Trefftz co-chain calculus

Area of Science:

• Computational mathematics
• Numerical analysis
• Electromagnetics

Background:

• Linear transmission problems are fundamental in various scientific fields.
• Discretization methods are crucial for solving these problems numerically.
• Domain decomposition offers a powerful strategy for handling complex geometries and different physical domains.

Purpose of the Study:

• To develop a unified framework for discretizing linear transmission problems.
• To couple distinct numerical methods across different spatial domains.
• To provide a robust approach for solving problems involving bounded and unbounded regions.

Main Methods:

• Utilizing a mesh-based discrete co-chain model (e.g., finite element exterior calculus) in bounded domains.
• Employing a meshless Trefftz-Galerkin approach with special solutions in unbounded domains.
• Developing a unified coupling strategy based on discrete Hodge operators.

Main Results:

• A novel method for coupling diverse discretizations across domain interfaces.
• Derivation of the resulting linear system of equations from discrete Hodge operators.
• Demonstration of the approach on a frequency-domain eddy-current problem.

Conclusions:

• The proposed unified framework effectively couples different discretization techniques.
• The method provides accurate numerical results for complex electromagnetic problems.
• This approach offers a flexible and powerful tool for solving linear transmission problems.