Unconditional superconvergence analysis of an &ITH&IT1-galerkin mixed finite element method for nonlinear Sobolev equations

An efficient H-1-Galerkin mixed finite element method (MFEM) is presented with EQ(1)(rot) and zero order Raviart-Thomas elements for the nonlinear Sobolev equations. On one hand, the existence and uniqueness of the solutions of the semidiscrete approximation scheme are proved and the super close results of order O(h(2)) for the original variable u in a broken H-1 norm and the auxiliary variable (q)over-right-arrow = a(u)del u(t) + b(u)del u in H(div;Omega) norm are deduced without the boundedness of the numerical solution in L-infinity-norm. Conversely, a linearized Crank-Nicolson fully discrete scheme with the unconditional super close property O(h(2) + tau(2) ) is also developed through a new approach, while previous literature always require certain time step conditions (see the references below). Finally, a numerical experiment is included to illustrate the feasibility of the proposed method. Here h is the subdivision parameter and tau is the time step.