Finite Element Methods For Two Classes Of Multi-term Time-fractional Partial Differential Equations

In this paper,we first give the definition of several common types of fractional inte-grals and derivatives,then we introduce the Sobolev space and some common inequalities,and basic theories about finite element methods.And then,based on the classical L1 scheme discrete time-fractional derivative operator,and using bilinear finite element and linear triangular finite element respectively to approximate the spatial direction of multi-term time-fractional wave equation and a class of multi-term time-fractional Cattaneo equation,an unconditionally stable fully-discrete approximate scheme for two kinds of equations is established.Furthermore,rigorous proofs are given here for convergence in L2-norm and superclose properties in H1-norm with order O(h2 + ?3-a)(1<a<2),where h and ? are the spatial size and time step,respectively.At the same time,theoretical analysis of global superconvergence in H1-norm is derived by using interpolation postpro-cessing technique.At last,with the help of numerical examples to verify the correctness of theoretical analysis.