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Scalable solvers for complex electromagnetics problems

Santiago Badia, Alberto F. Martín, Marc Olm

TL;DR

The paper tackles robust, scalable solutions for 3D Maxwell problems with high material contrasts by extending BDDC to curl-conforming spaces using a physics-based partition of interface objects. A change of basis and a perturbation-augmented preconditioner maintain robustness without spectral solvers, while a relaxed rpbbddc variant handles truly heterogeneous materials through averaged coefficients. Extensive numerical experiments across homogeneous, multi-material, and heterogeneous regimes—including realistic 3D cases—demonstrate excellent weak scalability and robustness to coefficient jumps. The approach preserves the simplicity of standard BDDC and is implemented in FEMPAR, enabling scalable solvers for large-scale electromagnetics simulations.

Abstract

In this work, we present scalable balancing domain decomposition by constraints methods for linear systems arising from arbitrary order edge finite element discretizations of multi-material and heterogeneous 3D problems. In order to enforce the continuity across subdomains of the method, we use a partition of the interface objects (edges and faces) into sub-objects determined by the variation of the physical coefficients of the problem. For multi-material problems, a constant coefficient condition is enough to define this sub-partition of the objects. For arbitrarily heterogeneous problems, a relaxed version of the method is defined, where we only require that the maximal contrast of the physical coefficient in each object is smaller than a predefined threshold. Besides, the addition of perturbation terms to the preconditioner is empirically shown to be effective in order to deal with the case where the two coefficients of the model problem jump simultaneously across the interface. The new method, in contrast to existing approaches for problems in curl-conforming spaces does not require spectral information whilst providing robustness with regard to coefficient jumps and heterogeneous materials. A detailed set of numerical experiments, which includes the application of the preconditioner to 3D realistic cases, shows excellent weak scalability properties of the implementation of the proposed algorithms.

Scalable solvers for complex electromagnetics problems

TL;DR

The paper tackles robust, scalable solutions for 3D Maxwell problems with high material contrasts by extending BDDC to curl-conforming spaces using a physics-based partition of interface objects. A change of basis and a perturbation-augmented preconditioner maintain robustness without spectral solvers, while a relaxed rpbbddc variant handles truly heterogeneous materials through averaged coefficients. Extensive numerical experiments across homogeneous, multi-material, and heterogeneous regimes—including realistic 3D cases—demonstrate excellent weak scalability and robustness to coefficient jumps. The approach preserves the simplicity of standard BDDC and is implemented in FEMPAR, enabling scalable solvers for large-scale electromagnetics simulations.

Abstract

In this work, we present scalable balancing domain decomposition by constraints methods for linear systems arising from arbitrary order edge finite element discretizations of multi-material and heterogeneous 3D problems. In order to enforce the continuity across subdomains of the method, we use a partition of the interface objects (edges and faces) into sub-objects determined by the variation of the physical coefficients of the problem. For multi-material problems, a constant coefficient condition is enough to define this sub-partition of the objects. For arbitrarily heterogeneous problems, a relaxed version of the method is defined, where we only require that the maximal contrast of the physical coefficient in each object is smaller than a predefined threshold. Besides, the addition of perturbation terms to the preconditioner is empirically shown to be effective in order to deal with the case where the two coefficients of the model problem jump simultaneously across the interface. The new method, in contrast to existing approaches for problems in curl-conforming spaces does not require spectral information whilst providing robustness with regard to coefficient jumps and heterogeneous materials. A detailed set of numerical experiments, which includes the application of the preconditioner to 3D realistic cases, shows excellent weak scalability properties of the implementation of the proposed algorithms.

Paper Structure

This paper contains 25 sections, 2 theorems, 36 equations, 18 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Domain partition
  4. Finite Element spaces
  5. Objects
  6. Physics-Based BDDC
  7. Change of basis
  8. Preconditioner
  9. Perturbed pbbddc preconditioner
  10. Relaxed pbbddc
  11. Implementation aspects
  12. Coarse edge partition
  13. Change of basis
  14. bddc constraints
  15. Building ${ {\Theta}^r_{\rm pb}}$
  16. ...and 10 more sections

Key Result

Proposition 4.1

The expression $R_i \widehat{\mathcal{Q}} \boldsymbol{u} = {\mathcal{{Q}}}_i \boldsymbol{u}^i$ holds, where $\boldsymbol{u} \in \widehat{X}_h$ and $\boldsymbol{u}^i \in {X}_h^i$.

Figures (18)

  • Figure 1: Standard (old) and new basis dof for 3D hexahedra second order edge fe over $E$. Additional depicted dof values are affected by the change of basis for $E$, while the remaining dof are invariant under the change of basis.
  • Figure 2: Partitions for scalar coefficients described by $\log(\alpha)= 3 \sin( 3\pi x)$ and $\log(\beta)= 3 \sin( 3\pi y)$ with an initial $3\times 3\times 3$ partition of the unit cube.
  • Figure 3: Aggregation of cells into subsets based on $log(\beta)= 3 \sin( 3\pi y)$ with threshold $r=10^3$.
  • Figure 4: Weak scalability results for first order edge fe with a constant distribution of materials for different local problem $\frac{H}{h}$ sizes. Subdomain problem sizes in \ref{['fig-homosizes']} are given for comparison purposes against the coarse problem size.
  • Figure 5: Weak scalability results for different order edge fe with a constant distribution of materials and a local problem $\frac{H}{h}=10$. Subdomain problem sizes in \ref{['fig-homord_sizes']} are given for comparison purposes against the coarse problem size.
  • ...and 13 more figures

Theorems & Definitions (7)

  • Remark 2.1
  • Remark 3.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Remark 4.3