Domain Decomposition Parallel Algorithms For Integro-Differential Equation

Domain decomposition methods are effective way to solve the large scale problem into small scale problems,making the computation more easier.Because of this,since 50's of last century,even before the advent of the parallel computers,domain decomposition methods have been applied on sequential computers.With the development of parallel algorithms,since the first 80s,this kind of method has flourished.Now,super parallel computers have been used in energy industry like nuclear industry and experiment, oil industry,biology research,weather forecast,etc.Domain decomposition methods are always employed in tow context.First,the division of large scale problems into several subproblems by the means of domain decomposition is a way to introduce parallelism into large scale problems,and thus the computing time can be shorten.Second,many problems involve more than one mathematical model,each poses on a different domain so that domain decomposition occurs naturally,and so the parallelism is achieved.Because domain decomposition method can turn a large scale problem into smaller ones,turn complicated boundary condition into easier ones,there are very much research work on it.Domain decomposition methods divide the domain into several sub-domains,so the problem can be solved on the sub-domains,its advantage is:First,it takes large scale problem into small scale problems and saves computations.Second,it can use quasi-uniform grid on sub-domains not on the whole domain,and it can use different discrete procedures to solve problem on sub-domains,which is flexible to many Anomalous problems like boiler combustion,roll designation.Third,it can take different mathematical models on different sub-domains to fit the problems,like oil,gas simulation.Forth,the procedure is parallel,can be computed independently on subdomains.When we apply domain decomposition methods to solve a mathematical model, we firstly divide the domain into several sub-domain according to the features of the model or the geometry of the domain.Then we solve the subproblems independently on their own sub- domains respectively.When finding the solution of a partial differential equation,we must have known its boundary condition.However,domain decomposition is an artificial division.Then,for a sub-domain,there is at least a part of boundary with unknown boundary condition,that is to say,the inner-domain boundary conditions are unknown.Our task is to give the boundary conditions to the interfaces of the sub-domains or give the boundary conditions to the interfaces in the procedure though the boundary conditions can not be given apparently.The explicit/implicit domain decomposition method is a kind of method that we give the inner-domain boundary conditions explicitly.We know,if we use an explicit method,the procedure can achieve parallelism naturally.However,there is a stable constraint for the explicit method and the time step is constrained too.So, if we want to march time on,the explicit method will need more steps than implicit method.And thus,it will cost more time to find the solution.Implicit method is unconditionally stable and there is no constraint to time step.While at each time level,we must solve a large,global system of equations.When the mesh is refined, the equation systems become larger at the same time.Solving this kind of equation system also cost much time.The explicit/implicit method include both of the advantage. It use simple,explicit calculations on the boundaries between sub-domains to predict the inner domain boundary condition.Then the equation on the whole domain is decomposed into several equations on sub-domains.When computing, the large,global equation system turns into several smaller ones.So the parallelism is achieved.The explicit nature of the inner-domain boundary conditions induces a time step limitation that is necessary to preserve stability,but this constraint is less severe than that which comes with a fully explicit method.There has been much work on the explicit/implicit domain decomposition methods.In Ref.[1],Dawson and his co-worker employed domain decomposition finite difference method,give the inner-domain boundary conditions explicitly and get an optimal l^∞normed error estimates.Based on this method,In Ref.[2]Q.Du and his co-workers proposed an efficient domain decomposition finite difference method, giving the inner-domain boundary conditions by the help of solutions from the last a few levels.In Ref.[3],Dawson and Dupont applied a an explicit/implicit domain decomposition finite method,defining a function to predict inner-domain boundary conditions,and derive an L² normed error estimates.Then,based on this method, in Ref.[4],Dawson and Dupont used explicit/implicit domain decomposition based on block- centered finite differences and get an l² normed error estimates.This paper includes two parts,in chapter 1,we give preparation knowledge,in chapter 2,we consider the domain decomposition method for integro-differential equations,the function value on the inner-domain boundary is given from the former level,and is parallel,we use new method for error estimates,define a operator including the inner-boundary information to delete the H^1/2,and get the optimal error estimates.