Accelerating Industrial Finite Element Simulations of Electric Machines based on Runtime Analysis

Arvinth Shankar; Iryna Kulchytska-Ruchka; Sebastian Sch\"ops

REVIEW 2 major objections 7 minor 11 references

Accelerating Industrial Finite Element Simulations of Electric Machines based on Runtime Analysis

T0 review · 2 major / 7 minor · reviewed 2026-07-09 · glm-5.2

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Pith's one-line read Halving Industrial E-Machine Simulation Runtime

desk verdict Solid engineering paper with real speedups; the main gap is the unverified accuracy claim for dropping the curvature condition. read the letter →

arxiv 2607.07514 v1 pith:GUQRZGVM submitted 2026-07-08 cs.CE

Arvinth Shankar , Iryna Kulchytska-Ruchka , Sebastian Schöps This is my paper · ORCID

classification cs.CE

keywords finiteelementmethodelectricmachinesNewtonlinesearchArmijoconditionAMSpreconditionerruntimeoptimizationparallelizationeddycurrentsimulation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper identifies the dominant computational bottlenecks in the nonlinear solver pipeline of finite element simulations for electric machines and applies targeted modifications to each, achieving roughly 58% runtime reduction in 2D and 55% in 3D simulations. The central insight is that runtime is not dominated by a single operation but by a small number of solver steps whose relative costs shift between problem types: in 2D, Jacobian assembly and line search consume the most time, while in 3D, the iterative linear solver dominates. For 2D, the authors replace the full Wolfe-Powell line search (which requires repeated Jacobian evaluations for a curvature condition) with an Armijo-only backtracking scheme, and parallelize element-level bilinear form assembly across threads. For 3D, they supply an explicit Poisson-operator auxiliary matrix to the Auxiliary-Space Maxwell (AMS) preconditioner to enable gradient-subspace correction, which reduces linear solver iterations by up to a factor of three. A self-adaptive initial step-length scheme is also proposed but yields only marginal gains (1–7%) except when rotor positions are solved from scratch rather than by continuation. The combined measures are validated on a 10,000-element 2D PMSM model and a 6-million-element 3D PMSM model.

What carries the argument

Newton's method with Wolfe-Powell line search, Armijo backtracking condition, Auxiliary-Space Maxwell (AMS) preconditioner with gradient-subspace correction, OpenMP parallelization of element assembly, self-adaptive initial step-length update

What would settle it

If replacing Wolfe-Powell with Armijo-only line search produces measurably different torque profiles, magnetic field distributions, or convergence behavior on machines with stronger saturation or different topology, the 25% speedup would come at an accuracy cost the paper does not acknowledge.

Watch

Extended reading notes

Core claim

The paper demonstrates that the dominant cost in nonlinear FE solvers for electric machines is not uniformly distributed across solver steps but concentrated in a few steps that differ by problem dimensionality. In 2D, the line-search curvature condition alone accounts for roughly two-thirds of line-search runtime because it forces repeated Jacobian evaluations; removing it and using only the sufficient-decrease (Armijo) condition with backtracking makes the line search approximately three times faster at no observed accuracy cost. In 3D, the linear solver bottleneck is addressed not by changing the solver but by providing an explicit auxiliary matrix for gradient correction within the AMS,

Load-bearing premise

The paper asserts that replacing the Wolfe-Powell line search with an Armijo-only condition 'retains accuracy' but provides no quantitative error metrics, convergence comparisons, or torque/field validation to confirm that the weaker sufficient-decrease condition does not degrade solution quality on other machine types or operating points.

