  In the present work we investigate the chemotactically and proteolytically
driven tissue invasion by cancer cells. The model employed is a system of
advection-reaction-diffusion equations that features the role of the serine
protease urokinase-type plasminogen activator. The analytical and numerical
study of this system constitutes a challenge due to the merging, emerging, and
travelling concentrations that the solutions exhibit.
  Classical numerical methods applied to this system necessitate very fine
discretization grids to resolve these dynamics in an accurate way. To reduce
the computational cost without sacrificing the accuracy of the solution, we
apply adaptive mesh refinement techniques, in particular h-refinement. Extended
numerical experiments exhibit that this approach provides with a higher order,
stable, and robust numerical method for this system. We elaborate on several
mesh refinement criteria and compare the results with the ones in the
literature.
  We prove, for a simpler version of this model, $L^\infty$ bounds for the
solutions, we study the stability of its conditional steady states, and
conclude that it can serve as a test case for further development of mesh
refinement techniques for cancer invasion simulations.

Numerical Analysis;Cell Behavior