Optimized Schwarz Methods For A Parabolic Interface Problem With A Nonlinear Jump Condition

For a parabolic interface problem with a nonlinear jump interface condition that describes the concentration of ions in chemical sensors,we propose several optimized Schwarz(waveform relaxation)methods.They decompose the space-time domain(1,1)×(0,T ] into two subdomains(1,0)×(0,T ] and(0,1)×(0,T ],and convert the heterogeneous problem into solving homogeneous problems on each subdomain iteratively.We achieve two purposes by designing the transmission conditions on the interface: 1.The global nonlinear problem is converted into two linear subproblems;2.Fast convergence of subdomain iterations,hence the overall calculations are extraordinarily reduced.More precisely,we propose three linear transmission conditions,including a Robin transmission condition,a prescaled Robin transmission condition and a two-sided Robin transmission condition.Using Fourier analysis,we derive for each case the convergence factors,which allow us to obtain the optimized transmission parameters using optimization technique.Furthermore,we also obtain the estimates of the corresponding convergence factors in asymptotic sense.Theoretical analysis shows that the two-sided Robin transmission condition leads to a temporal mesh independent convergence rate,that is to say,the convergence of subdomain iteration does not depend on the time stepsize.However,the optimized Schwarz methods using the above transmission conditions need to update the discrete matrices at each step of subdomain iteration.In order to overcome this deficiency,we made modifications on the above mentioned transmission conditions,and provide some theoretical analysis.The theoretical results are illustrated by using two examples in the literature.