Error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space

摘要

Variational time discretization schemes are getting of increasing importancefor the accurate numerical approximation of transient phenomena. Theapplicability and value of mixed finite element methods (MFEM) in space forsimulating transport processes have been demonstrated in a wide class of works.We consider a family of continuous Galerkin-Petrov time discretization schemesthat is combined with a mixed finite element (MFE) approximation of the spatialvariables. The existence and uniqueness of the semidiscrete approximation andof the fully discrete solution are established. For this, theBanach-Ne\v{c}as-Babu\v{s}ka theorem is applied in a non-standard way. Errorestimates with explicit rates of convergence are proved for the scalar andvector-valued variable. An optimal order estimate in space and time is provedby duality techniques for the scalar variable. The convergence rates areanalyzed and illustrated by numerical experiments, also on stochasticallyperturbed meshes.