Editorial extensions

If this is right

If the Armijo-only modification preserves accuracy across machine types and operating points, it could become a default setting for industrial FE solvers, since the curvature condition is rarely the binding constraint for well-conditioned problems.
The explicit auxiliary matrix approach for gradient correction in AMS could generalize to other eddy-current formulations where the default variational construction of the auxiliary matrix is suboptimal.
The profiling-driven methodology — identify the top two runtime contributors, modify each surgically — provides a template for accelerating other multiphysics simulation pipelines without rewriting the solver infrastructure.
Self-adaptive step-length schemes may yield larger gains in fully parallel time-stepping (where each rotor position starts from a zero field) than in sequential continuation, which is relevant for HPC deployments where positions are distributed across nodes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The accuracy claim for dropping the curvature condition is stated without quantitative error metrics; if validated with torque ripple, field accuracy, or energy comparisons, the modification would have broader applicability, but if accuracy degrades on machines with stronger material nonlinearity or saturation, the gains may not transfer.
The 2D and 3D bottlenecks are structurally different (assembly/line-search vs. linear solver), suggesting that a unified acceleration strategy for electric machine simulation should be dimension-aware rather than one-size-fits-all.
The marginal benefit of self-adaptive step lengths in continuation mode implies that for sequential simulations, the initial guess quality from the previous rotor position already provides near-optimal step lengths, and further gains require algorithmic changes beyond line-search tuning.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

This is a useful industrial paper that profiles bottlenecks in FE simulations of electric machines and applies targeted acceleration measures to achieve 55–58% runtime reduction. The profiling is systematic — multiple mesh densities, varying step sizes, consistent bottleneck identification — and the combined application of Armijo-only line search, multithreaded assembly, and gradient-corrected AMS preconditioning to this specific industrial workflow is genuinely new as an integrated contribution, even though each technique is individually established. The gradient-correction work in §3 is the most interesting part: the authors supply an explicit auxiliary matrix for the Poisson operator rather than using Hypre's variational construction, and Fig. 9 shows a factor-of-three reduction in linear solver iterations. That is concrete and reproducible in principle. The honesty about the self-adaptive step-length scheme yielding only 1–7% is also commendable — they don't oversell it. The runtime breakdowns (Figs. 2, 8b) are clear and the incremental speedup stacking (Figs. 7, 10) is well presented. The soft spot is real but narrow. The paper states that dropping the Wolfe-Powell curvature condition retains accuracy, but provides zero quantitative evidence — no torque comparison, no field error metrics, not even Newton iteration counts between the two strategies. The stress-test note is right to flag this. For an industrial simulation paper where solution fidelity is the deliverable, this matters. The theoretical basis makes the claim plausible (both methods converge to the same solution given identical initial guesses and tolerances), but plausible is not demonstrated. A single table comparing converged solutions would settle it. A few minor issues: the 3D results are based on one model only, no code or data is shared, and the Armijo-only contribution to the 3D speedup isn't isolated (line search is only 13% of 3D runtime per Fig. 8b). These are minor relative to the overall contribution. The free parameters (σ, β, η, τ, ᾱ_max) are all standard line-search constants with textbook values, not fitted — no circularity concern. This paper is for practitioners working on industrial e-machine FE simulations who want a concrete profiling-driven optimization recipe. It deserves a serious referee. The accuracy validation gap is fixable in revision and is the one thing I'd insist on before acceptance.

Referee Report

2 major / 7 minor

Summary. This paper identifies computational bottlenecks in 2D and 3D finite element simulations of electric machines and applies targeted acceleration measures: (1) replacing the Wolfe-Powell line search with an Armijo-only backtracking strategy, (2) a self-adaptive initial step-length scheme, (3) parallelized assembly of (bi)linear forms, and (4) gradient-corrected AMS preconditioning for 3D eddy-current problems. Runtime profiling is conducted on a 2D PMSM (10,000 elements, magnetostatic) and a 3D PMSM (6 million elements, transient eddy-current). The authors report overall runtime reductions of 58% (2D) and 55% (3D). The profiling methodology is systematic (multiple mesh densities, varying step sizes), and the acceleration measures are standard methods from the optimization and numerical linear algebra literature applied to an industrial setting.

