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arxiv: 1906.11436 · v1 · pith:PM5WVKIEnew · submitted 2019-06-27 · 🧮 math.NA · cs.NA

Adaptive First-Order System Least-Squares Finite Element Methods for Second Order Elliptic Equations in Non-Divergence Form

Weifeng Qiu , Shun Zhang This is my paper

Pith reviewed 2026-05-25 14:58 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords least-squares finite element methodnon-divergence formadaptive finite elementa posteriori error estimateelliptic partial differential equationfirst-order systemC0 finite elementsweighted least-squares
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The pith

First-order least-squares finite element methods solve second-order elliptic equations in non-divergence form with optimal a priori and a posteriori error estimates using standard C0 elements.

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

The paper recasts a second-order elliptic PDE in non-divergence form as a first-order system by treating the gradient of the unknown as an auxiliary variable. Two least-squares finite element discretizations are then constructed with continuous piecewise linear elements: a basic L2 version and a weighted version that incorporates a mesh-dependent weight. Under the sole assumption that the PDE admits a unique solution, both versions are shown to deliver optimal convergence rates together with reliable a posteriori error indicators that support adaptive mesh refinement. An additional but weaker regularity condition on the differential operator further guarantees convergence of the weighted method in standard norms, and L2 error bounds are derived for each formulation.

Core claim

By introducing the gradient as a new variable, the non-divergence equation becomes a first-order system whose least-squares weak form is stable without integration by parts. The resulting L2-LSFEM and weighted-LSFEM, both using simple C0 linear elements, admit optimal a priori and a posteriori error estimates whenever the original PDE has a unique solution. The weighted formulation additionally converges in standard norms under a regularity assumption weaker than the usual one, while L2-error estimates hold for both methods.

What carries the argument

First-order least-squares formulation of the non-divergence elliptic equation obtained by introducing the gradient of the solution as an auxiliary unknown, yielding a stable weak form discretized by C0 finite elements.

If this is right

  • Built-in a posteriori estimators drive reliable adaptive mesh refinement for both smooth and singular solutions.
  • Optimal convergence holds for problems whose coefficients may be non-smooth or even degenerate.
  • L2-error estimates are available for the solution and its gradient in both the L2-LSFEM and weighted-LSFEM.
  • The weighted method recovers standard-norm convergence rates once a mild operator-regularity condition is met.

Where Pith is reading between the lines

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

  • The same first-order reformulation may apply directly to other equations lacking divergence structure, such as certain linear transport or obstacle problems.
  • The weaker regularity hypothesis broadens the range of admissible coefficients beyond those required by classical variational approaches.
  • Adaptive refinement guided by the least-squares residuals could reduce degrees of freedom needed for high-dimensional or geometrically complex domains.
  • Avoidance of integration by parts removes the need for specialized nonconforming or discontinuous Galerkin spaces commonly used for non-divergence problems.

Load-bearing premise

The partial differential equation has a unique solution.

What would settle it

A concrete non-divergence elliptic problem that lacks a unique solution, for which the computed error indicators or observed convergence rates violate the stated optimal bounds.

read the original abstract

This paper studies adaptive first-order least-squares finite element methods for second-order elliptic partial differential equations in non-divergence form. Unlike the classical finite element method which uses weak formulations of PDEs not applicable for the non-divergence equation, the first-order least-squares formulations naturally have stable weak forms without using integration by parts, allow simple finite element approximation spaces, and have build-in a posteriori error estimators for adaptive mesh refinements. The non-divergence equation is first written as a system of first-order equations by introducing the gradient as a new variable. Then two versions of least-squares finite element methods using simple $C^0$ finite elements are developed in the paper, one is the $L^2$-LSFEM which uses linear elements, the other is the weighted-LSFEM with a mesh-dependent weight to ensure the optimal convergence. Under a very mild assumption that the PDE has a unique solution, optimal a priori and a posteriori error estimates are proved. With an extra assumption on the operator regularity which is weaker than traditionally assumed, convergences in standard norms for the weighted-LSFEM are also discussed. $L^2$-error estimates are derived for both formulations. We perform extensive numerical experiments for smooth, non-smooth, and even degenerate coefficients on smooth and singular solutions to test the accuracy and efficiency of the proposed methods.

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

1 major / 3 minor

Summary. The manuscript develops adaptive first-order system least-squares finite element methods for second-order elliptic PDEs in non-divergence form. The PDE is rewritten as a first-order system by introducing the gradient variable. Two methods are proposed: an L²-LSFEM using linear C⁰ elements and a weighted-LSFEM with mesh-dependent weighting. Under the assumption that the continuous problem has a unique solution, optimal a priori and a posteriori error estimates are proved for both; an additional (weaker-than-usual) regularity assumption on the operator is used to obtain convergence in standard norms for the weighted variant. L²-error estimates are also derived, and numerical experiments are reported for smooth/non-smooth/degenerate coefficients on smooth and singular solutions.

