pith. sign in

arxiv: 2605.19039 · v1 · pith:4OAYQFOOnew · submitted 2026-05-18 · 🧮 math.NA · cs.NA

A Priori Error Analysis of a High-Order Selective Discontinuous Galerkin Method for Elliptic Interface Problems

Fang Liu , Haroun Meghaichi , Xu Zhang This is my paper

Pith reviewed 2026-05-20 07:36 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords selective discontinuous Galerkinelliptic interface problemshybrid immersed finite elementa priori error analysisunfitted Cartesian mesheshigh-order methods
0
0 comments X

The pith

A selective discontinuous Galerkin method for elliptic interface problems achieves optimal error estimates while reducing computational cost.

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

This paper introduces a high-order selective discontinuous Galerkin method for elliptic interface problems discretized on interface-unfitted Cartesian meshes. The approach applies the discontinuous Galerkin formulation only on interface elements and the continuous Galerkin formulation on all other elements. A new hybrid immersed finite element space is constructed using high-order Frenet immersed finite element basis functions to maintain local conformity. The authors prove that this space achieves optimal approximation under h-refinement and establish the well-posedness of the resulting scheme. A priori error estimates are then derived in the energy norm and the L2 norm, with numerical examples confirming the theory.

Core claim

The central claim is that the hybrid immersed finite element space, built from high-order Frenet IFE basis functions, supports a selective DG-CG scheme that is well-posed and attains optimal a priori error bounds in energy and L2 norms for elliptic interface problems on unfitted meshes.

What carries the argument

The hybrid immersed finite element (HIFE) space on interface elements, which inherits conformity and approximation properties from the high-order Frenet IFE basis functions.

If this is right

  • The computational cost of the method is significantly lower than a full discontinuous Galerkin approach and comparable to continuous Galerkin methods.
  • Optimal convergence rates are guaranteed under h-refinement for the hybrid space.
  • The scheme remains stable and well-posed for interface problems.
  • Error estimates hold in both the energy norm and the L2 norm.

Where Pith is reading between the lines

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

  • This selective approach could extend to other interface problems such as Stokes flow or elasticity by adapting the hybrid space.
  • Large-scale simulations might benefit from the reduced degrees of freedom without sacrificing accuracy.
  • Further analysis could address adaptive mesh refinement or higher-order time integration.

Load-bearing premise

The high-order Frenet IFE basis functions from prior work supply the local conformity and approximation properties required for the hybrid space on interface elements.

What would settle it

Numerical experiments that fail to show the predicted optimal convergence rates in the energy or L2 norm under successive h-refinement would disprove the error estimates.

read the original abstract

This paper develops a high-order selective discontinuous Galerkin (SDG) method for solving elliptic interface problems on interface-unfitted Cartesian meshes. This method applies the discontinuous Galerkin (DG) formulation on interface elements and the continuous Galerkin (CG) formulation elsewhere. Correspondingly, we construct a new, locally conforming, hybrid immersed finite element (HIFE) space based on the high-order Frenet IFE basis functions of [1]. Compared with the DG method, the computational cost of this SDG method is significantly reduced and remains comparable to that of the CG method. We prove that the new HIFE space achieves optimal approximation under $h$-refinement, and we establish the well-posedness of the SDG scheme. {\it A priori} error estimates are derived in the energy and $L^2$ norms. Numerical examples are provided to verify the theoretical analysis.

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 / 1 minor

Summary. The paper develops a high-order selective discontinuous Galerkin (SDG) method for elliptic interface problems on unfitted Cartesian meshes. It applies the DG formulation only on interface elements and the CG formulation elsewhere, while constructing a new locally conforming hybrid immersed finite element (HIFE) space based on the high-order Frenet IFE basis functions from prior work [1]. The authors claim to prove that this HIFE space achieves optimal approximation under h-refinement, establish well-posedness of the SDG scheme, and derive a priori error estimates in the energy and L2 norms, supported by numerical examples.

