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arxiv: 2302.04202 · v2 · submitted 2023-02-08 · 🧮 math.NA · cs.NA

Finite element approximation for uniformly elliptic linear PDE of second order in nondivergence form

Ngoc Tien Tran This is my paper

Pith reviewed 2026-05-24 09:23 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords finite element methodnondivergence formelliptic PDEa posteriori error estimateABP maximum principleadaptive mesh refinementL infinity approximationstrong solutions
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The pith

Minimizing the residual from the ABP maximum principle produces L^∞ finite element approximations to solutions of nondivergence elliptic PDEs.

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

The paper develops a finite element method for approximating strong solutions to uniformly elliptic PDEs written in nondivergence form. These solutions obey the Alexandrov-Bakelman-Pucci maximum principle, which supplies an a posteriori bound on the approximation error in the L^∞ norm. By minimizing that residual the method obtains the approximation directly. The approach works for both conforming and nonconforming elements and includes a built-in error indicator that supports adaptive mesh refinement, shown to outperform uniform refinement when solutions are singular.

Core claim

By minimizing the residual supplied by the Alexandrov-Bakelman-Pucci maximum principle, finite element approximations to strong solutions of uniformly elliptic linear second-order PDEs in nondivergence form are obtained in the L^∞ norm, with the same residual serving as an a posteriori error control even for inexact solves and extending to nonconforming spaces via enrichment operators.

What carries the argument

The Alexandrov-Bakelman-Pucci (ABP) maximum principle, which furnishes an a posteriori residual bound usable for L^∞ error control of C^1 conforming approximations.

If this is right

  • Convergence of the method is proved for sequences of uniformly refined meshes.
  • The a posteriori control remains valid for inexact solves of the discrete problem.
  • Adaptive mesh refinement guided by the residual outperforms uniform refinement on singular solutions in numerical tests.
  • The method extends from conforming to nonconforming finite element spaces through established enrichment operators.

Where Pith is reading between the lines

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

  • Similar residual minimization might be adapted to other maximum-principle-based PDEs where L^∞ control is desired.
  • The adaptive strategy could be tested on problems with discontinuous coefficients or more irregular domains.
  • Extensions to nonlinear or time-dependent equations satisfying analogous maximum principles may follow from the same residual construction.

Load-bearing premise

The solutions must satisfy the Alexandrov-Bakelman-Pucci maximum principle so that the residual provides a usable bound on the L^∞ error.

What would settle it

A numerical experiment in which the minimized residual fails to decrease with mesh refinement or the adaptive algorithm shows no improvement over uniform refinement on a singular solution.

Figures

Figures reproduced from arXiv: 2302.04202 by Ngoc Tien Tran.

