Space-time finite element methods stabilized using bubble function spaces

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.