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arxiv: 1907.06991 · v1 · pith:FDJA336Mnew · submitted 2019-07-15 · 🧮 math.NA · cs.NA

Adaptive Flux-Only Least-Squares Finite Element Methods for Linear Transport Equations

Qunjie Liu , Shun Zhang This is my paper

Pith reviewed 2026-05-24 21:40 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords least-squares finite element methodflux-only formulationlinear transport equationdiscontinuous solutionsadaptive mesh refinementRaviart-Thomas elementsa posteriori error estimates
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The pith

Flux-only least-squares finite element methods for linear transport equations eliminate the solution variable to approximate discontinuous solutions with fewer degrees of freedom.

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

The paper develops two flux-only LSFEMs for linear hyperbolic transport problems whose solutions can jump. It introduces a flux variable to separate continuity requirements and then uses Raviart-Thomas elements to remove the solution from the discrete system entirely. The resulting methods require fewer unknowns than the earlier mixed formulation while still delivering existence, uniqueness, and both a priori and a posteriori error bounds. Adaptive refinement driven by the least-squares estimators produces accurate solutions even when element edges do not coincide with discontinuities.

Core claim

Two flux-only least-squares finite element methods are obtained for the linear transport equation by reformulating it around a flux variable whose normal component remains continuous across interfaces. Raviart-Thomas spaces supply enough degrees of freedom to approximate both the flux and its divergence, permitting elimination of the solution variable; the solution is recovered afterward by a simple post-processing step. The resulting schemes use fewer degrees of freedom than the prior mixed LSFEM, admit rigorous existence-uniqueness and error analysis, and support adaptive mesh refinement that remains effective without alignment to solution discontinuities.

What carries the argument

Flux-only formulations obtained by eliminating the solution variable through Raviart-Thomas mixed-element approximation of flux and divergence.

If this is right

  • Existence and uniqueness hold for the discrete flux-only problems.
  • A priori and a posteriori error estimates are available for both flux and recovered solution.
  • Adaptive refinement driven by the least-squares estimators converges without requiring mesh alignment to discontinuities.
  • Piecewise-constant reconstruction of the solution keeps overshooting mild.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same elimination technique could apply to other first-order hyperbolic systems whose fluxes satisfy interface continuity.
  • Higher-order post-processing recoveries might be derived to restore optimal rates for smooth regions.
  • Direct comparison of total degrees of freedom against standard discontinuous Galerkin schemes would quantify the savings.
  • The a posteriori estimators could be reused inside goal-oriented adaptivity for quantities of interest such as outflow integrals.

Load-bearing premise

The normal component of the flux remains continuous across every mesh interface even when the solution itself jumps.

What would settle it

A sequence of adaptive computations on successively refined meshes that fails to reduce the least-squares functional or produces persistent large overshoots for a known discontinuous solution would falsify the claims.

Figures

Figures reproduced from arXiv: 1907.06991 by Qunjie Liu, Shun Zhang.

