  A Lagrangian-type numerical scheme called the "comoving mesh method" or CMM
is developed for numerically solving certain classes of moving boundary
problems which include, for example, the classical Hele-Shaw flow problem and
the well-known mean curvature flow problem. This finite element scheme exploits
the idea that the normal velocity field of the moving boundary can be extended
throughout the entire domain of definition of the problem using, for instance,
the Laplace operator. Then, the boundary as well as the finite element mesh of
the domain are easily updated at every time step by moving the nodal points
along this velocity field. The feasibility of the method, highlighting its
practicality, is illustrated through various numerical experiments. Also, in
order to examine the accuracy of the proposed scheme, the experimental order of
convergences between the numerical and manufactured solutions for these
examples are also calculated.
