A Finite Difference Non-Overlapping Domain Decomposition Method For Numerical Solution Of The Time Fractional Slow Diffusion Equation

In this paper we introduce a finite difference non-overlapping domain de-composition method for numerical solution of the time fractional slow diffusion equation.The one-space-dimensional model is described as follows:For some T>0,where ??(0,1);f,?v are given,which satisfy the compatibility:?(0)=v(0)=0.Note that u(x,t)satisfies the zero initial condition on its domain of definition.In this model we use the classical definition of fractional derivative pro-posed by Caputo.When it comes to solving partial differential equations,it is well acknowl-edged that parallel computing can significantly improve the solving efficiency.Thus,it is natural to develop methods for numerically approximating solutions to partial differential equations on parallel computers.Domain decomposition procedures,however,can be used to divide the domain over which the problem is defined into several smaller ones,so as to allow us to consider and solve the completely independent sub-domain problems on parallel.It is worth noting that even on sequential computers such procedure is also useful,because it allows us to take different time or spatial steps on different sub-domains,especially on those we care about.As a result,we can focus on the domains we are interested in while reduce the cost on other domains.The main difficulty of domain decomposition lies in the fact that the smaller 'sub-domain problems'must be coupled in some way.That is to say,the procedure involves defining values on the sub-domain boundaries and piecing the solutions together in a reasonable way in order to obtain a reliable approximation to the exact solution.Based on the analysis above,this paper is mainly devoted to solve the time fractional slow diffusion equation by proposing a finite difference method combined with the technique of domain decomposition.The methods given allow us to break the work into several sub-domains to deal with.We discuss the algorithms for one and two space dimensional problems.Here we mainly focus on intervals and the regular rectangles.Furthermore,we consider the cases in which time and space steps are different on different sub-domains.We give the existence and stability analysis.Also,discrete maximum norm error estimates are derived for each one of the cases.When our methods are designed to improve the computational efficiency and to focus on the domains we care about,they are also proven to be accurate and reliable,with approximation rate reaching O(?+ h2 + H3)(? refers to the time step).According to the outline stated above,the rest of the article is organized as follows:Chapter 1:Introduce the research background and the method of domain decomposition.Review the current study from both overseas and domestic researchers.Chapter 2:Describe several forms of the de.finition of fractional deriva-tives and their approximation formulas.In order to introduce some of the approximation properties,we present them as preliminary lemmas.Chapter 3:A finite difference non-overlapping domain decomposition method for the one dimensional problem.We give the existence,stability analysis and the error estimate for the numerical solution.Moreover,consider the case where different time and space steps are used on different sub-domains.Discuss the case with multiple sub-domains.Error estimates are given for each one of the cases.Chapter 4:We carried out the corresponding study on the two dimension-al model.Firstly we present the basic domain decomposition finite difference scheme,and then we derive the error estimate.Next,consider the spatial-ly varying time steps domain decomposition procedure,and its convergence analysis is given.Chapter 5:Carry out numerical experiments to verify the theoretical re-sults obtained in this paper.We give one and two dimensional examples,and the computational results are in accordance with our theoretical conclusions.