Comparison between adaptive and uniform discontinuous Galerkin simulations in dry 2D bubble experiments

An error indicator is introduced for a warm air bubble test case which allows one to compare the accuracy between different choices of refinement regions without knowing the exact solution. Essentially this is done by comparing features of the solution that are strongly sensitive to spatial resolution. For the rising warm air bubble the additional error by using adaptivity is smaller than 1% of the total numerical error if the average number of elements used for the adaptive simulation is about a factor of two times smaller than the number used for the simulation with the uniform fine-resolution grid. Correspondingly the adaptive simulation is almost two times faster than the uniform simulation. Furthermore the adaptive simulation is more accurate than a uniform simulation when both use the same CPU-time.