High-Accuracy Analysis Of Two-Grid Methods For Nonlinear Parabolic Equations

In this paper,the high-accuracy analysis of two-grid methods(TGMs)are es-tablished for the nonlinear parabolic equations.From the points of Galerkin finite element method,new mixed finite element method and H1-Galerkin mixed finite element method view,the superclose and the global superconvergence results are derived,respectively.To begin with,we take the bilinear finite element for example to develop the two-grid algorithms for the problem in the Backward Euler(B-E)and Crank-Nicolson(C-N)fully-discrete TGM schemes.Based on the combination of the interpolation and Ritz projection,and derivative transformation techniques,the superclose estimates of order 0(h2+H4+?)and order 0(h2+H4+?2)of the original variable u in the H1-norm are deduced under a weaker hypothesis of ut €H2(?)instead of ut ?H3(?)required in previous literature.Then,the corresponding global superconvergence results of ordererO(h2+H4+?)and order O(h2+H4+?2)are obtained for the above two schemes,respectively,through the terpolated postprocessing approach.At the same time,a extrapolation methodoh of the bilinear finite element mehtod is constrtruted and a higher accuracy error timate of ordererO(h2+H6+?)is derived on the coarse meshes for the B-E-E heme.xt,through introducing the auxiliary variables q?= ?u and q? = ?u,e high-accuracy properties of a new mixed finite element method and an H1-lerkin mixed finite element method of the TGM are studied by the finite el-ent pairs Q11/Q01 × Q10 and Q11/Q10 × Q01,respectively.The superclose eatimates of order O(h2+H4+?)and the corresponding global superconvergence timates of ordererO(h2+H4+?)of the original variable in the H1-norm and the auxiliary variable p? in the L2-norm are deduced for the B-E fully-discrete scheme of the TGM,respectively.Finally,some numerical results are provided to confirm the above theoret-ical analysis,and also show that the proposed TGMs are indeed very effective numerical methods and the computing cost are only of 1/2 and 1/3 compared to the traditional finite element methods for the B-E and C-N fully discrete schemes,respectively.