuals affected by HD and explore where and why gaps may exist between evidence-based medicine and consumer behaviour and expectations.

Energy norm a-posteriori error estimation for hp-adaptive discontinuous Galerkin methods for elliptic problems in three dimensions

We develop the energy norm a-posteriori error estimation for hp-version discontinuous Galerkin (DG) discretizations of elliptic boundary-value problems on 1-irregularly, isotropically refined affine hexahedral meshes in three dimensions. We derive a reliable and efficient indicator for the errors measured in terms of the natural energy norm. The ratio of the efficiency and reliability constants is independent of the local mesh sizes and weakly depending on the polynomial degrees. In our analysis we make use of an hp-version averaging operator in three dimensions, which we explicitly construct and analyze. We use our error indicator in an hp-adaptive refinement algorithm and illustrate its practical performance in a series of numerical examples. Our numerical results indicate that exponential rates of convergence are achieved for problems with smooth solutions, as well as for problems with isotropic corner singularities.

A class of domain decomposition preconditioners for hp-discontinuous Galerkin finite element methods

In this article we address the question of efficiently solving the algebraic linear system of equations arising from the discretization of a symmetric, elliptic boundary value problem using hp-version discontinuous Galerkin finite element methods. In particular, we introduce a class of domain decomposition preconditioners based on the Schwarz framework, and prove bounds on the condition number of the resulting iteration operators. Numerical results confirming the theoretical estimates are also presented.