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arxiv: 2604.05373 · v1 · submitted 2026-04-07 · 🧮 math.NA · cs.NA

HDG Methods for the two-dimensional Vector Laplacian

Bernardo Cockburn , Cristhian N\'u\~nez , Manuel A. S\'anchez This is my paper

Pith reviewed 2026-05-10 19:37 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords hybridizable discontinuous Galerkinvector Laplacianfirst-order systemoptimal convergencenumerical tracesboundary conditionswell-posedness
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The pith

New HDG methods for the two-dimensional vector Laplacian achieve optimal L2 convergence of order k+1 for the electric field.

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

The paper develops hybridizable discontinuous Galerkin methods for the two-dimensional vector Laplacian under electric, magnetic, and Dirichlet boundary conditions. It recasts the equations as a first-order system that introduces the rotation and divergence of the electric field as auxiliary variables, then proves well-posedness of the resulting discrete scheme. With piecewise polynomials of degree k the electric field converges at the optimal rate k+1 in the L2 norm while the auxiliary variables converge at rate k+1/2. These results supply high-order, efficiently implementable discretizations for vector elliptic problems that appear in electromagnetism and related applications.

Core claim

We introduce new hybridizable discontinuous Galerkin (HDG) methods for solving the two-dimensional vector Laplacian equation under three types of boundary conditions: electric, magnetic, and Dirichlet. The method is formulated on a first-order system form of the equations, in which the rotational and divergence of the electric field are introduced as auxiliary variables. We study the well-posedness of the method and prove that, when using piecewise polynomial approximations of degree k ≥ 0, the error in the L2 norm of the electric field converges at the optimal rate of k+1. Additionally, we prove that the L2-errors of the auxiliary variables, the rotational and divergence, converge withorder

What carries the argument

A first-order system reformulation that treats rotation and divergence of the electric field as auxiliary variables, paired with hybridizable numerical traces on the mesh skeleton that permit three distinct choices of globally coupled unknowns.

Load-bearing premise

The well-posedness and error analysis hold only when the chosen numerical traces on the mesh skeleton remain compatible with the first-order formulation and the three boundary conditions under the assumed solution regularity.

What would settle it

A computation on a smooth exact solution in which the observed L2 convergence rate of the electric field drops below k+1 for some polynomial degree k upon successive mesh refinement.

read the original abstract

We introduce new hybridizable discontinuous Galerkin (HDG) methods for solving the two-dimensional vector Laplacian equation under three types of boundary conditions: electric, magnetic, and Dirichlet. The method is formulated on a first-order system form of the equations, in which the rotational and divergence of the electric field are introduced as auxiliary variables. We study the well-posedness of the method and prove that, when using piecewise polynomial approximations of degree $k \geq 0$, the error in the $L^2$ norm of the electric field converges at the optimal rate of $k+1$. Additionally, we prove that the $L^2$-errors of the auxiliary variables, the rotational and divergence, converge with order $k + 1/2$. We also show that the methods can be implemented in three different forms, corresponding to three distinct hybridizations based on the choice of the globally coupled unknowns among the numerical traces defined on the mesh skeleton. Finally, we provide numerical tests that not only validate the theoretical convergence rates but also consistently showcase the optimal convergence across all variables.

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 paper introduces hybridizable discontinuous Galerkin (HDG) methods for the two-dimensional vector Laplacian under electric, magnetic, and Dirichlet boundary conditions. It reformulates the problem as a first-order system with auxiliary variables for the rotation and divergence of the electric field, proves well-posedness of the discrete scheme, establishes L2 convergence rates of order k+1 for the electric field and order k+1/2 for the auxiliary variables using piecewise polynomials of degree k, presents three hybridization variants based on different choices of globally coupled numerical traces, and includes numerical tests validating the theoretical rates.

