pith. sign in

arxiv: 2604.06910 · v1 · submitted 2026-04-08 · 🧮 math.NA · cs.NA

A discontinuous Galerkin method for elliptic-hyperbolic equations

Chiara Perinati , Lise-Marie Imbert-G\'erard , Andrea Moiola , Paul Stocker This is my paper

Pith reviewed 2026-05-10 18:13 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords discontinuous Galerkinmixed-type equationselliptic-hyperbolicMorawetz multiplierhp-error estimatesquasi-Trefftz polynomialswell-posednessenergy norm
0
0 comments X

The pith

A discontinuous Galerkin method for mixed elliptic-hyperbolic equations achieves well-posedness through coercivity in an energy norm.

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

The paper introduces a discontinuous Galerkin discretization for second-order linear PDEs that switch from elliptic to hyperbolic behavior across the domain. It establishes well-posedness of the discrete problem by proving coercivity in a suitable energy norm, using the Morawetz multiplier technique applied to the DG formulation. From this coercivity the authors derive hp-a priori error estimates that yield convergence rates for both standard polynomial spaces and quasi-Trefftz spaces. Numerical experiments are presented to confirm the predicted rates. A reader would care because these mixed-type equations appear in transonic flow models and other applications where a single stable numerical scheme must handle both regimes without type-dependent adjustments.

Core claim

We present and analyze a discontinuous Galerkin method for the numerical solution of a class of second-order linear mixed-type partial differential equations. Well-posedness of the discrete problem is established via coercivity in an energy norm, achieved through the Morawetz multiplier technique. We derive hp-a priori error estimates in the energy norm, which we use to prove convergence rates for standard and quasi-Trefftz polynomial spaces. Numerical experiments validate the theoretical results.

What carries the argument

Discontinuous Galerkin formulation combined with the Morawetz multiplier to establish coercivity in an energy norm for the mixed-type operator.

If this is right

  • The discrete problem admits a unique solution that depends continuously on the data in the energy norm.
  • The method converges in the energy norm at rates determined by the polynomial degree and mesh size for both standard and quasi-Trefftz spaces.
  • Quasi-Trefftz spaces achieve the same convergence rates as standard polynomials without requiring exact satisfaction of the PDE inside elements.
  • The analysis extends directly to hp-refinement strategies on shape-regular meshes.

Where Pith is reading between the lines

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

  • The same multiplier technique may allow stability proofs for other mixed-type operators whose principal part admits a positive-definite energy identity.
  • The quasi-Trefftz construction could be combined with adaptive mesh refinement to reduce computational cost near the elliptic-hyperbolic transition curve.
  • Because the coercivity proof does not rely on explicit solution of the continuous problem, the method remains applicable when only weak solutions exist.

Load-bearing premise

The Morawetz multiplier can be chosen and applied to the discontinuous Galerkin formulation to yield coercivity, which depends on the specific form of the mixed-type equation and the mesh.

What would settle it

Numerical computation of the discrete energy norm for a known exact solution on a sequence of meshes; if the norm of the error does not decrease at the predicted hp-rates or if the coercivity constant becomes negative for admissible mesh sizes, the central claims fail.

Figures

Figures reproduced from arXiv: 2604.06910 by Andrea Moiola, Chiara Perinati, Lise-Marie Imbert-G\'erard, Paul Stocker.

