Optimal error estimates of two mixed finite element methods for parabolic integro-differential equations with nonsmooth initial data
The Cauchy-Dirichlet mixed problem, with homogeneous boundary condition and L\(^{2}\)-nonsmooth initial data, for parabolic linear integro-differential equations is investigated for finite element approximation of the solution. Firstly, some a priori bounds and regularity results are discussed in order to be used in the proof of the main theorems. A new mixed finite element Galerkin method is proposed proving the error estimate with optimal convergence rate, as the main result of the paper. Introducing the extended mixed Ritz-Volterra projections, the corresponding error estimates are discussed in order to be used for obtaining the proof of the main result. A superconvergence property is obtained as a consequence of this main result. Moreover, the L\(^{2}\)-error estimates for the standard mixed method with optimal order of convergence are obtained under general settings. The optimal L\(^{2}\)-error estimates are derived without using parabolic type duality techniques or resolvent operators, being an improvement of the known results and an extension from smooth to nonsmooth initial data.
scientific article; zbMATH DE number 6176981
scientific article