Error indicators for vector functions on adaptive grids

We consider grid adaptation based on an interpolation error estimate for the solution to a system of ordinary differential equations (ODE). Dealing with the interpolation error estimates, the problem of adaptation to a vector solution is of particular interest, since it is unclear how to choose a scalar key function for the adaptation. In our work, the effective technique is developed to analyze a standard h-refinement algorithm. This technique is implemented to study the grid refinement procedure for the vector solution of ODE. We discuss how the results of the adaptation depend on the choice of the key function. We show that the key function should be consistent with the refinement criterion used in the problem. We also discuss how to render the standard refinement procedure more effective in case that a quasiuniform grid is generated as a result of the adaptation due to a wrong choice of the key function.