Variational mesh adaptation II: error estimates and monitor functions

The key to the success of a variational mesh adaptation method is to define a proper monitor function which controls mesh adaptation. In this paper we study the choice of the monitor function for the variational adaptive mesh method developed in the previous work [J. Comput. Phys. 174 (2001) 924]. Two types of monitor functions, scalar matrix and non-scalar matrix ones, are defined based on asymptotic estimates of interpolation error obtained using the interpolation theory of finite element methods. The choice of the adaptation intensity parameter is also discussed for each of these monitor functions. Asymptotic bounds on interpolation error are obtained for adaptive meshes that satisfy the regularity and equidistribution conditions. Two-dimensional numerical results are given to verify the theoretical findings.