Reduced relative entropy techniques for a priori analysis of multiphase problems in elastodynamics

We give an a priori analysis of a semi-discrete discontinuous Galerkin scheme approximating solutions to a model of multiphase elastodynamics which involves an energy density depending not only on the strain but also the strain gradient. A key component in the analysis is the reduced relative entropy stability framework developed in Giesselmann (SIAM J Math Anal 46(5):3518–3539, 2014). The estimate we derive is optimal in the L∞(0,T;dG) norm for the strain and the L2(0,T;dG) norm for the velocity, where dG is an appropriate mesh dependent H1-like space.

Brief behavioural activation for adolescent depression: working with complexity and risk

Given the long-term negative outcomes associated with depression in adolescence, there is a pressing need to develop brief, evidence based treatments that are accessible to more young people experiencing low mood. Behavioural Activation (BA) is an effective treatment for adult depression, however little research has focused on the use of BA with depressed adolescents, particularly with briefer forms of BA. In this article we outline an adaptation of brief Behavioral Activation Treatment of Depression (BATD) designed for adolescents and delivered in eight sessions (Brief BA). This case example illustrates how a structured, brief intervention was useful for a depressed young person with a number of complicating and risk factors.