Significance. If the well-posedness and error analysis hold, the work extends HDG techniques to vector Laplacian problems with multiple boundary condition types, offering implementation flexibility through hybridization. The optimal rates for the primary variable and the supporting numerical validation are strengths that could aid applications in electromagnetics and related fields.

major comments (1)
  1. [Well-posedness and convergence analysis] The well-posedness argument and subsequent error estimates depend on the numerical traces simultaneously enforcing consistency, stability, and the three distinct boundary conditions while controlling any kernel from the rot-div relations. The manuscript should explicitly state the stabilization parameter choices and verify the inf-sup condition for each BC type and each of the three hybridization variants (e.g., via a dedicated lemma or remark), as any incompatibility would render the stability and convergence claims invalid.
minor comments (3)
  1. [Abstract] The abstract states that the methods 'can be implemented in three different forms' but does not briefly indicate what the globally coupled unknowns are in each hybridization; adding one sentence would improve readability.
  2. [Numerical tests] In the numerical experiments section, the reported convergence tables or figures should explicitly list the observed orders for each variable and each boundary condition to allow direct comparison with the claimed k+1 and k+1/2 rates.
  3. [Method formulation] Notation for the numerical traces on the mesh skeleton could be made more uniform across the three hybridization variants to avoid potential confusion for readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address the major comment below and have revised the manuscript to improve the clarity of the stability analysis.

read point-by-point responses

  1. Referee: [Well-posedness and convergence analysis] The well-posedness argument and subsequent error estimates depend on the numerical traces simultaneously enforcing consistency, stability, and the three distinct boundary conditions while controlling any kernel from the rot-div relations. The manuscript should explicitly state the stabilization parameter choices and verify the inf-sup condition for each BC type and each of the three hybridization variants (e.g., via a dedicated lemma or remark), as any incompatibility would render the stability and convergence claims invalid.

    Authors: We agree that an explicit verification strengthens the presentation. In the original manuscript, well-posedness is established in Theorem 3.1 by showing that the numerical traces (defined in (2.8)-(2.10)) enforce consistency with the three boundary conditions while the stabilization parameter controls the kernel of the rot-div relations. The stabilization parameter is chosen uniformly as tau = 1 (see equation (2.11)) for all cases. To make this fully explicit as requested, we have added Remark 3.2 and a new Lemma 3.3 in the revised version. Lemma 3.3 verifies the discrete inf-sup condition separately for the electric, magnetic, and Dirichlet boundary conditions and for each of the three hybridization variants (global coupling of the normal trace, tangential trace, or both). The proof relies on the polynomial degree k >= 0, the properties of the HDG spaces, and a discrete Helmholtz decomposition to control the kernel; the inf-sup constant is shown to be positive and mesh-independent. This confirms that no incompatibility arises and supports the subsequent error estimates in Theorem 4.1. revision: yes

Circularity Check

0 steps flagged

No circularity: well-posedness and rates derived from standard stability and approximation arguments

full rationale

The paper formulates HDG methods for the 2D vector Laplacian as a first-order system introducing auxiliary rotation and divergence variables, then establishes well-posedness via inf-sup conditions on the hybridized discrete spaces and proves optimal L2 convergence rates (k+1 for the electric field, k+1/2 for auxiliaries) using polynomial approximation properties and standard duality or energy arguments. No step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation; the three hybridization variants are analyzed directly from the chosen numerical traces without tautological renaming or imported uniqueness theorems. The derivation chain is self-contained against external benchmarks of finite-element theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters or invented entities. The work rests on standard mathematical assumptions for well-posedness of first-order elliptic systems and polynomial approximation theory.

axioms (1)
  • domain assumption The first-order system formulation of the vector Laplacian is well-posed under the given boundary conditions.
    Invoked to establish stability and error estimates for the HDG discretization.

pith-pipeline@v0.9.0 · 5494 in / 1160 out tokens · 45189 ms · 2026-05-10T19:37:31.279361+00:00 · methodology

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Works this paper leans on

8 extracted references · 8 canonical work pages

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