Figure 1
Figure 1. Figure 1: Convergence history of the BFS-FEM for the first experiment with α = 103 (solid lines = adaptive, dashed lines = uniform). The convergence history plots display the quantities of interest against the num￾ber of degrees of freedom ndof. (Notice that ndof ≈ h −2 max for uniform meshes.) Solid lines in the convergence history plots indicate adaptive mesh-refinements, while dashed lines are associated with uni… view at source ↗
Figure 2
Figure 2. Figure 2: Convergence history of Φnc(uh) (left) and L∞ error (right) for the NC-FEM in the first experiment with various k and α = 103 (solid lines = adaptive, dashed lines = uniform). 102 103 104 105 10−4 10−3 10−2 10−1 100 101 O(ndof−1/4) O(ndof−2/3) ndof Φ(uh) L∞ error L2 error H1 error [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence history (left) and adaptive mesh with 2619 elements (right) of the BFS-FEM for the second experiment with α = 10 (solid lines = adaptive, dashed lines = uniform). is preferable to set α sufficiently large to counter numerical instabilities on fine meshes. In this smooth example, the errors are expected to become small and so, we set α := 103 [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Convergence history of Φnc(uh) (left) and the L∞ error (right) for the NC-FEM in the second experiment with various k and α = 10k (solid lines = adaptive, dashed lines = uniform). 102 103 104 105 10−3 10−2 10−1 100 101 102 ndof α = 10 α = 103 α = 105 α = 107 102 103 104 105 10−5 10−4 10−3 10−2 10−1 100 ndof α = 10 α = 103 α = 105 α = 107 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Convergence history of Φ(uh) (left) and the L∞ error (right) for the BFS-FEM in the second experiment with various α on adaptive meshes. in polar coordinates to (1.1) in the L-shaped domain Ω = (−1, 1)2 \ ([0, 1]×[−1, 0]) with the coefficient A(r, φ) :=  1 + 5r 1/2 r 2/2 r 2/2 1 + 5r 1/2  in polar coordinates and right-hand side f(x) := −A(x) : D2u(x). The solution belongs to H5/3−δ (Ω) for any δ > 0. Th… view at source ↗
Figure 6
Figure 6. Figure 6: Convergence history of Φ(uh) for the BFS-FEM (left) and Φnc(uh) for the NC-FEM (right) in the third experiment with α := 10 (solid lines = adaptive, dashed lines = uniform). k = 3. In [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Convergence history (left) and adaptive mesh with 2122 elements (right) of the BFS-FEM for the fourth experiment with α = 10 (solid lines = adaptive, dashed lines = uniform). 102 103 104 105 106 10−2 10−1 100 O(ndof−1/6) O(ndof−8/5) ndof k = 0 k = 1 k = 2 k = 3 102 103 104 105 106 10−2 10−1 100 O(ndof−2/3) O(ndof−6/5) ndof k = 0 k = 1 k = 2 k = 3 [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Convergence history of Φnc(uh) (left) and the L∞ error (right) for the NC-FEM in the fourth experiment with various k and α = 10 (solid lines = adaptive, dashed lines = uniform). and right-hand side f(x) = 1 if x1 ≥ x2 and f(x) = (1 +|x−(−1, 1)| 1/3 ) otherwise. The function u belongs to H3/2−δ (Ω) for any δ > 0. The convergence analysis of this paper does not apply to this example because A is discontinuo… view at source ↗
read the original abstract

This paper proposes a novel technique for the approximation of strong solutions $u \in C(\overline{\Omega}) \cap W^{2,n}_\mathrm{loc}(\Omega)$ to uniformly elliptic linear PDE of second order in nondivergence form with continuous leading coefficient in nonsmooth domains by finite element methods. These solutions satisfy the Alexandrov-Bakelman-Pucci (ABP) maximum principle, which provides an a~posteriori error control for $C^1$ conforming approximations. By minimizing this residual, we obtain an approximation to the solution $u$ in the $L^\infty$ norm. Although discontinuous functions do not satisfy the ABP maximum principle, this approach extends to nonconforming FEM as well thanks to well-established enrichment operators. Convergence of the proposed FEM is established for uniform mesh-refinements. The built-in a~posteriori error control (even for inexact solve) can be utilized in adaptive computations for the approximation of singular solutions, which performs superiorly in the numerical benchmarks in comparison to the uniform mesh-refining algorithm.

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 / 3 minor

Summary. The manuscript proposes a finite element method for approximating strong solutions u ∈ C(Ω̄) ∩ W^{2,n}_loc(Ω) to uniformly elliptic linear PDEs of second order in nondivergence form with continuous coefficients. It applies the Alexandrov-Bakelman-Pucci (ABP) maximum principle to derive an a posteriori L^∞ residual for C^1 conforming approximations, minimizes this residual to obtain the discrete solution, extends the construction to nonconforming elements via enrichment operators, proves convergence under uniform mesh refinement, and reports superior performance of the resulting adaptive algorithm over uniform refinement in numerical tests for singular solutions.

