A method of domain decomposition for generalized Stokes problem is developed. The condition of compatibility on the interfaces between each subdomain is satisfied weakly by introducing a Lagrange multiplier technique. In the computational domain, the finite element meshes can be non-coincident at the interfaces. The Petrov-Galerkin formulation is applied to solve the generalized Stokes problem in each subdomain, and the Lagrange multiplier problem on the interfaces is solved using an algorithm of conjugated gradient. Velocity, density (or pressure) and Lagrange multiplier are approached in the spaces of continue piecewise linear functions. Associated with the above domain decomposition method, a local a posteriori error estimation has been constructed. The localization of the error estimation is based on solving local problems. The local error estimators respectively for velocity, density and Lagrange multiplier, are defined in the space of discontinue quadratic bump functions. At the end, some numerical results on the adaptive meshes are also given based on local a posteriori error estimation.