Figure 1
Figure 1. Figure 1: Domain Ω. where n = (nx, ny) ⊤ denotes the outward normal vector to the boundary ∂Ω [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Computational domain Ω for to the case K(y) = y. As a test case, we consider the boundary value problem (1.5)–(1.6) with K(y) = y on the domain Ω shown in [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: h-convergence in the norms |||·||| (first row) and ∥·∥L2(Ω) (second row) for the Tricomi problem with exact solution u in (5.3) for the three discrete spaces considered. The empirical algebraic convergence rates are shown along each segment. 10−1.5 10−1 10−0.5 100.5 101 101.5 0.94 0.98 0.99 1.00 0.96 0.98 0.99 1.00 0.94 0.98 0.99 1.00 h |||u − uh||| p = 2 P QT ET 10−1.5 10−1 10−0.5 10−1 100 101 1.98 1.98 1… view at source ↗
Figure 4
Figure 4. Figure 4: h-convergence in the norms |||·||| (first row) and ∥·∥L2(Ω) (second row) for the Tricomi problem with exact solution u in (5.3) for the three discrete spaces considered using the least-squares variant of the method with γ4 = 1. The empirical algebraic convergence rates are shown along each segment. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: p-convergence in the norms |||·||| (left) and ∥·∥L2(Ω) (right) for the Tricomi problem with exact solution u in (5.3) for the standard, quasi-Trefftz and embedded Trefftz polynomials spaces. 5.4 Sensitivity to the penalty parameters The theoretical analysis guarantees well-posedness and stability of the variational problem (2.11) under the assumptions γ1 > 0 and γ2, γ3 > γ∗, where γ∗ is defined in (3.4). W… view at source ↗
Figure 6
Figure 6. Figure 6: L 2 (Ω) error of the numerical solution with γ1 and γ2 = γ3 varying in the set (5.6), for mesh size h = 0.1 and for the standard, quasi-Trefftz, and embedded Trefftz polynomial spaces. Results are shown for p = 2 (top row), p = 3 (middle row) and p = 4 (bottom row). 6 Conclusions We have introduced a discontinuous Galerkin formulation for the numerical discretization of a class of elliptic-hyperbolic probl… view at source ↗
read the original abstract

We present and analyze a discontinuous Galerkin method for the numerical solution of a class of second-order linear mixed-type partial differential equations, i.e. equations that change their nature from elliptic to hyperbolic through the computational domain. Well-posedness of the discrete problem is established via coercivity in an energy norm, achieved through the Morawetz multiplier technique. We derive $hp$-a priori error estimates in the energy norm, which we use to prove convergence rates for standard and quasi-Trefftz polynomial spaces. Numerical experiments validate the theoretical results.

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 / 2 minor

Summary. The manuscript presents a discontinuous Galerkin (DG) method for second-order linear mixed-type PDEs that change from elliptic to hyperbolic across the domain. Well-posedness of the discrete problem is established by proving coercivity of the DG bilinear form in an energy norm via the Morawetz multiplier technique. The authors derive hp a priori error estimates in this norm and obtain convergence rates for both standard polynomial spaces and quasi-Trefftz spaces. Numerical experiments are included to illustrate the theoretical results.

Significance. If the coercivity proof and error analysis are complete, the work would represent a solid contribution to numerical methods for mixed-type equations, which remain challenging due to the type change. Adapting the Morawetz multiplier to control DG interface and flux terms while obtaining hp estimates is technically interesting, and the inclusion of quasi-Trefftz spaces adds a useful dimension. The numerical validation provides practical support, though the overall impact depends on the rigor of the discrete coercivity argument.

major comments (1)
  1. [Section 3 (Coercivity and Well-posedness)] The central well-posedness claim rests on coercivity of the DG form after application of the Morawetz multiplier. The analysis must explicitly verify that all volume integrals, numerical-flux consistency terms on interior faces, and jumps across the elliptic-hyperbolic interface produce non-negative contributions that can be absorbed into the energy norm with a constant independent of h and p. Without a detailed accounting of these discrete terms (particularly the transition-interface contributions when test functions are discontinuous), the coercivity constant may not be uniform, which would invalidate the subsequent hp-error estimates.
minor comments (2)
  1. [Section 2 (Formulation)] The notation for the broken energy norm and the precise definition of the numerical fluxes at the elliptic-hyperbolic interface should be stated more explicitly, perhaps with a dedicated subsection, to improve readability for readers outside the immediate DG community.
  2. [Section 5 (Numerical Results)] In the numerical experiments, several convergence plots lack clear labeling of the polynomial degrees or mesh sizes used; adding this information would make the validation of the hp-rates easier to follow.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment on the coercivity analysis. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Section 3 (Coercivity and Well-posedness)] The central well-posedness claim rests on coercivity of the DG form after application of the Morawetz multiplier. The analysis must explicitly verify that all volume integrals, numerical-flux consistency terms on interior faces, and jumps across the elliptic-hyperbolic interface produce non-negative contributions that can be absorbed into the energy norm with a constant independent of h and p. Without a detailed accounting of these discrete terms (particularly the transition-interface contributions when test functions are discontinuous), the coercivity constant may not be uniform, which would invalidate the subsequent hp-error estimates.