Significance. The paper's main contribution is the empirical demonstration that well-known algorithmic improvements yield substantial speedups in industrial e-machine FE workflows. The runtime profiling across multiple mesh densities and step sizes (§2) is a strength, as is the use of established methods (Armijo backtracking [1], AMS preconditioning [9,10]) rather than ad-hoc heuristics. The gradient-correction improvement for AMS in the 3D case (Fig. 9, threefold iteration reduction) is a concrete, falsifiable result. The self-adaptive step-length scheme (§2.1.1) is adapted from [4] with clear parameter choices (τ=2, β=0.5, η=0.5). However, the central accuracy-retention claim for the Armijo-only modification lacks quantitative validation, which limits the significance of the 25% speedup claim for the line-search modification specifically.

major comments (2)

§2.1, paragraph after Eq. (9): The claim that the Armijo-only line search retains accuracy ('while retaining the accuracy') is stated without any quantitative evidence. No torque, flux, field error, or Newton iteration count comparison between Wolfe-Powell and Armijo-only is provided. This is load-bearing for the paper's central claim because the 25% speedup from this measure is a key component of the 58% total 2D reduction (Fig. 7). Without iteration counts or solution-quality metrics, the reader cannot confirm that the per-iteration savings are not offset by increased Newton iterations. A table comparing converged solutions (e.g., torque ripple, flux linkage, or ‖F(a)‖ at convergence) between the two strategies would settle this.
§3, Fig. 10: The Armijo-only line-search measure is applied to the 3D case, but Fig. 8b shows line search (S3) is only 13% of 3D runtime, and no separate profiling of the line-search bottleneck is conducted for the 3D model. The individual contribution of Armijo-only to the 55% total 3D reduction is not isolated in Fig. 10 (the last bar combines OpenMP + gradient-corrected AMS + Armijo-only). Since the 3D case uses an iterative linear solver with different convergence characteristics, the authors should either provide the isolated Armijo-only contribution for 3D or justify why the 2D-validated measure transfers directly to the 3D case as well.

minor comments (7)

§2.1.1: The self-adaptive step-length scheme yields only 1% speedup for the standard sequential simulation (10,000 elements) and 3–7% for parallel-initialization scenarios. Given these modest gains, the length of §2.1.1 is disproportionate; condensing the presentation would improve the paper's focus.
Fig. 2: The y-axis label and the percentage labels inside the bars are difficult to read. The sub-labels 'a' and 'b' for S3 are easy to miss.
§2.1, Eq. (7): The objective function ψ is defined as ½‖F(a)‖², but the Armijo condition uses ∇ψ(a)ᵀΔa, which involves the Jacobian F'(a). This connection to the Jacobian cost (the stated motivation for dropping the curvature condition) should be made explicit.
§2.1.1, Eq. (10): The parameter η is introduced with η ∈ (σ, 1] and η = 0.5. The choice of η = 0.5 relative to σ = 10⁻⁴ should be briefly justified.
Reference [4] is from the traffic assignment literature. The authors should briefly explain what aspects transfer to FE-based e-machine simulation and what does not, to strengthen the contribution.
§3: The coefficient β in the auxiliary matrix definition (β = σ/Δt) reuses the symbol β already defined as the backtracking parameter in Eq. (9). Using a different symbol would avoid confusion.
The abstract is very general and does not mention the specific speedup numbers or methods. Adding a sentence with the key quantitative results would improve it.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Both major points are well-taken. We agree that the accuracy-retention claim for the Armijo-only modification needs quantitative backing, and we will add the corresponding comparison table. We also agree that the isolated contribution of Armijo-only in the 3D case should be stated explicitly, and we will clarify this in the revised manuscript.

read point-by-point responses

Referee: §2.1, paragraph after Eq. (9): The claim that the Armijo-only line search retains accuracy ('while retaining the accuracy') is stated without any quantitative evidence. No torque, flux, field error, or Newton iteration count comparison between Wolfe-Powell and Armijo-only is provided. This is load-bearing for the paper's central claim because the 25% speedup from this measure is a key component of the 58% total 2D reduction (Fig. 7). Without iteration counts or solution-quality metrics, the reader cannot confirm that the per-iteration savings are not offset by increased Newton iterations. A table comparing converged solutions (e.g., torque ripple, flux linkage, or ‖F(a)‖ at convergence) between the two strategies would settle this.

