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arxiv: 2512.14219 · v3 · submitted 2025-12-16 · 🧮 math.NA · cs.NA

Analysis of a finite element method for second order uniformly elliptic PDEs in non-divergence form

Weifeng Qiu This is my paper

Pith reviewed 2026-05-16 22:16 UTC · model grok-4.3

classification 🧮 math.NA cs.NA MSC 65N3035J15
keywords finite element methodnon-divergence formuniformly elliptic PDEHamilton-Jacobi-Bellman equationW^{2,p} estimatesoptimal convergenceconvex polyhedra
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The pith

A finite element method for uniformly elliptic PDEs in non-divergence form proves well-posedness in W^{2,p} and optimal convergence for 1 < p ≤ 2 on convex polyhedra.

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

The paper introduces a finite element method applicable to second-order linear uniformly elliptic PDEs written in non-divergence form and to uniformly elliptic Hamilton-Jacobi-Bellman equations. It establishes existence and uniqueness of strong solutions belonging to the Sobolev space W^{2,p}(Ω) and demonstrates that the discrete approximations converge to these solutions at the optimal rate measured in a discrete counterpart of the W^{2,p} norm. The results hold for convex polyhedral domains in two and three space dimensions when the integrability index satisfies 1 < p ≤ 2; a narrower interval for p is obtained when the domain is a non-convex polygon in two dimensions. The analysis further relaxes the continuity assumptions previously imposed on the coefficients of the HJB equation.

Core claim

We prove the well-posedness of strong solution in W^{2,p}(Ω) and optimal convergence in discrete W^{2,p}-norm of the finite element approximation to the strong solution for 1<p≤2 on convex polyhedra in R^d (d=2,3). If the domain is a two dimensional non-convex polygon, p is valid in a more restricted region. Furthermore, we relax the assumptions on the continuity of coefficients of the HJB equation.

What carries the argument

The proposed finite element discretization for non-divergence form equations, which directly discretizes second-order derivatives to obtain discrete W^{2,p} estimates under uniform ellipticity.

If this is right

  • The same discretization applies equally to linear non-divergence PDEs and to nonlinear HJB equations.
  • Optimal convergence holds in the discrete W^{2,p} norm for the full range 1 < p ≤ 2 on convex domains in two and three dimensions.
  • Coefficient continuity requirements for the HJB equation are weaker than those used in prior analyses.
  • The method remains valid on non-convex polygonal domains in two dimensions provided p lies in a narrower subinterval of (1,2].

Where Pith is reading between the lines

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

  • The relaxation of continuity assumptions on coefficients may allow the method to handle problems arising from stochastic control with merely measurable or discontinuous data.
  • Extension of the analysis to three-dimensional non-convex domains or to higher-order equations would require new regularity estimates.
  • Implementation on adaptive meshes could be tested to see whether the discrete W^{2,p} norm still yields the same optimal rates.

Load-bearing premise

The coefficients satisfy uniform ellipticity and the domain is a convex polyhedron (or a restricted range of p is used when the domain is a non-convex polygon).

What would settle it

Numerical computation of the discrete W^{2,p} error for a known strong solution on a convex polyhedron with p=2; the observed rate matching the predicted optimal order would support the claim while a strictly lower rate would falsify it.

read the original abstract

We propose one finite element method for both second order linear uniformly elliptic PDE in non-divergence form and the uniformly elliptic Hamilton-Jacobi-Bellman (HJB) equation. For both linear elliptic PDE in non-divergence form and the HJB equation, we prove the well-posedness of strong solution in $W^{2,p}(\Omega)$ and optimal convergence in discrete $W^{2,p}$-norm of the finite element approximation to the strong solution for $1<p\leq 2$ on convex polyhedra in $\mathbb{R}^{d}$ ($d=2,3$). If the domain is a two dimensional non-convex polygon, $p$ is valid in a more restricted region. Furthermore, we relax the assumptions on the continuity of coefficients of the HJB equation, which have been widely used in literature.

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

0 major / 3 minor

Summary. The paper proposes a single finite element method for both linear second-order uniformly elliptic PDEs in non-divergence form and uniformly elliptic Hamilton-Jacobi-Bellman equations. It proves well-posedness of strong solutions in W^{2,p}(Ω) together with optimal convergence of the FEM approximation in a discrete W^{2,p}-norm, for 1 < p ≤ 2 on convex polyhedra in R^d (d=2,3); a restricted range of p is stated for non-convex 2D polygons. The analysis relaxes the usual continuity requirements on the coefficients of the HJB equation.

Significance. If the stated well-posedness and convergence results hold, the work supplies a unified FEM framework for non-divergence-form elliptic problems that covers both linear and nonlinear (HJB) cases under standard uniform ellipticity and domain-convexity hypotheses. The relaxation of coefficient continuity for the HJB equation and the use of a discrete W^{2,p} norm that inherits the continuous regularity are potentially useful extensions of existing theory for applications such as stochastic control.

minor comments (3)
  1. [Abstract / Introduction] The abstract and introduction should explicitly define or reference the precise discrete W^{2,p} norm employed for the error analysis, as this quantity is central to the convergence statement.
  2. [Abstract] For the non-convex 2D polygon case, the restricted range of admissible p should be stated with a concrete interval or condition rather than the phrase 'more restricted region'.
  3. [Introduction] The manuscript would benefit from a short comparison paragraph (in the introduction or a dedicated subsection) that situates the relaxed coefficient assumption against the continuity hypotheses used in prior HJB-FEM literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation of minor revision. The report correctly identifies the unified treatment of linear non-divergence elliptic PDEs and HJB equations, the well-posedness in W^{2,p}, the optimal convergence in the discrete W^{2,p} norm, and the relaxation of coefficient continuity assumptions.

Circularity Check

0 steps flagged

No significant circularity; results rest on standard elliptic regularity

full rationale

The derivation invokes uniform ellipticity plus convexity of the polyhedral domain to obtain W^{2,p} regularity via Calderón-Zygmund theory, then constructs a discrete W^{2,p} norm and consistency estimate that inherit the same regularity class. These steps are standard in non-divergence FEM analysis and do not reduce any claim to a self-definition, fitted parameter renamed as prediction, or self-citation chain. The paper explicitly treats the regularity hypotheses as given external inputs rather than deriving them internally. No load-bearing step collapses by construction to the paper's own ansatz or data fit.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard Sobolev-space theory for elliptic operators and finite-element approximation properties; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption Uniform ellipticity and boundedness of coefficients
    Invoked to guarantee well-posedness in W^{2,p}.
  • standard math Standard finite-element approximation theory on polyhedra
    Used to obtain optimal convergence rates in the discrete W^{2,p} norm.

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Reference graph

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