Space-Time Finite Element Method For The Semi-Linear Parabolic Equations

The continuous space-time Galerkin FEM for the semi-linear parabolic problem are discussed.The existence and uniqueness of the weak solution are given by the technique of combining the FEM and finite difference method.And the error estimates in maximum norm in time,L₂norm in space,namely,the L^∞(L²)error estimates are proved.And the error estimates in maximum norm in time and H¹norm in space are discussed.In the mean time ,the numerical simulations are presented to illustrate the efficiency of the continuous spacetime form.At the end part of the article,the error estimates are given for the semi-linear problem integral differential differential equations using the linearized technique.