Several Efficient Finite Element Methods For The Parabolic Equation

In this paper,the efficient numerical algorithms are constructed for the parabolic equation based on the finite element method.First,the operator splitting method via finite element method is presented to solve the high dimensional heat equation,which is decomposed into a series of one dimensional problem.Hence,the complexity of the problem is reduced.Besides,the stability analysis and error estimate of the algorithm are presented,and the efficiency of the algorithm is indicated by the numerical experiments.For the Allen-Cahn equation,we apply the classical operator splitting method to decompose it into linear and nonlinear subproblems.Among of them,the linear problem can be solved by the finite element method,and the nonlinear problem is solved analytically.The classical operator splitting scheme is easy to solve,but the schemes have low convergence rate.Therefore,we consider to combine the operator splitting scheme with Parareal method to obtain the higher order numerical scheme.In addition,to relax the restriction on time step of the operator splitting scheme,we consider to construct the stabilization operator splitting scheme.By add to the stabilization item,the restriction on the time step can be released.