  In this work we study a DPG method for an ultra-weak variational formulation
of a reaction-diffusion problem. We improve existing a priori convergence
results by sharpening an approximation result for the numerical flux. By
duality arguments we show that higher convergence rates for the scalar field
variable are obtained if the polynomial order of the corresponding
approximation space is increased by one. Furthermore, we introduce a simple
elementwise postprocessing of the solution and prove superconvergence.
Numerical experiments indicate that the obtained results are valid beyond the
underlying model problem.
