Domain decomposition methods and local adaptive mesh refinement strategies are applied to a system of nonlinear parabolic partial differential equations in two space dimensions. Solutions are calculated on local space-time meshes using a domain decomposition finite element algorithm in space and an extrapolation algorithm in time. Different time steps are used in different spatial regions based on a domain decomposition method. A new multiplicative Schwarz algorithm is used to coordinate solutions between overlapping grids. Extrapolation methods based on either a linearly implicit mid-point rule or a linearly implicit Euler method are used to integrate in time. Extrapolation methods are a better fit than BDF (implicit multi-step methods based on backward difference formulas) methods in our context because local time stepping in different spatial regions precludes history information The problem is solved on a uniform mesh with high order elements using one- or two-level domain decomposition methods. A posteriori error estimator controls a local h-refinement strategy. Elements that violate the error criteria are grouped together to form clusters. A fine mesh with a piecewise polygonal shape is formed for each cluster. The problem is resolved on the fine meshes until the accuracy requirement is achieved. Compared to other local adaptive refinement methods, our methods possess a simpler data structure, require less memory, and use nearly minimal fine meshes which follow closely to the regions of high error Domain decomposition methods also provide a natural way of parallelizing, i.e. problems on different sub-domains can be sent to different processors. Communications between different sub-domains will be carried out through the master processor (68)