On Two Conservative HDG Schemes for Nonlinear Klein-Gordon Equation

Amiya Kumar Pani; Sangita Yadav; Shipra Gupta

REVIEW 3 major objections 5 minor 32 references

On Two Conservative HDG Schemes for Nonlinear Klein-Gordon Equation

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves optimal O(h^{k+1}) HDG convergence for the nonlinear Klein-Gordon equation, O(h^{k+2}) after post-processing, and a fully discrete scheme that conserves energy while keeping error O(h^{k+1} + (Δt)^2).

desk verdict The semidiscrete HDG analysis and energy conservation are solid and new, but the fully discrete error claims rest on a false lemma and an unstated fourth-order time regularity assumption. read the letter →

arxiv 2411.15572 v1 pith:XY7OBPUP submitted 2024-11-23 math.NA cs.NA

Shipra Gupta , Amiya Kumar Pani , Sangita Yadav This is my paper · ORCID

classification math.NAcs.NA MSC 65N1265N22

keywords HDGhybridizablediscontinuousGalerkinKlein-Gordonequationenergyconservationapriorierrorestimatessuperconvergencepost-processingnonlinearwave

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper analyzes two hybridizable discontinuous Galerkin (HDG) schemes for the nonlinear Klein-Gordon equation u_tt − Δu + f(u) = 0 with f(u) = u³ − u, a locally Lipschitz nonlinearity. It proves that the semidiscrete approximation converges at the optimal spatial rate O($h^{{k+1}}$) for both the displacement and the flux when piecewise polynomials of degree k are used, and that a cheap element-by-element post-processing of the displacement improves this to O($h^{{k+2}}$) for k ≥ 1. For the time discretization, a second-order conservative finite-difference scheme is shown to conserve a discrete energy exactly while keeping the overall error at O($h^{{k+1}}$ + (Δt)²); a non-conservative variant that is linear at each time step achieves the same rates. A second HDG variant, with displacement polynomials of degree k+1, gives O($h^{{k+2}}$) for the displacement. Numerical experiments with polynomial degrees 1 to 3 confirm the predicted convergence orders and show energy errors at machine precision.

What carries the argument

The load-bearing machinery is the HDG spatial discretization together with its projection-based error analysis. The method seeks a triple (u_h, q_h, û_h) of piecewise polynomials of degree k, where q_h approximates the flux ∇u, u_h the displacement, and û_h a single-valued trace on the mesh faces; the numerical flux is defined by q̂_h·n = q_h·n − τ(u_h − û_h). The HDG projection (Π_W, Π_V) is defined locally by L² orthogonality conditions and a trace condition on element faces, and it transfers the regularity of the exact solution into sharp O($h^{{k+1}}$) projection errors. The discrete energy identity supplies the boundedness that tames the locally Lipschitz nonlinear term, and for the fully discrete scheme the symmetric quotient F($U^{{n+1}}$,$U^{{n−1}}$) = (F($U^{{n+1}}$) − F($U^{{n−1}}$))/($U^{{n+1}}$ − $U^{{n−1}}$) with F(s) = (1−s²)²/4 makes the discrete energy exactly conserved.

What would settle it

Take the fully discrete conservative scheme on the unit square with the manufactured solution u(x,y,t) = $t^{{3.2}}$ sin(πx) sin(πy), whose fourth time derivative is not square-integrable near t = 0; if the observed temporal order of convergence is still 2 in Δt, the u_tttt hypothesis is not necessary, whereas a drop below 2 would show that the missing hypothesis is real. A complementary test is to solve the same problem on a domain with a re-entrant corner, where the spatial solution is only $H^{{k+1}}$; the postprocessed O($h^{{k+2}}$) rate should degrade if it depends on elliptic regularity.

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Extended reading notes

Core claim

The paper establishes that the HDG framework, which rewrites the second-order wave equation as a first-order system in a displacement variable u_h and a flux variable q_h together with a single-valued trace û_h on element faces, can handle the locally Lipschitz nonlinearity f(u) = u³ − u without any global Lipschitz condition. The key step is the discrete energy identity E(t) = 1/2(‖u_ht‖² + ‖q_h‖² + ‖u_h − û_h‖²₊ + 2(F(u_h),1)), with F(s) = (1−s²)²/4, which is conserved exactly and yields the L^p bounds needed to control the nonlinear term. On the strength of this identity and the HDG projection, the paper proves optimal O($h^{{k+1}}$) error estimates for displacement and flux, superconvergence O($h^{{k+2}}$) after local post-processing for k ≥ 1, and a fully discrete error bound O($h^{{k+1}}$ + (Δt)²) for a conservative time-stepping scheme that preserves the discrete energy exactly. The analysis also covers a non-conservative linear-per-step scheme and a mixed-degree variant, and the claims are backed by numerical experiments.

Load-bearing premise

For the fully discrete error bound to hold, the exact solution must have a square-integrable fourth time derivative, a regularity condition that is used in the Taylor remainder estimates but not stated in the theorem; if the solution is less smooth in time, the (Δt)² part of the claimed convergence is not proven.