Significance. If the proofs of well-posedness and optimal approximation hold without degradation from the hybrid assembly, the SDG method would offer a practical efficiency gain over full DG approaches while retaining optimal convergence rates for interface problems. This could be significant for high-order simulations on unfitted meshes, as the computational cost is stated to remain comparable to CG methods.

major comments (1)
  1. Abstract and the section on HIFE space construction: the central claim that the new HIFE space achieves optimal approximation under h-refinement rests on transferring local conformity and approximation properties from the high-order Frenet IFE basis functions of [1]. The hybrid selective formulation (CG away from the interface, DG on interface elements) may introduce additional consistency or jump terms at the boundaries between CG and DG regions that are not explicitly addressed; without a detailed check that these terms do not degrade the inherited estimates, the optimality proof does not follow directly.
minor comments (1)
  1. The abstract refers to 'the new HIFE space' and 'the SDG scheme' without a brief equation or diagram summarizing the hybrid formulation; adding a short display of the selective weak form would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We address the major comment in detail below and outline the revisions we will make to strengthen the presentation of the approximation results.

read point-by-point responses

  1. Referee: Abstract and the section on HIFE space construction: the central claim that the new HIFE space achieves optimal approximation under h-refinement rests on transferring local conformity and approximation properties from the high-order Frenet IFE basis functions of [1]. The hybrid selective formulation (CG away from the interface, DG on interface elements) may introduce additional consistency or jump terms at the boundaries between CG and DG regions that are not explicitly addressed; without a detailed check that these terms do not degrade the inherited estimates, the optimality proof does not follow directly.

    Authors: We appreciate this observation, which highlights an important point for clarity. The manuscript does establish local conformity of the hybrid HIFE space by construction from the Frenet IFE basis functions of [1], and the selective formulation ensures that DG is applied only on interface elements while CG elements remain continuous. However, we agree that the potential consistency and jump terms arising precisely at the interfaces between CG and DG regions warrant an explicit verification to confirm they do not degrade the inherited approximation properties. In the revised version, we will expand the HIFE space construction section with a dedicated paragraph (or short subsection) that analyzes these boundary terms. We will show that, because the HIFE space is designed to be continuous across CG-DG element faces (via the local conformity) and the DG penalty terms are localized to interface elements, the additional terms are bounded by the standard approximation error of the underlying IFE basis and do not affect the optimal rates under h-refinement. This addition will make the transfer of approximation properties fully rigorous without altering the main theorems. revision: yes

Circularity Check

0 steps flagged

Minor reliance on prior basis construction from [1] does not create circularity in the error analysis

full rationale

The paper constructs a hybrid HIFE space based on high-order Frenet IFE basis functions from referenced prior work [1] and claims to prove optimal approximation properties for this new space under h-refinement, followed by well-posedness and a priori error estimates for the SDG scheme. The derivation applies standard finite-element approximation and stability arguments to the selective CG/DG formulation on unfitted meshes. No equations or claims in the abstract or description reduce the target error bounds to quantities fitted or defined inside this paper itself. The cited basis properties from [1] supply local conformity and approximation order as an external input rather than a self-referential loop, and the hybrid assembly is presented as a new construction whose properties are established here. This qualifies as a normal self-citation that is not load-bearing for the central result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the approximation properties of the Frenet IFE basis from prior work and standard regularity assumptions for elliptic interface problems; no free parameters or new physical entities are introduced.

axioms (1)
  • domain assumption The interface is sufficiently smooth and the elliptic coefficients satisfy standard regularity conditions so that optimal approximation rates hold for the IFE space.
    Invoked to guarantee the h-refinement optimality stated in the abstract.
invented entities (1)
  • Hybrid immersed finite element (HIFE) space no independent evidence
    purpose: To enable a locally conforming hybrid formulation that mixes DG and CG while respecting the unfitted interface.
    New space constructed from high-order Frenet IFE basis functions; no independent falsifiable evidence supplied beyond the construction itself.

pith-pipeline@v0.9.0 · 5686 in / 1356 out tokens · 32220 ms · 2026-05-20T07:36:52.930140+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 1 internal anchor

  1. [1]

