A High-Order Nodal Galerkin Formulation for the Müller Equation: Bypassing Divergence Conformity via Kernel Cancellation
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913056589611008 |
|---|---|
| author | Luo, Yao |
| author_facet | Luo, Yao |
| contents | The Müller boundary integral equation for penetrable electromagnetic scattering is conventionally discretized using divergence-conforming basis functions, a restriction inherited from the PMCHWT framework. This paper demonstrates that this constraint can be bypassed. The double-gradient operator in the Müller formulation acts on the kernel difference $φ_a - φ_i$, so that the $\mathcal{O}(R^{-3})$ hypersingularity cancels identically, reducing the operators to weakly singular $\mathcal{O}(R^{-1})$ kernels. Exploiting this cancellation, we develop a nodal, high-order Galerkin formulation using $\mathrm{P}_2$ isoparametric shape functions on curved manifolds. The surface vector field is constructed via a metric-weighted orthonormal tangent frame. The singular integrals are evaluated by Sauter--Schwab quadrature, and a Morton-ordered Block Jacobi preconditioner is introduced. By capturing the dominant near-field interactions within geometrically clustered diagonal blocks, it yields robust, superlinear GMRES convergence under extreme material and geometric parameters. Validation against semi-analytical EBCM references confirms high-order spatial accuracy and optical-theorem satisfaction to high precision. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21181 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A High-Order Nodal Galerkin Formulation for the Müller Equation: Bypassing Divergence Conformity via Kernel Cancellation Luo, Yao Computational Physics The Müller boundary integral equation for penetrable electromagnetic scattering is conventionally discretized using divergence-conforming basis functions, a restriction inherited from the PMCHWT framework. This paper demonstrates that this constraint can be bypassed. The double-gradient operator in the Müller formulation acts on the kernel difference $φ_a - φ_i$, so that the $\mathcal{O}(R^{-3})$ hypersingularity cancels identically, reducing the operators to weakly singular $\mathcal{O}(R^{-1})$ kernels. Exploiting this cancellation, we develop a nodal, high-order Galerkin formulation using $\mathrm{P}_2$ isoparametric shape functions on curved manifolds. The surface vector field is constructed via a metric-weighted orthonormal tangent frame. The singular integrals are evaluated by Sauter--Schwab quadrature, and a Morton-ordered Block Jacobi preconditioner is introduced. By capturing the dominant near-field interactions within geometrically clustered diagonal blocks, it yields robust, superlinear GMRES convergence under extreme material and geometric parameters. Validation against semi-analytical EBCM references confirms high-order spatial accuracy and optical-theorem satisfaction to high precision. |
| title | A High-Order Nodal Galerkin Formulation for the Müller Equation: Bypassing Divergence Conformity via Kernel Cancellation |
| topic | Computational Physics |
| url | https://arxiv.org/abs/2604.21181 |