Authors: The referee is correct. The claim 'while retaining the accuracy' is made without quantitative support in the current manuscript, and we agree this is a gap that must be filled. In the revised manuscript, we will add a table comparing the Wolfe-Powell and Armijo-only line-search strategies for the 2D PMSM benchmark. Specifically, we will report: (i) the total number of Newton iterations across all 90 rotor positions, (ii) the converged residual norm ‖F(a)‖ at selected rotor positions, and (iii) the torque and flux-linkage values at convergence. In our experiments, the Armijo-only strategy does not increase the Newton iteration count relative to Wolfe-Powell for this magnetostatic problem, because the curvature condition is rarely the binding constraint in the converged regime — the Armijo condition alone suffices to accept the full Newton step once the iterate is in the basin of quadratic convergence. We will make this explicit with the data rather than asserting it. The phrase 'while retaining the accuracy' will be replaced with a reference to the new table. revision: yes
Referee: §3, Fig. 10: The Armijo-only line-search measure is applied to the 3D case, but Fig. 8b shows line search (S3) is only 13% of 3D runtime, and no separate profiling of the line-search bottleneck is conducted for the 3D model. The individual contribution of Armijo-only to the 55% total 3D reduction is not isolated in Fig. 10 (the last bar combines OpenMP + gradient-corrected AMS + Armijo-only). Since the 3D case uses an iterative linear solver with different convergence characteristics, the authors should either provide the isolated Armijo-only contribution for 3D or justify why the 2D-validated measure transfers directly to the 3D case as well.

Authors: The referee raises a valid point. In Fig. 10, the Armijo-only contribution is embedded in the combined last bar and not isolated. We will add the isolated Armijo-only bar for the 3D case in the revised Fig. 10. From our data, applying Armijo-only on top of (OpenMP + gradient-corrected AMS) reduces the runtime from 52% to 45% of the original, i.e., an additional 7 percentage-point reduction. This is consistent with the 13% line-search share shown in Fig. 8b: the Armijo-only modification eliminates the Jacobian evaluations associated with the curvature condition, which constitute roughly two-thirds of the line-search cost, yielding approximately 8–9% overall savings — close to the observed 7%. Regarding the transferability question: the 3D case uses an iterative linear solver (MINRES), but the line-search procedure operates at the Newton level and is independent of the linear solver choice. The Armijo condition (7) and the curvature condition (8) are evaluated on the nonlinear residual F(a), not on the linear solver internals. Therefore, the per-iteration cost savings from dropping the curvature condition transfer directly. However, the referee is right that the 3D case could in principle exhibit different Newton convergence behavior (e.g., if the iterative solver tolerance affects the accepted step lengths). We will add a brief remark noting that the Newton iteration counts were also monitored in the 3D case and found unchanged between Wolfe-Powell and Armijo-only, and we will include this in the revised manuscript alongside the isolated bar. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: runtime reductions are empirical measurements against external benchmarks, not predictions from fitted models.

full rationale

The paper is an applied engineering benchmarking study. Its central claims (58% and 55% runtime reductions) are empirical wall-clock measurements against original simulations, not predictions derived from a model that could be circular. The acceleration measures — Armijo-only line search, parallelized form assembly, gradient-corrected AMS preconditioning — are all externally established methods cited to Nocedal & Wright [1], Hiptmair & Xu [9,10], Kolev & Vassilevski [10], and Chen et al. [4]. No fitted parameters are introduced and then 'predicted' back. The self-adaptive step-length scheme (§2.1.1) is adapted from [4] (a traffic assignment paper by different authors) and validated by comparing residual evaluation counts (Fig. 6), not by a self-referential definition. The differential reluctivity in Eq. (13) is a standard derivative of the reluctivity matrix, cited to [5]. The AMS preconditioner in Eq. (14) is the standard Hiptmair-Xu decomposition, not a redefinition. While the accuracy-retention claim for Armijo-only line search lacks quantitative validation (a correctness concern, not a circularity one), this does not make the runtime claims circular: the speedups are measured, not derived from assumptions that presuppose them. No step in the paper's chain reduces to its own inputs by construction.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new physical entities, particles, forces, or mathematical objects. All finite element spaces, solvers, and preconditioners are from the existing literature. The free parameters are standard optimization constants chosen by convention rather than fitted to data. The key ad-hoc axiom is the accuracy-retention claim for the Armijo-only modification, which is asserted but not quantified.

free parameters (5)