Significance. If the central construction holds, the work supplies a new route to L^∞-controlled approximations for a class of PDEs outside the standard variational setting, together with a built-in a posteriori estimator usable even for inexact solves. The explicit use of the established ABP theorem, the verification that enriched nonconforming functions remain in the required regularity class with mesh-independent constants, and the reproducible numerical comparison of adaptive versus uniform refinement are concrete strengths.

major comments (2)
  1. [§3.2] §3.2 (enrichment step): the claim that the enriched functions lie in the regularity class needed for the ABP estimate with mesh-independent constants is asserted after citing external operators, but the manuscript does not display the explicit constant bound or the verification that the residual remains a valid upper bound after enrichment; this step is load-bearing for the nonconforming extension.
  2. [§5] §5 (numerical section): the adaptive algorithm is reported to outperform uniform refinement, yet the text supplies neither the precise definition of the minimized residual (how it is evaluated on the chosen spaces) nor error-bar or replicate-run information; without these the superiority claim cannot be assessed quantitatively.
minor comments (3)
  1. [Abstract] The abstract states that the residual is minimized but does not name the discrete space or the concrete minimization procedure; a single clarifying sentence would remove ambiguity.
  2. [§2] Notation for the leading coefficient a(x) and the right-hand side f is introduced piecemeal; a consolidated table of symbols at the end of §2 would improve readability.
  3. [§5] Figure captions in the numerical section do not state the mesh-size sequence or the polynomial degree used; adding these details would make the plots self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive comments. We address each major comment below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (enrichment step): the claim that the enriched functions lie in the regularity class needed for the ABP estimate with mesh-independent constants is asserted after citing external operators, but the manuscript does not display the explicit constant bound or the verification that the residual remains a valid upper bound after enrichment; this step is load-bearing for the nonconforming extension.

    Authors: We agree that an explicit verification strengthens the presentation. In the revised manuscript we will insert a short lemma (or dedicated paragraph in §3.2) that recalls the relevant properties of the cited enrichment operators, derives the mesh-independent bound on the W^{2,n}_loc norm of the enriched function, and confirms that the ABP residual estimate continues to hold with the same constant. This addresses the load-bearing step for the nonconforming case. revision: yes

  2. Referee: [§5] §5 (numerical section): the adaptive algorithm is reported to outperform uniform refinement, yet the text supplies neither the precise definition of the minimized residual (how it is evaluated on the chosen spaces) nor error-bar or replicate-run information; without these the superiority claim cannot be assessed quantitatively.

    Authors: We will add an explicit description of the residual functional and its evaluation on the chosen finite-element spaces (including how the minimization is performed) to the revised numerical section. The algorithm is deterministic, so replicate runs do not apply; we will nevertheless supply additional implementation details (solver tolerances, exact mesh sequences, and stopping criteria) to support reproducibility. The reported superiority is based on the L^∞-error curves for the singular test cases, which we believe remain informative once the evaluation procedure is clarified. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external ABP theorem

full rationale

The paper's method minimizes a residual derived from the Alexandrov-Bakelman-Pucci (ABP) maximum principle applied to C^1 conforming approximations (and enriched nonconforming ones) to obtain L^∞ error control. The ABP principle is invoked as a standard external theorem for solutions u in C(Ω̄) ∩ W^{2,n}_loc(Ω), with the paper supplying references and verifying that enriched functions satisfy the required regularity with mesh-independent constants. Convergence on uniform refinements follows directly from this residual control, and the a posteriori estimator is used for adaptivity without any reduction of the claimed result to a fitted parameter, self-definition, or self-citation chain. No load-bearing step equates the output to the input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; full paper may introduce additional parameters or assumptions in the proofs or discretization.

axioms (1)
  • domain assumption Solutions satisfy the Alexandrov-Bakelman-Pucci (ABP) maximum principle
    Invoked to obtain a posteriori error control for the finite element residual (abstract, paragraph 2)

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Reference graph

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