Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems

We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction–diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and “balanced” norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the “balanced” norm is artificially introduced to capture the boundary layer contribution. We establish a uniform convergence of order k for the LDG method using the balanced norm with the local weighted L2 projection as well as an optimal convergence of order k+1 for the energy norm using the local Gauss–Radau projections. The numerical method, the layer structure as well as the used adaptive meshes are all discussed in a symmetry way. Numerical experiments are presented.

Singularly Perturbed Reaction–Diffusion Problems as First order systems

We consider a singularly perturbed reaction diffusion problem as a first order two-by-two system. Using piecewise discontinuous polynomials for the first component and $$H_{{{\,\mathrm{{div}}\,}}}$$ H div -conforming elements for the second component we provide a convergence analysis on layer adapted meshes and an optimal convergence order in a balanced norm that is comparable with a balanced $$H^2$$ H 2 -norm for the second order formulation.

Criminal accountability at what cost? Norm conflict, UN peace operations and the International Criminal Court

How do international organisations (IOs) balance norms that have conflicting prescriptions? In this article, we build on the literature on norm contestation and norm conflict to identify four ways in which IOs might respond to norm conflicts: (1) consistent norm prioritisation; (2) ad hoc norm prioritisation; (3) balanced norm reconciliation and (4) imbalanced norm reconciliation. How IOs are more likely to respond, we argue, depends on the salience of the norm conflict and the relative strength of the conflicting norms. We illustrate our argument by investigating the norm conflicts that the United Nations (UN) encountered between traditional UN peacekeeping norms and the norm of international criminal accountability in the context of assistance by UN peace operations to the International Criminal Court. Distinguishing between behaviour and discourse as well as headquarters and field levels, we argue that the UN’s response strategies have been largely successful at reducing the tensions generated by the norm conflicts, but that in the longer term they run the risk of both undermining the international criminal accountability norm and damaging the acceptance and credibility of UN peace operations.

Balanced-norm error estimates for sparse grid finite element methods applied to singularly perturbed reaction–diffusion problems

We consider a singularly perturbed linear reaction–diffusion problem posed on the unit square in two dimensions. Standard finite element analyses use an energy norm, but for problems of this type, this norm is too weak to capture adequately the behaviour of the boundary layers that appear in the solution. To address this deficiency, a stronger so-called ‘balanced’ norm has been considered recently by several researchers. In this paper we shall use two-scale and multiscale sparse grid finite element methods on a Shishkin mesh to solve the reaction–diffusion problem, and prove convergence of their computed solutions in the balanced norm.