Long Time Stability And Convergence Of Evolution Equations, The Approximate Solution

In this paper, linear and semilinear parabolic equation is considered, which possesses a global attractor. The existence of an attractor is one of the most important characteristics for a dissipative system. The longtime dynamics is completely determined by the attractor of the system.Firstly, we analyze the long-time and dynamics characteristics of discere system of heat conduction equation by finite difference method. The heat conduction equation uses two level discrete schemes. We obtain the long-time L2 normal and Hl normal of solution- of discrete system. Furthermore, we get the boundary infinite normal. The existence of a global attractor A for the discrete dynamical scheme is also proved.In following section, we make discrete semilinear parabolic by difference method and debate full discrete Euler implicit scheme. To obtain the long-time characteristics of solution, according to theorem 3.8.6 in reference [20]and properly assumption conditions. We also prove the existence of continuous Liapunov function and the bounded equilibrium sets. So we get a global attractor of discrete system.At last, we also debate the long-time error estimate of the discrete system. We prove that the error order is O(h2+r) in finite time, then we also prove that the result is true by using contractive mapping theory in long time under properly assumption conditions. Furthermore we obtain the convergence theory of the discrete system.