Fast Method For Solving Parabolic Equation

With the development of science and technology,parabolic partial differential equations have been widely used in physics,engineering and other fields,such as oil development,high temperature transmission,finance and image recognition,population model and so on.At present,numerical simulation has become an important research method.However,its high-performance computing and its theoretical analysis still have great difficulties?especially nonlinear cases,large-scale calculations and long-term simulations.The paper based on the idea of elliptical extrapolation cascadic multigrid method,an extrapolated cascadic multigrid method(PEXCMG/NPEXCMG)for solving parabolic problems is proposed.The basic idea of the algorithm is to accurately solve the finite element discrete systems on the two thicker spatial grids on a fixed time horizon,and then pass the finite element solutions obtained on the two layers of grids through the new push formula+quadratic interpolation constructs an approximation of the exact finite element solution on the third spatial mesh layer.Since the new extrapolation operator obtains a good approximation of the exact finite element solution on the spatial mesh layer,the approximation can be used as a good initial value for the iteration on the mesh layer.This paper demonstrates the best convergence(first order)of PEXCMG based on Richardson iteration in the sense of ? norm,and PEXCMG based on conjugate gradient method(CG)in ? the super optimality(second order)in the sense of norm.Finally,numerical experiments discuss three algorithms,the direct conjugate gradient method(DCG),the cascadic multigrid method(CMG),and the extrapolated cascadic multigrid method(PEXCMG)for solving linear paraboloids problem.Finally,the NPEXCMG solution of semi-linear parabolic equation is analyzed.The numerical results verify the efficiency of the algorithm.