    Authors: In Section 3 we derive coercivity by applying the Morawetz multiplier directly to the DG weak form. After integration by parts, the volume integrals are controlled by the energy norm using the sign properties of the coefficient that distinguishes elliptic and hyperbolic regions. The numerical-flux consistency terms on interior faces are shown to vanish or contribute non-negatively by the standard DG flux definitions (upwind in hyperbolic regions and central in elliptic regions). At the elliptic-hyperbolic interface we exploit the continuity of the multiplier across the type-change curve together with the jump penalization in the DG form; the resulting interface integrals are estimated via trace inequalities and absorb into the energy norm without introducing h- or p-dependent factors. The full expansion appears in (3.8)–(3.22), where each contribution is bounded separately and the final coercivity constant is independent of h and p. We therefore maintain that the existing argument already supplies the required accounting, although we can add a short clarifying paragraph on the interface handling if the referee considers the presentation insufficiently explicit. revision: partial

Circularity Check

0 steps flagged

No circularity: standard multiplier technique applied to DG formulation

full rationale

The central claim of well-posedness rests on applying the classical Morawetz multiplier to the discontinuous Galerkin bilinear form to obtain coercivity in an energy norm. This is a direct, non-self-referential derivation that accounts for volume terms, interface fluxes, and the elliptic-hyperbolic transition; the subsequent hp-error estimates then follow from the resulting coercivity plus standard polynomial approximation theory. No step reduces by construction to a fitted parameter, a self-citation chain, or a renamed input. The analysis is self-contained against external benchmarks (Morawetz identity, DG consistency/stability theory).

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method builds on the standard discontinuous Galerkin framework and the classical Morawetz multiplier technique from the literature on hyperbolic PDEs. No new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Existence of a suitable Morawetz multiplier for the given class of mixed-type equations.
    Used to establish coercivity of the discrete bilinear form.

pith-pipeline@v0.9.0 · 5390 in / 1262 out tokens · 41340 ms · 2026-05-10T18:13:55.382818+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages

  1. [1]

    D. N. Arnold. An interior penalty finite element method with discontinuous elements.SIAM J. Numer. Anal., 19(4):742–760, 1982

    work page 1982

  2. [2]

    A. K. Aziz, H. D. Han, and M. Schneider. A finite element method for a boundary value problem of mixed type.Numer. Math., 44(2):191–200, 1984

    work page 1984

  3. [3]

    A. K. Aziz and M. Schneider. On uniqueness of Frankl-Morawetz problem inR 2.Monatsh. Math., 85(4):265–276, 1978

    work page 1978

  4. [4]

    A. K. Aziz, M. Schneider, and A. Werschulz. A finite element method for the Tricomi problem. Numer. Math., 35(1):13–20, 1980

    work page 1980

  5. [5]

    Babuˇ ska and M

    L. Babuˇ ska and M. Suri. The optimal convergence rate of thep-version of the finite element method. SIAM J. Numer. Anal., 24(4):750–776, 1987

    work page 1987

  6. [6]

    Bers.Mathematical aspects of subsonic and transonic gas dynamics

    L. Bers.Mathematical aspects of subsonic and transonic gas dynamics. Courier Dover Publications, 2016

    work page 2016

  7. [7]

    Cangiani, Z

    A. Cangiani, Z. Dong, and E. H. Georgoulis.hp-version discontinuous Galerkin methods on essen- tially arbitrarily-shaped elements.Math. Comp., 91(333):1–35, 2021

    work page 2021

  8. [8]

    Cangiani, Z

    A. Cangiani, Z. Dong, E. H. Georgoulis, and P. Houston.hp-version discontinuous Galerkin methods on polygonal and polyhedral meshes. SpringerBriefs in Mathematics. Springer, Cham, 2017

    work page 2017

  9. [9]

    Cinquini Cibrario

    M. Cinquini Cibrario. Equazioni a derivate parziali di tipo misto.Rend. Sem. Mat. Fis. Milano, 25:18–40, 1953/54

    work page 1953

  10. [10]

    Dautray and J.-L

    R. Dautray and J.-L. Lions.Mathematical analysis and numerical methods for science and technology. Vol. 4. Springer-Verlag, Berlin, 1990. Integral equations and numerical methods

    work page 1990

  11. [11]

    D. A. Di Pietro and A. Ern.Mathematical aspects of discontinuous Galerkin methods, volume 69 of Math´ ematiques & Applications (Berlin). Springer, Heidelberg, 2012

    work page 2012

  12. [12]

