Schwarz Algorithms For The Time-harmonic Eddy Current Problem

In this thesis,we study Schwarz algorithms for the time-harmonic eddy current problem.We start from the H_C/E₁ formulation of the eddy current problem proposed by Rodriguez.We first use the variational principle to deduce the suitable coupling conditions of the problem,and get the complete and well posed time-harmonic eddy current model.The obtained coupling condition is the same as the one given by Rodriguez in his article[Rodriguez A A,Hernandez R V,SIAM Journal on Scientific Computing,31（4）（2009）,pp.3155-3157].We call the Schwarz algorithm with these coupling conditions as the transmission conditions the classical Schwarz algorithm.Using the Fourier analysis,we find that the classical Schwarz algorithm of the time-harmonic eddy problem is not convergent,which is similar to the result the classical Schwarz algorithm for the Laplace equation with non-overlapping domain decomposition does not converge.In order to obtain the convergent Schwarz algorithm,we propose to use the impedance boundary conditions as transmission conditions.Using the Fourier analysis and optimization techniques,we get the best choice of the free parameters in the impedance condition as α>0,α =-ai（a>0）or α= ā（1-i）（ā>0）,and give the asymptotic estimate on the corresponding convergence factors.We find that when the free parameters in the transmission condition are α>0 or α =-ai（ā>0）,the optimized convergence factor has an estimate of 1-O(k_max^-2),where kmax is the largest Fourier frequency involved in a uniform mesh.However,as the free parameter is given by α =ā（1-i）（ā>0）,the optimized convergence factor has an estimate of 1-O(k_max^-1).