Optimized Schwarz Algorithms For The Plate Bending Problem With Circular Geometry

In this paper,we study the optimized Schwarz algorithms for the plate bending problem with circular geometry.First,we prove that the classical Schwarz algorithm for the plate bending problem with circular geometry does not converge in the non-overlapping case and converges in the overlapping case.However it converges very slow,this is similar to the performance of the classical Schwarz algorithm for Laplace equation.Then we put forward a Robin type transmission condition where the optimal transmission operator is non-local and expensive to use.To tackle the problem,we approximate the Fourier symbol of the optimal operators by those corresponding to local operators and give the Robin-like,Ventcel-like and two-side Robin-like optimized transmission conditions.By optimizing the convergence factor obtained by Fourier transformation,we obtain the optimized parameters involved in the three transmission conditions above for both the overlapping and non-overlapping cases and give as well the corresponding asymptomatic convergence rate estimates.Finally we illustrate the theoretical results with numerical experiments.