    Ern and J.-L

    A. Ern and J.-L. Guermond. Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory.SIAM J. Numer. Anal., 44(2):753–778, 2006

    work page 2006

  13. [13]

    Ern and J.-L

    A. Ern and J.-L. Guermond. Discontinuous Galerkin methods for Friedrichs’ systems. II. Second- order elliptic PDEs.SIAM J. Numer. Anal., 44(6):2363–2388, 2006

    work page 2006

  14. [14]

    G. J. Fix and M. D. Gunzburger. On least squares approximations to indefinite problems of the mixed type.International Journal for Numerical Methods in Engineering, 12(3):453–469, 1978

    work page 1978

  15. [15]

    G. J. Fix and M. E. Gurtin. On patched variational principles with application to elliptic and mixed elliptic-hyperbolic problems.Numer. Math., 28(3):259–271, 1977. 23

    work page 1977

  16. [16]

    K. O. Friedrichs. Symmetric positive linear differential equations.Comm. Pure Appl. Math., 11:333– 418, 1958

    work page 1958

  17. [17]

    M. Y. Huang. On discontinuous finite element approximation for the solution of Tricomi’s problem. J. Comput. Math., 3(4):289–297, 1985

    work page 1985

  18. [18]

    Imbert-G´ erard

    L.-M. Imbert-G´ erard. Local Taylor-based polynomial quasi-Trefftz spaces for scalar linear equations. arXiv preprint arXiv:2505.18480, 2025

    work page arXiv 2025

  19. [19]

    Imbert-G´ erard, A

    L.-M. Imbert-G´ erard, A. Moiola, C. Perinati, and P. Stocker. Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems.IMA J. Numer. Anal., 45(6):3473–3506, 2025

    work page 2025

  20. [20]

    Jensen.Discontinuous Galerkin methods for Friedrichs systems with irregular solutions

    M. Jensen.Discontinuous Galerkin methods for Friedrichs systems with irregular solutions. PhD thesis, University of Oxford, 2004

    work page 2004

  21. [21]

    Katsanis

    T. Katsanis. Numerical solution of Tricomi equation using theory of symmetric positive differential equations.SIAM J. Numer. Anal., 6:236–253, 1969

    work page 1969

  22. [22]

    Y. S. Kim.Analytical and numerical methods for boundary value problems of mixed type. PhD thesis, Texas A&M University, 1990

    work page 1990

  23. [23]

    K¨ ubler

    J. K¨ ubler. On the spectrum of a mixed-type operator with applications to rotating wave solutions. Calc. Var. Partial Differential Equations, 62(2):Paper No. 50, 30, 2023

    work page 2023

  24. [24]

    Lederer, C

    P. Lederer, C. Lehrenfeld, P. Stocker, and I. Voulis. A unified framework for Trefftz-like discretization methods.arXiv preprint arxiv:2412.00806, 2024

    work page arXiv 2024

  25. [25]

    Lehrenfeld and P

    C. Lehrenfeld and P. Stocker. Embedded Trefftz discontinuous Galerkin methods.Internat. J. Numer. Methods Engrg., 124(17):3637–3661, 2023

    work page 2023

  26. [26]

    Lozinski

    A. Lozinski. A primal discontinuous Galerkin method with static condensation on very general meshes.Numer. Math., 143(3):583–604, 2019

    work page 2019

  27. [27]

    D. Lupo, C. S. Morawetz, and K. R. Payne. On closed boundary value problems for equations of mixed elliptic-hyperbolic type.Comm. Pure Appl. Math., 60(9):1319–1348, 2007

    work page 2007

  28. [28]

    Moiola and I

    A. Moiola and I. Perugia. A space-time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation.Numer. Math., 138(2):389–435, 2018

    work page 2018

  29. [29]

    C. S. Morawetz. A uniqueness theorem for Frankl’s problem.Comm. Pure Appl. Math., 7:697–703, 1954

    work page 1954

  30. [30]

    C. S. Morawetz. A weak solution for a system of equations of elliptic-hyperbolic type.Comm. Pure Appl. Math., 11:315–331, 1958

    work page 1958

  31. [31]

    C. S. Morawetz. The Dirichlet problem for the Tricomi equation.Comm. Pure Appl. Math., 23:587– 601, 1970

    work page 1970

  32. [32]

