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

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Scientific Computing

Background:

• Variational time discretization schemes are crucial for accurately simulating transient phenomena.
• Mixed finite element methods (FEM) are valuable for modeling transport processes in various applications.
• Existing methods require robust numerical schemes for both spatial and temporal approximations.

Purpose of the Study:

• To develop and analyze a novel family of continuous Galerkin-Petrov time discretization schemes.
• To combine these schemes with mixed FEM for spatial approximation in transient simulations.
• To establish theoretical foundations and error estimates for the proposed numerical methods.

Main Methods:

• Application of the Banach-Nečas-Babuška theorem in a non-standard manner to prove existence and uniqueness.
• Development of continuous Galerkin-Petrov time discretization.
• Utilizing mixed finite element methods for spatial discretization.
• Employing duality techniques for optimal order error estimation.

Main Results:

• Existence and uniqueness of semidiscrete and fully discrete solutions are rigorously established.
• Explicit rates of convergence for error estimates are derived for scalar and vector-valued variables.
• Optimal order error estimates in both space and time are proven for the scalar variable using duality arguments.
• Numerical experiments validate the convergence rates, including on stochastically perturbed meshes.

Conclusions:

• The proposed combination of continuous Galerkin-Petrov time schemes and mixed FEM provides a robust framework for transient phenomena simulation.
• The theoretical error estimates offer guarantees on the accuracy and convergence of the numerical solutions.
• The findings contribute to the advancement of accurate and efficient numerical methods in computational science and engineering.