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Isogeometric multi-patch preconditioner for time-dependent electromagnetic problems
Bernard Kapidani, Gabriele Loli, Giancarlo Sangalli, Mattia Tani, Rafael Vázquez
TL;DR
Inverting 1-form mass matrices poses computational challenges in multi-patch Maxwell discretizations. This paper combines single-patch preconditioners with additive Schwarz, and experiments on realistic problems confirm the theoretical findings and practical effectiveness of the proposed methods.
Problem
Efficiently solving linear systems from 1-form mass matrices is a central challenge in Maxwell discretizations spanning multi-patch domains.
Source: §1 · p. 2Method
The paper preconditions 1-form mass matrices with single-patch solvers, then uses them as local solvers in an additive Schwarz framework for multi-patch domains.
Source: §1 · p. 2, §5 · p. 23Results
Numerical experiments on realistic problems confirmed the theoretical findings and demonstrated practical effectiveness of the proposed methods.
Source: §5 · p. 23Takeaways & Limitations
The single-patch preconditioners and additive Schwarz framework facilitate highly parallel implementation, improving preconditioning efficiency.
Source: §5 · p. 23The analysis assumes magnetic permeability and electric permittivity are uniformly positive and bounded scalar fields.
Source: §1 · p. 2
Abstract
from arXiv · showhide
We build upon recent developments in high-order spline-based geometric methods for the three-dimensional initial boundary value problem of Maxwell's equations. Previous schemes were limited to single-patch geometries. To address the computational challenges associated with inverting mass matrices in multi-patch settings, we extend existing techniques for 0-forms mass and stiffness matrices to 1-forms mass matrices. Our approach leverages the tensor-product structure of spline spaces to construct a spectrally equivalent preconditioner for the 1-forms mass matrices. Specifically, we design an efficient preconditioner for single-patch domains that is robust with respect to both mesh size and spline degree. In the multi-patch case, this methodology is extended by integrating the single-patch preconditioner with an additive Schwarz approach, ensuring robustness and scalability with respect to the number of patches. We analyze the computational complexity of the proposed preconditioner and validate its robustness and efficiency through comprehensive numerical experiments.
1 Introduction
The paper targets efficient solution of 1-form mass-matrix systems arising in the Maxwell discretization introduced in [1], proposing preconditioners for single- and multi-patch domains. It builds on isogeometric differential-form methods and their discrete de Rham complexes.
Problem setting
- The target problem is Maxwell’s initial boundary value problem on a bounded three-dimensional space-time domain.The fields are represented using differential forms, with uniformly positive and bounded permeability and permittivity coefficients.Source: §1 · pp. 1–2
Background
- Isogeometric analysis represents fields with splines and supports discrete de Rham complexes for tensor-product B-splines, enabling Galerkin discretizations of Maxwell’s equations [2, 3, 4, 5, 6, 7].The framework has also been applied to plasma physics [8] and pointwise divergence-free fluid methods.Source: §1 · p. 2
Related work
- Prior multi-patch work combined isogeometric de Rham complexes with discontinuous Galerkin methods to ensure compatibility across conforming patches.Earlier B-spline approaches used a dual staggered grid or adapted the de Rham complex for mimetic discretization.Source: §1 · p. 2
- The dual-complex method in [20] avoids explicit dual-mesh construction through a polynomial-degree transformation and discretizes Hodge stars to connect primal and dual complexes.Its convergence was analyzed for elliptic problems in [20], and [1] used the structure for a high-order explicit-in-time Maxwell approximation.Source: §1 · p. 2
2 Discrete spline differential forms and formulations
In 3D, the paper sets up spline differential-form spaces on single- and multi-patch domains and presents continuous and symmetric discontinuous Galerkin formulations. It identifies how their mass matrices and energy properties depend on the formulation.
