An optimization based domain decomposition method for partial differential equations

AbstractAn optimization-based domain decomposition method for the solution of partial differential equations is presented. The crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains; the constraints are the partial differential equations. The existence of optimal solutions for the optimization problem is shown as is the convergence to the exact solution of the given problem. We then derive an optimality system of partial differential equations from which solutions of the domain decomposition problem may be determined. Finite element approximations to solutions of the optimality system are defined and analyzed as well as an eminently parallelizable gradient method for solving the optimality system. Then, the results of some numerical experiments and some concluding remarks are given. The latter includes the extension of the method to nonlinear problems such as the Navier-Stokes equations