Figure 1
Figure 1. Figure 1: Initial mesh for all examples with a (0, 1)2 domain Since we have two formulations for σh and two recoveries for uh, there are four variants to find the numerical solution pair (uh, σh). We use LSFEMi-j to denote them, with i denoting the method to get σh and j denoting the method to recover uh. For example, LSFEM1-2 means we use LSFEM1 to get σh and use the second recovery to get uh. For almost all our nu… view at source ↗
Figure 2
Figure 2. Figure 2: Global smooth solution: convergence histories on uniformly refined meshes [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Peterson example 5.4. An example with a piecewise smooth solution, on a matching grid Consider the following problem: Ω = (0, 1)2 with β = (1/ √ 2, 1/ √ 2)T . The inflow boundary is {x = 0, y ∈ (0, 1)} ∪ {x ∈ (0, 1), y = 0}, i.e., the west and south boundaries of the domain. Let γ = 1. Choose f and g such that the exact solution u is u =    sin(x + y) if y > x, cos(x + y) if y < x. We choose an init… view at source ↗
Figure 4
Figure 4. Figure 4: Piecewise smooth solution with a matching mesh: convergence histories for on uniformly refined meshes [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Piecewise continuous problem on a non-matching mesh: the initial mesh [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Piecewise continuous problem on a non-matching uniform mesh [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Piecewise continuous problem on a non-matching mesh: adaptively refined meshes after several iterations [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Piecewise continuous problem on a non-matching mesh: convergence histories on adaptive refined meshes [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Piecewise continuous problem on a non-matching adaptive mesh [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Piecewise smooth solution on a non-matching grid: convergence histories on uniformly refined meshes [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Piecewise smooth solution on a non-matching grid [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Curved transport problem: initial mesh(left), numerical solution (LSFEM 1-1) on an almost uniform mesh [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Curved transport problem: projected numerical solutions on an almost uniform mesh (left), convergence [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Curved transport problem: adaptively refined meshes after several iterations (left), convergence behaviors on [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Curved transport problem: reduction of overshootings by adaptive LSFEM (left), projection solution on an [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Transient layer problem: adaptive convergence behaviors [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Transient layer problem  = 0.01 For  = 0.01, we show the numerical results in Figs. 17 and 16. The behaviors of the methods are very similar to the global continuous solution case. For  = 10−10, we show the numerical results in Figs. 16 and 18. The behaviors of the methods are very similar to the piecewise smooth solution example with a non-matching grid, the example 5.6. The order of convergence of ku… view at source ↗
Figure 18
Figure 18. Figure 18: Transient layer problem  = 10−10 [11]. We do not observe smearing effect for the bona fide least-squares methods developed in this paper. 6. Concluding Remarks In this paper, two flux-only least-squares finite element methods (LSFEM) for the linear hyperbolic transport problem are developed. We first reformulate the linear transport equation into a flux-solution system, then eliminate the solution from t… view at source ↗
read the original abstract

In this paper, two flux-only least-squares finite element methods (LSFEM) for the linear hyperbolic transport problem are developed. The transport equation often has discontinuous solutions and discontinuous inflow boundary conditions, but the normal component of the flux across the mesh interfaces is continuous. In traditional LSFEMs, the continuous finite element space is used to approximate the solution. This will cause unnecessary error around the discontinuity and serious overshooting. In arXiv:1807.01524 [math.NA], we reformulate the equation by introducing a new flux variable to separate the continuity requirements of the flux and the solution. Realizing that the Raviart-Thomas mixed element space has enough degrees of freedom to approximate both the flux and its divergence, we eliminate the solution from the system and get two flux-only formulations, and develop corresponding LSFEMs. The solution then is recovered by simple post-processing methods using its relation with the flux. These two versions of flux-only LSFEMs use less DOFs than the method we developed in arXiv:1807.01524 [math.NA]. Similar to the LSFEM developed in arXiv:1807.01524 [math.NA], both flux-only LSFEMs can handle discontinuous solutions better than the traditional continuous polynomial approximations. We show the existence, uniqueness, a priori and a posteriori error estimates of the proposed methods. With adaptive mesh refinements driven by the least-squares a posteriori error estimators, the solution can be accurately approximated even when the mesh is not aligned with discontinuity. The overshooting phenomenon is very mild if a piecewise constant reconstruction of the solution is used. Extensive numerical tests are done to show the effectiveness of the methods developed in the paper.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops two flux-only least-squares finite element methods for the linear hyperbolic transport equation. By introducing a flux variable to separate continuity requirements and using Raviart-Thomas elements to eliminate the solution variable, the methods reduce degrees of freedom relative to the authors' prior formulation. Existence, uniqueness, a priori and a posteriori error estimates are shown, and adaptive refinement driven by the least-squares estimators is demonstrated to accurately approximate discontinuous solutions even on meshes not aligned with discontinuities, with mild overshooting under piecewise constant reconstruction.