    Adjerid, S., Lin, T., Meghaichi, H.: A high order geometry conforming immersed finite element for elliptic interface problems. Comput. Methods Appl. Mech. Engrg.420, 116703–21 (2024) https://doi.org/10.1016/j.cma.2023.116703 19

    work page doi:10.1016/j.cma.2023.116703 2024

  2. [2]

    Prentice-Hall Series in Automatic Computation, p

    Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall Series in Automatic Computation, p. 306. Prentice-Hall, New Jersey (1973)

    work page 1973

  3. [3]

    Studies in Mathematics and its Applications, vol

    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, vol. Vol. 4, p. 530. North-Holland Publishing Co., Amsterdam-New York-Oxford (1978)

    work page 1978

  4. [4]

    Texts in Applied Mathematics, vol

    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, p. 294. Springer, New York (1994). https://doi.org/ 10.1007/978-1-4757-4338-8

    work page doi:10.1007/978-1-4757-4338-8 1994

  5. [5]

    Cockburn, B., Hou, S., Shu, C.-W.: The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comp.54(190), 545–581 (1990) https://doi.org/10.2307/2008501

    work page doi:10.2307/2008501 1990

  6. [6]

    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinu- ous Galerkin methods for elliptic problems. SIAM J. Numer. Anal.39(5), 1749–1779 (2001/02) https://doi.org/10.1137/S0036142901384162

    work page doi:10.1137/s0036142901384162 2001

  7. [7]

    Frontiers in Applied Mathematics, vol

    Rivi` ere, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. Frontiers in Applied Mathematics, vol. 35, p. 190. Society for Industrial and Applied Mathematics, Philadelphia (2008). https://doi.org/10.1137/1.9780898717440

    work page doi:10.1137/1.9780898717440 2008

  8. [8]

    Lin, T., Sheen, D., Zhang, X.: A nonconforming immersed finite element method for elliptic interface problems. J. Sci. Comput.79(1), 442–463 (2019) https://doi.org/10. 1007/s10915-018-0865-9

    work page 2019

  9. [9]

    He, X., Lin, T., Lin, Y.: A selective immersed discontinuous Galerkin method for elliptic interface problems. Math. Methods Appl. Sci.37(7), 983–1002 (2014) https://doi.org/ 10.1002/mma.2856

    work page doi:10.1002/mma.2856 2014

  10. [10]

    Computing (Arch

    Babuska, I.: The finite element method for elliptic equations with discontinuous coef- ficients. Computing (Arch. Elektron. Rechnen)5, 207–213 (1970) https://doi.org/10. 1007/bf02248021

    work page 1970

  11. [11]

    Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math.79(2), 175–202 (1998) https://doi.org/10. 1007/s002110050336

    work page 1998

  12. [12]

    Li, Z.: The immersed interface method using a finite element formulation. Appl. Numer. Math.27(3), 253–267 (1998) https://doi.org/10.1016/S0168-9274(98)00015-4

    work page doi:10.1016/s0168-9274(98)00015-4 1998

  13. [13]

    In: Minev, P., Lin, Y

    Lin, T., Lin, Y., Rogers, R., Ryan, M.L.: A rectangular immersed finite element space for interface problems. In: Minev, P., Lin, Y. (eds.) Scientific Computing and Applications (Kananaskis, AB, 2000). Advances In Computation: Theory And Practice, vol. 7, pp. 107–114. Nova Sci. Publ., Huntington, NY (2001)

    work page 2000

  14. [14]

    He, X., Lin, T., Lin, Y.: Approximation capability of a bilinear immersed finite element space. Numer. Methods Partial Differential Equations24(5), 1265–1300 (2008) https: //doi.org/10.1002/num.20318

    work page doi:10.1002/num.20318 2008

  15. [15]

    Li, Z., Lin, T., Lin, Y., Rogers, R.C.: An immersed finite element space and its approxi- mation capability. Numer. Methods Partial Differential Equations20(3), 338–367 (2004) https://doi.org/10.1002/num.10092

    work page doi:10.1002/num.10092 2004

  16. [16]

    Lin, T., Lin, Y., Zhang, X.: Partially penalized immersed finite element methods for elliptic interface problems. SIAM J. Numer. Anal.53(2), 1121–1144 (2015) https://doi. org/10.1137/130912700

    work page doi:10.1137/130912700 2015

  17. [17]

