Are there alternative discretization approaches for the MFIE that could potentially outperform the Petrov-Galerkin method in terms of computational efficiency or accuracy?
Yes, there are several alternative discretization approaches for the magnetic field integral equation (MFIE) that could potentially outperform the Petrov-Galerkin method in terms of computational efficiency or accuracy. One such approach is the use of higher-order boundary elements, which can provide better accuracy with fewer degrees of freedom compared to the lowest-order Raviart-Thomas elements used in the Petrov-Galerkin method. Higher-order elements can capture the solution's behavior more effectively, especially in regions with high gradients or singularities.
Another alternative is the use of hybrid methods that combine boundary element methods with finite element methods (FEM). These hybrid approaches can leverage the strengths of both methods, allowing for more flexible handling of complex geometries and potentially improving convergence rates. For instance, using FEM in the interior domain while applying BEM on the boundary can lead to more efficient computations, particularly in large-scale simulations.
Additionally, the implementation of adaptive mesh refinement techniques can enhance the performance of discretization methods. By dynamically refining the mesh in regions where the solution exhibits rapid changes, one can achieve higher accuracy without a significant increase in computational cost.
Finally, the use of fast boundary element methods, such as the Fast Multipole Method (FMM) or the Hierarchical Matrix (H-matrix) techniques, can significantly reduce the computational burden associated with the matrix-vector products required in BEMs. These methods can improve the efficiency of the numerical solution process, making them attractive alternatives to the traditional Petrov-Galerkin discretization scheme.