Error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space
Provides rigorous error analysis for a variational time-stepping method combined with mixed finite elements, benefiting researchers in numerical analysis of transient transport processes.
The paper develops and analyzes a combined continuous Galerkin-Petrov time discretization and mixed finite element spatial discretization for parabolic problems, proving existence, uniqueness, and optimal-order error estimates with explicit convergence rates, validated by numerical experiments.
Variational time discretization schemes are getting of increasing importance for the accurate numerical approximation of transient phenomena. The applicability and value of mixed finite element methods (MFEM) in space for simulating transport processes have been demonstrated in a wide class of works. We consider a family of continuous Galerkin-Petrov time discretization schemes that is combined with a mixed finite element (MFE) approximation of the spatial variables. The existence and uniqueness of the semidiscrete approximation and of the fully discrete solution are established. For this, the Banach-Nečas-Babuška theorem is applied in a non-standard way. Error estimates with explicit rates of convergence are proved for the scalar and vector-valued variable. An optimal order estimate in space and time is proved by duality techniques for the scalar variable. The convergence rates are analyzed and illustrated by numerical experiments, also on stochastically perturbed meshes.