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Space-time finite element methods stabilized using bubble function spaces.

Ioannis Toulopoulos^1,2

¹Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria.

Applicable Analysis

|June 16, 2020

Summary

This study introduces a stabilized space-time finite element method for linear parabolic evolution problems. The method ensures stability and provides accurate error estimates for complex simulations.

Keywords:

Parabolic initial-boundary value problems bubble stabilization optimal convergence rates space-time finite element methods

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Area of Science:

Numerical analysis
Computational mathematics
Partial differential equations

Background:

Linear parabolic evolution problems are fundamental in various scientific and engineering fields.
Existing numerical methods often face challenges in simultaneously discretizing space and time.
Stabilization techniques are crucial for improving the accuracy and reliability of numerical solutions.

Purpose of the Study:

To develop and analyze a stabilized space-time finite element method for linear parabolic evolution problems.
To provide a unified framework for space-time discretization using finite element techniques.
To rigorously prove the stability and derive error estimates for the proposed method.

Main Methods:

A space-time variational formulation is employed for unified discretization.
Stabilization terms are incorporated using classical bubble spaces.
Stability is proven with respect to a mesh-dependent norm.
A priori error estimates are derived.

Main Results:

The stabilized space-time finite element method demonstrates stability.
Theoretical error estimates are established for the discretization process.
Numerical examples validate the derived theoretical estimates.
The method allows for simultaneous discretization in both space and time.

Conclusions:

The proposed stabilized space-time finite element method is effective for linear parabolic evolution problems.
The method offers a robust and accurate approach to numerical simulation.
The theoretical analysis and numerical results confirm the method's validity and performance.