The Space-Time Finite Element Method for Parabolic Problems

Adaptive space-time finite element method, continuous in space but discontinuous in time for semi-linear parabolic problems is discussed. The approach is based on a combination of finite element and finite difference techniques. The existence and uniqueness of the weak solution are proved without any assumptions on choice of the space-time meshes. Basic error estimates in L (L²) norm, that is maximum-norm in time, L²-norm in space are obtained. The numerical results are given in the last part and the analysis between theoretic and experimental results are obtained.