    C. S. Morawetz.Lectures on nonlinear waves and shocks, volume 67 ofTata Institute of Fundamental Research Lectures on Mathematics and Physics. Tata Institute of Fundamental Research, Bombay; Springer-Verlag, Berlin-New York, 1981. Notes by P. S. Datti

    work page 1981

  33. [33]

    C. S. Morawetz. Mixed equations and transonic flow.J. Hyperbolic Differ. Equ., 1(1):1–26, 2004

    work page 2004

  34. [34]

    T. H. Otway.Elliptic-hyperbolic partial differential equations. SpringerBriefs in Mathematics. Springer, Cham, 2015. A mini-course in geometric and quasilinear methods

    work page 2015

  35. [35]

    K. R. Payne. Weak well posedness of the Dirichlet problem for equations of mixed ellptic-hyperbolic type.Le Matematiche, 60(2):315–327, 2005

    work page 2005

  36. [36]

    K. R. Payne. Multiplier methods for mixed type equations.Int. J. Appl. Math. Stat, 8:58–75, 2007

    work page 2007

  37. [37]

    Perinati

    C. Perinati. A quasi-Trefftz discontinuous Galerkin method for the homogeneous diffusion-advection- reaction equation with piecewise-smooth coefficients. Master’s thesis, University of Pavia, 2023. 24

    work page 2023

  38. [38]

    Perinati.Discontinuous Galerkin methods with Trefftz-type spaces

    C. Perinati.Discontinuous Galerkin methods with Trefftz-type spaces. PhD thesis, University of Pavia, in preparation

    work page

  39. [39]

    Perinati, L.-M

    C. Perinati, L.-M. Imbert-G´ erard, A. Moiola, and P. Stocker. Replication data for: A discontinu- ous Galerkin method for elliptic–hyperbolic equations.Zenodo, 2026.https://doi.org/10.5281/ zenodo.18998989

    work page 2026

  40. [40]

    M. H. Protter. Uniqueness theorems for the Tricomi problem.J. Rational Mech. Anal., 2:107–114, 1953

    work page 1953

  41. [41]

    M. H. Protter. Uniqueness theorems for the Tricomi problem. II.J. Rational Mech. Anal., 4:721–732, 1955

    work page 1955

  42. [42]

    J. M. Rassias.Lecture Notes on Mixed Type Partial Differential Equations. Number 4 in G - Reference, Information and Interdisciplinary Subjects Series. World Scientific, 1990

    work page 1990

  43. [43]

    J. L. Ripley and F. Pretorius. Gravitational collapse in Einstein dilation-Gauss-Bonnet gravity. Classical Quantum Gravity, 36(13):134001, 33, 2019

    work page 2019

  44. [44]

    J. L. Ripley and F. Pretorius. Hyperbolicity in spherical gravitational collapse in a Horndeski theory. Phys. Rev. D, 99(8):084014, 8, 2019

    work page 2019

  45. [45]

    Sch¨ oberl

    J. Sch¨ oberl. C++ 11 implementation of finite elements in NGSolve.Institute for analysis and scientific computing, Vienna University of Technology, 30, 2014

    work page 2014

  46. [46]

    Schwab.p- andhp-finite element methods

    C. Schwab.p- andhp-finite element methods. Numerical Mathematics and Scientific Computation. The Clarendon Press, Oxford University Press, New York, 1998. Theory and applications in solid and fluid mechanics

    work page 1998

  47. [47]

    P. Sermer. A Galerkin method for elliptic-hyperbolic type equations.SIAM J. Numer. Anal., 20(3):471–484, 1983

    work page 1983

  48. [48]

    E. M. Stein.Singular integrals and differentiability properties of functions, volume 30 ofPrinceton Mathematical Series. Princeton University Press, Princeton, NJ, 1970

    work page 1970

  49. [49]

    P. Stocker. NGSTrefftz: Add-on to NGSolve for Trefftz methods.J. Open Source Softw., 7(71):4135, 2022

    work page 2022

  50. [50]

    J. A. Trangenstein. A finite element method for the Tricomi problem in the elliptic region.SIAM J. Numer. Anal., 14(6):1066–1077, 1977

    work page 1977

  51. [51]

    F. G. Tricomi.Sulle equazioni lineari alle derivate parziali di 2°ordine, di tipo misto. Tipografia della Reale Accademia nazionale dei Lincei, 1923. 25

    work page 1923