Finite Element Methods For Neutral Delay Reaction-Diffusion Equations And Delay Navier-Stokes Equations

Delay reaction-diffusion equations(DRDEs)and delay Navier-Stokes equations(DNS-Es)are two representative types of delay partial differential equations(PFEs),which have wide applications in physics,mechanics,biology,automatic control theory,fluid dynamics and other scientific fields.The solutions of delay equations depend not only on the present state of the described system but also on the historical state.Hence,delay equations can de-scribe a wider range of models than equations without delay,and more accurately describe the development of objective things.In general,the exact solutions of delay equations are hard to achieve,and hence constructing efficient,accurate and stable numerical methods to solve this kinds of equations are interesting topics for many scholars.On the other hand,finite element methods(FEMs)for the PDEs without delay are accurate and stable,which maintain a good performance while solving some particular delay PDEs.In view of this,we are going to extend FEMs to solve neutral reaction-diffusion equations with constant delay(NRDEDs),neutral reaction-diffusion equations with piecewise continuous arguments(N-RDEPCAs)and incompressible Navier-Stokes equations(INSEs)with time-variable delay,and analyse the numerical properties,such as convergence and stability.At first,we introduce the research background and current states of DRDEs and DNS-Es,give a brief review to FEMs for solving delay PDEs,and summarize the research moti-vation and main research contents of this paper.To solve the two-dimensional NRDEDs,semi-discrete FEMs and fully discrete Crank-Nicolson FEMs are constructed.Under some appropriate conditions and the sense of L~2and H~1norm,we derived the stability and convergence orders for FEMs,and provide numerical experiments to further verify the effectiveness and convergence orders of the methods.For the two-dimensional NRDEDs,we discuss the asymptotical stability of their theo-retical solutions and finite element solutions.By constructing suitable Lyapunov function,we obtain the asymptotical stability conditions for the solutions of NRDEDs.And un-der these stability conditions,we prove that the semi-discrete and fully discrete FEMs are asymptotical-stability-preserving for NRDEDs.With a numerical example,a further illus-tration is given to show the obtained stability results.To solve the NRDEPCAs,semi-discrete and fully discrete FEMs are constructed..Un-der the appropriate conditions and the sense of L~2and H~1norm,we analyse the convergence orders of semi-discrete FEMs and a class of one-parameter fully discrete FEMs with param-eterθ(0≤θ≤1),prove the unique solvability of the fully discrete FEMs,and provide a numerical experiment to verify the effectiveness and convergence orders of the methods.For the NRDEPCAs,we discuss the asymptotical stability of their theoretical solutions and finite element solutions.Firstly,by the method of separation of variables,we obtain the expression of the theoretical solutions of NRDEPCAs,and by this,we further obtain the asymptotical stability conditions for the theoretical solutions.Then,we prove that the semi-discrete and fully discrete FEMs are asymptotically stable under some suitable conditions,and we further illustrate the obtained theoretical results with a numerical example.FEMs are extended to solve the two-dimensional INSEs with time-variable delay.We derive the spatial semi-discrete FEMs,and analyse the error estimate of the velocity by the bounds of the analytical solutions and some appropriate assumptions.Through the relation-ship between velocity error and pressure error,we further derive the pressure error estimate.By implicit Euler method,implicit-explicit Euler method,two-step backward differentiation formula(BDF2)method,implicit-explicit BDF2 method and the semi-discrete FEMs,we obtain four kinds of fully discrete FEMs,and provide a numerical experiment to confirm the theoretical results.Finally,we briefly summarize the work of this paper,and based on this,we further address several typical issues for further research.