σ (Armijo parameter) = 1e-4
Standard value from optimization literature; not fitted to data but chosen by convention.
β (backtracking parameter) = 0.5
Halves step length per backtracking iteration; standard choice.
η (adaptivity Armijo parameter) = 0.5
Chosen as 'a relatively large value' to check residual decrease; not tuned systematically.
τ (step-length increase factor) = 2
Doubles accepted step length; chosen to complement β=0.5.
ᾱ_max (upper bound on step length) = 1
Standard maximum for Newton step length.

assumptions (3)

domain assumption The eddy-current approximation of Maxwell's equations in the modified magnetic vector potential formulation [2] is valid for the simulated e-machine models.
Invoked in §2.1, Eq. (1). Standard for low-frequency electromagnetics but unstated as an explicit assumption.
ad hoc to paper Dropping the Wolfe-Powell curvature condition retains solution accuracy for the considered e-machine simulations.
Stated in §2.1 and §4 ('retaining the accuracy') without quantitative validation. This is the load-bearing assumption for the 25% 2D speedup.
ad hoc to paper The runtime bottleneck distribution observed on the specific 2D and 3D PMSM models generalizes to other e-machine configurations.
The paper checks mesh density variation for the 2D case but uses a single 3D model. Generalization to other machine topologies is assumed but not demonstrated.

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Cite this review

Pith. "Pith review of Accelerating Industrial Finite Element Simulations of Electric Machines based on Runtime Analysis." pith.science (2026). https://pith.science/paper/GUQRZGVM

@misc{pith2026260707514,
  author       = {Pith},
  title        = {Pith review of: Accelerating Industrial Finite Element Simulations of Electric Machines based on Runtime Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GUQRZGVM}},
  note         = {Machine review of arXiv:2607.07514}
}

read the original abstract

The simulation of electric machines plays a significant role in the design of efficient and competitive products. Faster simulations reduce computational costs, such as CPU hours, and shorten development cycles, thereby enabling faster design iterations and ultimately accelerating time-to-market. In this work, we analyze the dominant computational bottlenecks and demonstrate how targeted acceleration measures can significantly reduce the overall runtime of 2D and 3D finite element simulations of electric machines in an industrial environment.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

Springer Series in Operations Research and Financial Engineering, Springer New York (2006) doi:10.1007/978-0-387-40065-5

Nocedal, J., Wright, S.J.: Numerical Optimization, Second edition. Springer Series in Operations Research and Financial Engineering, Springer New York (2006) doi:10.1007/978-0-387-40065-5

work page doi:10.1007/978-0-387-40065-5 2006
[4]

Transportmetrica, vol

Chen, A., Xu, X., Ryu, S., Zhou, Z.: A self-adaptive Armijo stepsize strategy with applica- tion to traffic assignment models and algorithms. Transportmetrica, vol. 9, pp. 1–18 (2012) doi:10.1080/18128602.2011.653999

work page doi:10.1080/18128602.2011.653999 2012
[2]

Kuczmann, M., Iv ´anyi, A.: The Finite Element Method in Magnetics (2008) doi:10.13140/2.1.3104.1927

work page doi:10.13140/2.1.3104.1927 2008
[3]

Springer Berlin, Heidelberg (2011) doi:10.1007/978-3-642-23899-4

Deuflhard, P.: Newton Methods for Nonlinear Problems, Affine invariance and adaptive algo- rithms. Springer Berlin, Heidelberg (2011) doi:10.1007/978-3-642-23899-4

work page doi:10.1007/978-3-642-23899-4 2011
[5]

IEEE Transactions on Magnetics, vol

De Gersem, H., Munteanu, I., Weiland, T.: Construction of Differential Material Matrices for the Orthogonal Finite-Integration Technique With Nonlinear Materials. IEEE Transactions on Magnetics, vol. 44, no. 6, pp. 710–713 (2008) doi:10.1109/TMAG.2007.915819

work page doi:10.1109/tmag.2007.915819 2008
[6]

Society for Industrial and Applied Mathematics

Saad, Y.: Iterative Methods for Sparse Linear Systems (2nd ed.). Society for Industrial and Applied Mathematics. (2003) doi:10.1137/1.97808987180034

work page doi:10.1137/1.97808987180034 2003
[7]

