Space-Time Finite Element Methods For Fourth Order Parabolic Equations

Discontinuous space-time finite element methods deal with the spatial and the temporal variables simutaneously.And high order of accuracy in both directions are achieved. At the same time,this kind of methods is highly adaptive,and suitable for dealing with complex discontinuous problems.The idea of combining the discontinuous finite element methods with mixed finite element methods is considered for parabolic differential equations with several different boundary conditions. For each fourth order parabolic equations, the mixed discontinuous space-time finite element scheme is constructed by lowering the order of the equation by mixed finite element. And the equation is discretized by space-time finite element method ,continuous in space but discontinuous in time . The stability ,uniqueness and existence , error estimates of the approximate solution are proved for linear problems. Brouwer fixed points theorem is used to prove the uniqueness of the approximate solution of semi-linear parabolic problems and the error estimate for time in max norm and space in L₂ norm is obtained. For nonlinear parabolic equation, optimal error estimate in L_∞(L₂) norm is proved. In the meantime ,several numerical results are presented to demonstrate the feasibility and effectiveness of this kind of the algorithm .