Significance. If the well-posedness results, error estimates, and numerical behavior hold, the work provides a computationally lighter approach to discontinuous transport problems that avoids the overshooting typical of continuous polynomial approximations while retaining adaptive capability; the reduction in DOFs and the post-processing recovery step are concrete practical advantages.

major comments (2)
  1. [Abstract and well-posedness analysis] The central reformulation and elimination step rest on the continuity of the normal flux component across interfaces (stated in the abstract as the enabling property). The manuscript should explicitly verify in the well-posedness section that this property is preserved under the linear transport operator even when inflow data and solutions are discontinuous, as this is load-bearing for the Raviart-Thomas elimination.
  2. [A posteriori estimates and adaptive algorithm] The a posteriori estimator is used to drive adaptive refinement that succeeds on non-aligned meshes. The proof of reliability/efficiency should be checked against the post-processing recovery step, because any consistency error introduced by the simple reconstruction could affect the estimator's ability to detect discontinuities.
minor comments (2)
  1. The abstract states that the new methods use fewer DOFs than the formulation in arXiv:1807.01524; a short table or remark quantifying the DOF reduction for representative polynomial degrees would strengthen the comparison.
  2. [Numerical experiments] Numerical tests are described as 'extensive'; ensure that all tables report both error norms and observed convergence rates so that the a priori estimates can be directly compared with the computed rates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and the detailed comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and well-posedness analysis] The central reformulation and elimination step rest on the continuity of the normal flux component across interfaces (stated in the abstract as the enabling property). The manuscript should explicitly verify in the well-posedness section that this property is preserved under the linear transport operator even when inflow data and solutions are discontinuous, as this is load-bearing for the Raviart-Thomas elimination.

    Authors: The continuity of the normal flux component follows from the weak formulation of the transport equation, where the flux variable satisfies div(σ) in L², implying continuity of the normal trace in the appropriate sense, regardless of discontinuities in the solution or inflow data. This is implicitly used in the existence and uniqueness proof in Section 3. To make this explicit as suggested, we will add a short paragraph or remark in the well-posedness analysis section verifying this property under the given assumptions. revision: yes

  2. Referee: [A posteriori estimates and adaptive algorithm] The a posteriori estimator is used to drive adaptive refinement that succeeds on non-aligned meshes. The proof of reliability/efficiency should be checked against the post-processing recovery step, because any consistency error introduced by the simple reconstruction could affect the estimator's ability to detect discontinuities.

    Authors: The least-squares a posteriori estimator is based solely on the residual of the flux-only formulation and does not involve the post-processing recovery of the primal variable. The reliability and efficiency proofs in Section 4 are independent of the reconstruction step, which is applied only after the flux is computed for the purpose of visualization. We will include a clarifying statement in the manuscript to confirm that the estimator remains unaffected by the recovery procedure. revision: yes

Circularity Check

1 steps flagged

Self-citation to prior reformulation is load-bearing for flux-only methods but new analysis adds independent content

specific steps
  1. self citation load bearing [Abstract]
    "In arXiv:1807.01524 [math.NA], we reformulate the equation by introducing a new flux variable to separate the continuity requirements of the flux and the solution. Realizing that the Raviart-Thomas mixed element space has enough degrees of freedom to approximate both the flux and its divergence, we eliminate the solution from the system and get two flux-only formulations, and develop corresponding LSFEMs."

    The enabling reformulation that separates continuity requirements and permits elimination of the solution variable (the premise for all subsequent flux-only LSFEMs, error estimates, and adaptivity) is justified exclusively by citation to the authors' overlapping prior paper rather than re-derived or externally verified here.

full rationale

The paper's derivation of flux-only LSFEMs begins from a reformulation introduced in the authors' own prior work (arXiv:1807.01524), which supplies the separation of continuity requirements and the Raviart-Thomas elimination step. This self-citation is load-bearing for the core methodological reduction. However, the present manuscript independently establishes existence/uniqueness, a priori/a posteriori estimates, post-processing recovery, and adaptive refinement behavior. No step reduces a claimed prediction or theorem to a fitted input or tautological renaming within this paper alone. The central claims therefore retain grounding outside the cited prior work, producing only moderate circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that flux normal continuity holds across interfaces and on standard properties of Raviart-Thomas spaces; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The normal component of the flux across the mesh interfaces is continuous.
    Invoked in the abstract as the property that separates continuity requirements and enables the flux-only reformulation.

pith-pipeline@v0.9.0 · 5847 in / 1295 out tokens · 25699 ms · 2026-05-24T21:40:13.352521+00:00 · methodology

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Reference graph

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