2 Discrete spline differential forms and formulations
- The exterior derivative maps k-forms to (k + 1)-forms and satisfies d ◦ d = 0 in the L2 differential-form setting.The setup focuses on domains Ω ⊂ R3 and defines L2Λk(Ω) as the completion of smooth k-forms.Source: §2 · p. 2
- For perfect electrical conductor boundaries, the electric field and magnetic flux belong to zero-trace spaces, while the magnetic field and electric flux belong to unconstrained spaces.The discrete fields are selected from the corresponding discrete subspaces.Source: §2 · p. 3
2.1.1 Single patch
- Single-patch discrete 1-forms use tensor-product B-spline spaces, mapped from the parametric domain by a pullback involving the geometry Jacobian.The single-patch map is assumed bi-Lipschitz, elementwise smooth, and orientation-preserving.Source: §2.1.1 · p. 4
2.1.2 Multi-patch
- Multi-patch HΛ1-conforming spaces merge coincident interface degrees of freedom when trace spaces match, accounting for their orientations.For vanishing boundary conditions, patch-interface basis functions are retained.Source: §2.1.2 · p. 5
2.2 Continuous Galerkin multi-patch formulation
- The continuous Galerkin semi-discretization requires inversion of the 1-form mass matrix under explicit time marching and conserves energy [1].It uses conforming spline spaces and the specified discrete Hodge–star operators.Source: §2.2 · pp. 5–6
2.3 Symmetric discontinuous Galerkin formulation
- The symmetric discontinuous Galerkin formulation imposes no inter-patch continuity, making its mass matrices block diagonal.With central numerical fluxes, the semi-discrete scheme preserves electromagnetic energy.Source: §2.3 · p. 6
3 Preconditioning
This section develops tensor-product preconditioners for 1-forms mass matrices, extending single-patch methods to discontinuous and continuous multi-patch formulations, with additive Schwarz for the continuous case. It examines their robustness, scalability, and computational costs, including opportunities for reuse and parallel execution.
3.1 Single patch preconditioner
- The 1-forms preconditioner is spectrally equivalent to Mϵ with constants independent of the mesh and spline degree, yielding uniformly bounded PCG iterations.The condition-number bound tightens as A more closely approximates C.Source: §3.1 · pp. 7–8
3.1.1 Block diagonal preconditioner
- Choosing A = I gives a block-diagonal preconditioner whose Kronecker-structured systems can be solved efficiently.When F is the identity and ϵ = 1, this preconditioner coincides with Mϵ.Source: §3.1.1 · p. 8
3.1.2 Block Kronecker preconditioner
- For 2.5-dimensional geometries, the block Kronecker preconditioner uses separable approximations to address the reduced mass-matrix structure.It can also be effective when the off-diagonal entries Ck,ℓ are relatively small; diagonal coefficient entries use an iterative separable approximation, while off-diagonal entries use rank-1 CPD.Source: §3.1.2 · pp. 9–10
3.2 Symmetric discontinuous Galerkin preconditioner
- For the discontinuous Galerkin formulation, the mass matrix is block diagonal across patches, so single-patch preconditioners extend the established robustness to the multi-patch case.Each diagonal block corresponds to one patch’s mass matrix.Source: §3.2 · pp. 11–12
3.3 Continuous Galerkin multi-patch preconditioner
- For continuous multi-patch systems, additive Schwarz combines single-patch preconditioners on local spaces spanned by basis functions nonzero on each patch.The condition-number bound is independent of mesh sizes, patch count, and maximal adjacency and interaction numbers, but may depend on spline degree, geometry, and ϵ; supports can also create interactions between nonadjacent patches.Source: §3.3 · pp. 12–13
- Preconditioned iteration counts are mesh-independent, may depend on spline degree, and the condition number depends on maximum patch adjacency rather than total patch count.The stability constants are mesh-independent but may depend on p, F, and ε.Source: §3.3 · pp. 14–15
- The multi-patch results also hold when local solvers are replaced by spectrally equivalent preconditioners such as PI or PC.The established spectral equivalence preserves stable decomposition and local stability.Source: §3.3 · p. 15
3.4 Computational complexity
- Preconditioner setup is performed once in a time-marching scheme, and applying patchwise preconditioners supports a highly parallelizable implementation.The same preconditioner can be reused because the matrices remain the same at each time step, making setup negligible relative to repeated application.Source: §3.4 · pp. 15–16
- Single-patch preconditioner setup costs O(p^3N_dof^(1/3) + N_dof), while application costs O(pN_dof); PC’s dense eigenvector operations cost O(N_dof^(4/3)) theoretically but are fast in practice.PC setup also computes variable approximations at O(p^3N_dof) cost and generalized eigendecompositions at O(N_dof) cost.Source: §3.4 · pp. 15–16
4 Numerical results
PCG experiments for systems associated with Mϵ (ϵ = 1) assess the robustness and computational efficiency of PI and PC preconditioners. Across single- and multi-patch tests, both show robust iteration counts, with PC generally requiring fewer iterations; performance remains robust on the nonorthogonal twisted cable geometry.
