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

TL;DR

This paper analyzes the error and convergence of a combined continuous Galerkin-Petrov time discretization with mixed finite element methods for parabolic problems, providing explicit error estimates and numerical validation.

Contribution

It establishes existence, uniqueness, and explicit error estimates for a novel combined time-space discretization scheme for parabolic PDEs, including stochastic mesh considerations.

Findings

Proved convergence rates with explicit error bounds.

Demonstrated optimal order in space and time via duality.

Validated theoretical results with numerical experiments.

Abstract

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-Ne\v{c}as-Babu\v{s}ka 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.