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A Domain Decomposition Method Based on Weighted Interior Penalties for Advection-Diffusion-Reaction Problems

We propose a domain decomposition method for advection-diffusion-reaction equations based on Nitsche's transmission conditions. The advection-dominated case is stabilized using a continuous interior penalty approach based on the jumps in the gradient over element boundaries. We prove the convergence of the finite element solutions of the discrete problem to the exact solution and propose a parallelizable iterative method. The convergence of the resulting domain decomposition method is proved, and this result holds true uniformly with respect to the diffusion parameter. The numerical scheme that we propose here can thus be applied straightforwardly to diffusion-dominated, advection-dominated, and hyperbolic problems. Some numerical examples are presented in different flow regimes showing the influence of the stabilization parameter on the performance of the iterative method, and we compare our method with some other domain decomposition techniques for advection-diffusion equations.