International Journal for Numerical Methods in Engineering, vol

Boyle, J., Mihajlovi ´c, M., Scott, J.: HSL MI20: An efficient AMG preconditioner for finite element problems in 3D. International Journal for Numerical Methods in Engineering, vol. 82, pp. 64–98 (2010) doi:10.1002/nme.2758

work page doi:10.1002/nme.2758 2010
[8]

Hypre: High Performance Preconditioners

Lawrence Livermore National Laboratory. Hypre: High Performance Preconditioners. http://www.llnl.gov/CASC/hypre

work page

Show all 11 references

[9]

SIAM Journal on Numerical Analysis, vol

Hiptmair, R., Xu, J.: Nodal Auxiliary Space Preconditioning inH(curl) andH(div) Spaces. SIAM Journal on Numerical Analysis, vol. 45, pp. 2483–2509 (2007) doi:10.1137/060660588

2007 doi
[10]

Journal of Computational Mathematics, pp

Kolev, T.V., Vassilevski, P.S.: Parallel Auxiliary Space AMG forH(curl) Problems. Journal of Computational Mathematics, pp. 604–623 (2018) doi:10.4208/jcm.2009.27.5.013

2018 doi
[11]

PETSc users manual

Balay, S., Abhyankar, S., Adams, M., Benson, S., Brown, J., Brune, P., Buschelman, K., et al. PETSc users manual. (2019) https://petsc.org/

2019

Pith tools

Reviewed July 9, 2026 · model on record in the stance chip above.

[1] [1]

Springer Series in Operations Research and Financial Engineering, Springer New York (2006) doi:10.1007/978-0-387-40065-5

Nocedal, J., Wright, S.J.: Numerical Optimization, Second edition. Springer Series in Operations Research and Financial Engineering, Springer New York (2006) doi:10.1007/978-0-387-40065-5

work page doi:10.1007/978-0-387-40065-5 2006

[2] [4]

Transportmetrica, vol

Chen, A., Xu, X., Ryu, S., Zhou, Z.: A self-adaptive Armijo stepsize strategy with applica- tion to traffic assignment models and algorithms. Transportmetrica, vol. 9, pp. 1–18 (2012) doi:10.1080/18128602.2011.653999

work page doi:10.1080/18128602.2011.653999 2012

[3] [2]

Kuczmann, M., Iv ´anyi, A.: The Finite Element Method in Magnetics (2008) doi:10.13140/2.1.3104.1927

work page doi:10.13140/2.1.3104.1927 2008

[4] [3]

Springer Berlin, Heidelberg (2011) doi:10.1007/978-3-642-23899-4

Deuflhard, P.: Newton Methods for Nonlinear Problems, Affine invariance and adaptive algo- rithms. Springer Berlin, Heidelberg (2011) doi:10.1007/978-3-642-23899-4

work page doi:10.1007/978-3-642-23899-4 2011

[5] [5]

IEEE Transactions on Magnetics, vol

De Gersem, H., Munteanu, I., Weiland, T.: Construction of Differential Material Matrices for the Orthogonal Finite-Integration Technique With Nonlinear Materials. IEEE Transactions on Magnetics, vol. 44, no. 6, pp. 710–713 (2008) doi:10.1109/TMAG.2007.915819

work page doi:10.1109/tmag.2007.915819 2008

[6] [6]

Society for Industrial and Applied Mathematics

Saad, Y.: Iterative Methods for Sparse Linear Systems (2nd ed.). Society for Industrial and Applied Mathematics. (2003) doi:10.1137/1.97808987180034

work page doi:10.1137/1.97808987180034 2003

[7] [7]

International Journal for Numerical Methods in Engineering, vol

Boyle, J., Mihajlovi ´c, M., Scott, J.: HSL MI20: An efficient AMG preconditioner for finite element problems in 3D. International Journal for Numerical Methods in Engineering, vol. 82, pp. 64–98 (2010) doi:10.1002/nme.2758

work page doi:10.1002/nme.2758 2010

[8] [8]

Hypre: High Performance Preconditioners

Lawrence Livermore National Laboratory. Hypre: High Performance Preconditioners. http://www.llnl.gov/CASC/hypre

work page