The authors consider a diffusion-reaction parabolic equation in

$d$ (

$\le 3$) dimensions and continue their development of dynamic parallel Galerkin domain decomposition methods based on the work of

*C. N. Dawson* and

*T. F. Dupont* [Math. Comput. 58, No. 197, 21–34 (1992;

Zbl 0746.65072)] and the former work by

*K. Ma, T. Sun* and

*D. Yang*, [Numer. Methods Partial Differ. Equations 25, No. 5, 1167–1194 (2009;

Zbl 1173.65061)]. Here dynamically changing decompositions, grids in subdomains, and finite element spaces are possible, the aim being a better resolution in problems with sharp fronts and layers. They use implicit Galerkin in the subdomains and (two variants of) explicit flux calculations at inter-domain boundaries – from where a time step restriction arises. The bulk of the paper is devoted to a priori error estimates. The paper concludes with a report on numerical experiments on two-dimensional problems using bilinear elements which shows second order convergence, somewhat better

${L}_{2}$ errors than the method of Ma, Sun, and Yang [loc. cit.], but on the cost of decisively more computing time.