4.1 Example 1: Single patch preconditioners
- PCG iteration counts stayed below 24 for PI and 11 for PC across spline degrees and subdivisions, with about 6.6 million degrees of freedom at the finest single-patch level.The results indicate robustness to spline degree and subdivision count; PC outperformed PI.Source: §4.1 · p. 16, Table 2 · p. 17
- PC application time appeared to scale as Ndof despite the fast diagonalization method's O(Ndof^4/3) complexity, while PI setup was dominated by O(p^3 Ndof^1/3) univariate mass-matrix assembly.The O(Ndof) diagonal-scaling cost did not dominate PI setup; other timings aligned with theoretical complexities.Source: §4.1 · p. 18, Figure 2 · p. 18
4.2 Example 2: Continuous Galerkin multi-patch preconditioners
- PCG iteration counts were practically independent of sector and patch count, robust to subdivisions, and only mildly dependent on spline degree across the 2- to 5-sector continuous Galerkin tests.The largest problem had approximately 33 million degrees of freedom, and iteration counts remained particularly low with PC as the local solver.Source: §4.2 · p. 18, Table 3 · p. 20
- Multi-patch setup and PCG application wall-clock times exhibited scaling behavior consistent with the single-patch tests.The timings were reported for the ITER-inspired domains under the continuous Galerkin formulation.Source: §4.2 · p. 18, Figure 4 · p. 21
4.3 Example 3: Discontinuous Galerkin preconditioners for the ITER tokamak
- PC consistently required fewer PCG iterations than PI in the five-sector symmetric discontinuous Galerkin ITER test, while both were robust to spline degree and subdivisions.The reported quantity is the maximum iteration count across all patches; wall-clock trends were consistent with the preceding examples.Source: §4.3 · p. 22, Table 4 · p. 22, Figure 5 · p. 22
4.4 Example 4: Discontinuous Galerkin preconditioners for the twisted cable
- On the nonorthogonal twisted cable, PI and PC had very similar, robust iteration counts, with PC retaining a slight advantage despite a slightly higher count than in orthogonal cases.The largest problem had approximately 22 million degrees of freedom; wall-clock trends were consistent with previous tests.Source: §4.4 · pp. 22–23, Table 5 · p. 23
5 Conclusions
The paper introduces preconditioners for 1-form mass matrices arising in explicit Maxwell time integration, first for single-patch domains and then for multi-patch domains using additive Schwarz. Numerical experiments on realistic problems support the theoretical findings and demonstrate practical effectiveness.
5 Conclusions
- Two spectrally equivalent single-patch preconditioners are robust and computationally efficient for 1-form mass matrices in explicit Maxwell time integration.The multi-patch extension uses these preconditioners as local solvers within an additive Schwarz framework, remaining efficient, robust, and scalable with the total number of patches.Source: §5 · p. 23
- The single-patch preconditioners and additive Schwarz framework facilitate highly parallel implementation, improving preconditioning efficiency.Numerical experiments on realistic problems confirmed the theoretical findings and demonstrated practical effectiveness.Source: §5 · p. 23