An Equilibrated a Posteriori Error Estimator for Arbitrary-Order Nédélec Elements for Magnetostatic Problems

Area of Science:

• Computational Electromagnetics
• Numerical Analysis

Background:

• Magnetostatic problems are crucial in various engineering applications.
• Accurate error estimation is vital for reliable numerical solutions, particularly with finite element methods.
• Existing a posteriori error estimators for Nédélec elements often rely on constants or complex local problems.

Purpose of the Study:

• To develop a novel, constant-free a posteriori error estimator for Nédélec elements applicable to magnetostatic problems.
• To extend existing error estimation techniques to arbitrary-degree Nédélec elements.
• To ensure the estimator is computationally efficient and easy to implement.

Main Methods:

• The proposed estimator is based on the equilibration of the magnetic field.
• It involves solving small, parallelizable local problems on single elements, faces, and small node sets.
• The method is proven to be reliable and efficient through theoretical analysis.

Main Results:

• A novel a posteriori error estimator for Nédélec elements in magnetostatic problems has been developed.
• The estimator is constant-free, providing an upper bound on the error without generic constants.
• The method is applicable to Nédélec elements of arbitrary degree and simplifies local problem solving compared to previous approaches.

Conclusions:

• The developed error estimator is reliable and efficient for magnetostatic problems using Nédélec elements of any degree.
• The constant-free nature and simplified local problem structure offer significant advantages for computational efficiency.
• Numerical examples confirm the theoretical findings and demonstrate the practical utility of the estimator.