Ignoring the mesh convergence´can lead the design process in wrong direction. Although error indicators can be requested to evaluate the mesh convergence, the unaveraged stress contour plot could be used as an alternative in cases where error indicator is not available due to software capability or simply because an analyst forgets to request error output variable.
Adaptive remeshing generates a new mesh such as to refine/coarsen mesh depending upon the extent and distribution of discretization error. The process of adaptive meshing is automated and software does all of the mesh refinement in the regions where it is required. Adaptive meshing is a great tool for accuracy control.