Significance. If the error estimates are rigorously established under only the uniqueness assumption, the work would be a useful contribution to the numerical treatment of non-divergence-form elliptic problems, for which standard weak formulations are unavailable. The built-in a posteriori estimators, use of simple C⁰ spaces, and reported performance on degenerate coefficients are practical strengths. The derivation of L² estimates for both formulations adds value if the analysis is complete.

major comments (1)
  1. [Sections 3–4] Sections 3–4 (proofs of a priori/a posteriori estimates): the central claim is that uniqueness of the continuous solution alone yields optimal estimates for both LSFEM variants. For first-order system least-squares methods the least-squares functional must be shown coercive and continuous in the chosen norm with constants independent of the mesh and of the coefficient A(x). It is not evident whether the uniqueness hypothesis supplies a positive lower bound on the discrete inf-sup constant without further control on the ellipticity of A or on the specific structure of the first-order system; if the proofs derive coercivity solely from uniqueness, an explicit lemma or counter-example check would be required to support the claim.
minor comments (3)
  1. [Abstract, §1] Abstract and §1: the first-order system obtained by introducing the gradient variable should be written explicitly with an equation number so that the subsequent least-squares functionals can be referenced directly.
  2. [Numerical experiments] Numerical experiments section: while the abstract states that extensive tests were performed, the manuscript summary provides no tables, convergence rates, or efficiency metrics; inclusion of representative data (e.g., observed orders for the weighted method on degenerate coefficients) would allow readers to assess the practical performance.
  3. [§2] Notation: the precise definition of the mesh-dependent weight in the weighted-LSFEM and its dependence on the local mesh size should be stated once, with a clear reference when the weighted norm is introduced.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. Below we address the major comment point by point.

read point-by-point responses

  1. Referee: [Sections 3–4] Sections 3–4 (proofs of a priori/a posteriori estimates): the central claim is that uniqueness of the continuous solution alone yields optimal estimates for both LSFEM variants. For first-order system least-squares methods the least-squares functional must be shown coercive and continuous in the chosen norm with constants independent of the mesh and of the coefficient A(x). It is not evident whether the uniqueness hypothesis supplies a positive lower bound on the discrete inf-sup constant without further control on the ellipticity of A or on the specific structure of the first-order system; if the proofs derive coercivity solely from uniqueness, an explicit lemma or counter-example check would be required to support the claim.

    Authors: We appreciate the referee drawing attention to the role of the uniqueness assumption. In Sections 3 and 4 the coercivity of the least-squares functional (both standard L2 and mesh-weighted) is established precisely by using uniqueness of the continuous first-order system: if the functional vanishes then the pair (u, gradient) satisfies the homogeneous problem and must therefore be zero. The mesh-independent lower bound then follows from a standard compactness argument (weak convergence plus uniqueness implies strong convergence to zero). The resulting constant is independent of the mesh size h but does depend on the fixed coefficient A through the uniqueness hypothesis; this is the natural setting for a given PDE. The discrete inf-sup stability follows directly from the continuous coercivity together with the approximation properties of the C0 linear elements. To make the argument fully transparent we will insert an explicit lemma (new Lemma 3.1 or equivalent) that isolates the passage from uniqueness to coercivity/continuity, including a brief remark on the dependence on A. This revision will also cover the a-posteriori estimator equivalence. We believe the added lemma fully addresses the concern while preserving the paper’s stated hypotheses. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives a priori and a posteriori error estimates for L2-LSFEM and weighted-LSFEM formulations of the non-divergence elliptic problem by rewriting it as a first-order system and applying standard least-squares analysis. The central assumption (unique continuous solution) is used to establish coercivity and continuity of the least-squares functional with respect to the chosen norms; the proofs proceed via standard functional-analytic arguments rather than reducing any estimate to a fitted quantity or self-referential definition. No load-bearing step collapses by construction to its inputs, no self-citation chains are invoked to force uniqueness or coercivity, and the weighted-norm convergence discussion relies on an additional regularity assumption that is independent of the target result. The analysis is therefore self-contained against external benchmarks of FEM theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the domain assumption of unique solvability plus standard background results from finite-element theory; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The PDE has a unique solution
    Invoked explicitly as the mild assumption under which all optimal error estimates hold.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    math.NA 2019-07 unverdicted novelty 6.0

    Two flux-only LSFEMs for linear transport equations are proposed, with existence/uniqueness proofs, a priori/a posteriori error estimates, and adaptive refinement for improved discontinuity handling.

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