    Camp, B., Lin, T., Lin, Y., Sun, W.: Quadratic immersed finite element spaces and their approximation capabilities. Adv. Comput. Math.24(1-4), 81–112 (2006) https: //doi.org/10.1007/s10444-004-4139-8 20

    work page doi:10.1007/s10444-004-4139-8 2006

  18. [18]

    Applied Numerical Mathematics59(6), 1303–1321 (2009) https://doi.org/10.1016/j.apnum.2008.08.005

    Adjerid, S., Lin, T.: A p-th degree immersed finite element for boundary value prob- lems with discontinuous coefficients. Applied Numerical Mathematics59(6), 1303–1321 (2009) https://doi.org/10.1016/j.apnum.2008.08.005

    work page doi:10.1016/j.apnum.2008.08.005 2009

  19. [19]

    Adjerid, S., Ben-Romdhane, M., Lin, T.: Higher degree immersed finite element methods for second-order elliptic interface problems. Int. J. Numer. Anal. Model.11(3), 541–566 (2014)

    work page 2014

  20. [20]

    Adjerid, S., Guo, R., Lin, T.: High degree immersed finite element spaces by a least squares method. Int. J. Numer. Anal. Model.14(4-5), 604–626 (2017)

    work page 2017

  21. [21]

    Guo, R., Lin, T.: A higher degree immersed finite element method based on a Cauchy extension for elliptic interface problems. SIAM J. Numer. Anal.57(4), 1545–1573 (2019) https://doi.org/10.1137/18M121318X

    work page doi:10.1137/18m121318x 2019

  22. [22]

    Adjerid, S., Lin, T., Meghaichi, H.: The Frenet immersed finite element method for elliptic interface problems: an error analysis. Comput. Methods Appl. Mech. Engrg. 438, 117829–19 (2025) https://doi.org/10.1016/j.cma.2025.117829

    work page doi:10.1016/j.cma.2025.117829 2025

  23. [23]

    arXiv preprint arXiv:2510.12018 (2025)

    Adjerid, S., Lin, T., Meghaichi, H.: Construction of basis functions for the geometry conforming immersed finite element method. arXiv preprint arXiv:2510.12018 (2025)

    work page arXiv 2025

  24. [24]

    arXiv preprint arXiv:2512.23238 (2025)

    Lin, T., Lin, Y., Zhang, X.: Frenet immersed finite element spaces on triangular meshes. arXiv preprint arXiv:2512.23238 (2025)

    work page internal anchor Pith review arXiv 2025

  25. [25]

    He, X., Lin, T., Lin, Y.: The convergence of the bilinear and linear immersed finite element solutions to interface problems. Numer. Methods Partial Differential Equations 28(1), 312–330 (2012) https://doi.org/10.1002/num.20620

    work page doi:10.1002/num.20620 2012

  26. [26]

    Guo, R., Lin, T., Zhuang, Q.: Improved error estimation for the partially penalized immersed finite element methods for elliptic interface problems. Int. J. Numer. Anal. Model.16(4), 575–589 (2019)

    work page 2019

  27. [27]

    Lin, T., Yang, Q., Zhang, X.: A priori error estimates for some discontinuous Galerkin immersed finite element methods. J. Sci. Comput.65(3), 875–894 (2015) https://doi. org/10.1007/s10915-015-9989-3

    work page doi:10.1007/s10915-015-9989-3 2015

  28. [28]

    Adjerid, S., Babuˇ ska, I., Guo, R., Lin, T.: An enriched immersed finite element method for interface problems with nonhomogeneous jump conditions. Comput. Methods Appl. Mech. Engrg.404, 115770–37 (2023) https://doi.org/10.1016/j.cma.2022.115770

    work page doi:10.1016/j.cma.2022.115770 2023

  29. [29]

    submitted, arXiv:2509.12555 (2025) 21

    Guo, R., Zhang, X.: An enriched immersed finite element method for three-dimensional interface problems with nonhomogeneous jump. submitted, arXiv:2509.12555 (2025) 21

    work page arXiv 2025