Editorial extensions

If this is right

The semidiscrete HDG approximation converges at rate O(h^{k+1}) in L² for both displacement and flux, and the locally post-processed displacement converges at rate O(h^{k+2}) for polynomial degree k ≥ 1.
The fully discrete conservative scheme conserves the discrete energy exactly, E^{n+1/2} = E^{3/2}, and has error O(h^{k+1} + (Δt)²).
The non-conservative variant, which solves only linear systems at each time step, attains the same optimal error rates.
The mixed-degree HDG variant, with displacement polynomials of degree k+1 and flux polynomials of degree k, yields O(h^{k+2}) for the displacement and O(h^{k+1}) for the flux.
Numerical experiments with polynomial degrees 1 through 3 match the predicted convergence orders and give energy errors at machine precision.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension is to apply the same symmetric-quotient energy-conserving time discretization to other Hamiltonian wave equations with a polynomial potential, such as the sine-Gordon equation; the proof structure would likely carry over as long as the potential gives a priori L^p bounds.
The gap between the stated regularity in the fully discrete theorem and the regularity actually used in the Taylor remainders suggests that the (Δt)² estimate could potentially be proved under weaker time regularity by adapting the paper's own nonstandard energy technique, and a numerical test with rough-in-time data would reveal whether the missing hypothesis is essential.
Because the discrete energy is an exact invariant, the conservative scheme is a candidate for long-time integration of the Klein-Gordon equation; one could test this by measuring phase and amplitude errors over very long intervals.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The part of this paper worth keeping is the semidiscrete analysis. Extending the Cockburn-Fu-Hungria-Ji-Sanchez-Sayas HDG framework to the locally Lipschitz Klein-Gordon nonlinearity, with the Lp bounds that handle f(u)=u^3-u and a k+2 postprocessing result, is a genuine contribution. The energy-conservation identities in Theorems 2.1 and 5.1 check out, and the a priori estimates for the semidiscrete scheme are coherent. I would cite that part.

The fully discrete part is where the paper breaks. The stress-test note is right. Lemma 5.1 as stated is false: take Phi1=Psi1=M, Phi2=Psi2=0. The left side grows like M^6 while the right side only grows like M^4. That lemma is used in the uniqueness proof and in the error analysis, so the proof of Theorem 5.2 and Theorem 5.3 is not valid as written. The fix is probably repairable - use a quartic prefactor and the uniform L6 bounds from the energy conservation - but it has to be done.

More serious is the missing u_tttt regularity in Theorem 5.3. The bound (5.28) contains (Delta t)^3 integral ||u_tttt||^2 ds, and the theorem statement imposes no regularity beyond what was assumed for the semidiscrete case. The counterexample u(x,t)=t^(5/2) sin(pi x) sin(pi y) makes that integral infinite, so the claimed O((Delta t)^2) error bound is not established for the class of solutions the theorem claims to cover. This is not cosmetic. The same expansion is used in Theorem 5.4, so the non-conservative scheme has the same gap. The authors need to state the additional smoothness explicitly and check whether the problem data admit it.

Section 6 is also thin. The superconvergence proof for the second variant is a sketch, with multiple steps deferred to 'similarly' and 'combining all the terms.' The numerical tables are mostly believable but contain some impossible experimental order entries (e.g., jumps in error that cannot produce the reported EOC).

Bottom line: this is not a desk reject. The semidiscrete analysis is worth refereeing, and the fully discrete problems are likely repairable. But the current version does not support the abstract's fully discrete claims. Send it to a knowledgeable referee with instructions to focus on Sections 5 and 6, and ask for a revision that fixes Lemma 5.1 and states the missing regularity.

Referee Report

3 major / 5 minor

Summary. The paper develops and analyzes two hybridizable discontinuous Galerkin (HDG) schemes for the nonlinear Klein-Gordon equation u_tt - Δu + f(u) = 0 with f(u) = u^3 - u. The main claims are: optimal O(h^{k+1}) semidiscrete error for both displacement and flux, O(h^{k+2}) superconvergence for a local postprocessed displacement when k ≥ 1, discrete energy conservation for a fully discrete conservative scheme, fully discrete errors of order O(h^{k+1} + (Δt)^2) for both conservative and non-conservative time discretizations, and a variant HDG method with O(h^{k+2}) displacement error. The analysis uses HDG projection estimates, L^p bounds for the discrete solution to handle the locally Lipschitz nonlinearity, a Baker-type argument for the displacement, and a Brouwer fixed-point argument for discrete well-posedness. Numerical experiments for k = 1, 2, 3 and for several exact solutions are reported and confirm the predicted orders and energy conservation.

Significance. If the results are correct, the paper would be a useful contribution to HDG methods for nonlinear hyperbolic problems: it addresses a locally Lipschitz polynomial nonlinearity without imposing a global Lipschitz condition, proves discrete energy conservation, and provides postprocessing superconvergence together with fully discrete second-order time error estimates. The semidiscrete part is structurally substantial, and the L^p technique used to control the nonlinearity is a genuine strength. However, the fully discrete claims currently rest on an unstated fourth-order time regularity assumption and on a false algebraic inequality in Lemma 5.1, while the superconvergence proof for the variant in Section 6 is only sketched. No fitting of parameters or circularity was detected; the concerns are about technical correctness of key proof steps.

major comments (3)

[Section 5.1, Lemma 5.1] The inequality stated in Lemma 5.1 and displayed in Eq. (5.8) is false as written. Taking Φ1 = Ψ1 = M, Φ2 = Ψ2 = 0 gives a left-hand side behaving like ‖M^3 - M‖^2, which is of order M^6 for large M, whereas the right-hand side is of order M^4 since each difference factor contributes M^2 and the bracketed factor contributes M^2. This lemma is used in the uniqueness proof at Eq. (5.16) and in the error estimate at Eq. (5.30) to reduce the nonlinear term to a difference squared in L^6 times a bounded factor. The lemma is repairable by putting a quartic power in the bracketed factor and then using the uniform L^6 bounds from Theorem 5.1, but as stated the estimate is not valid and the subsequent proofs relying on it lack support.
[Theorem 5.3, Eq. (5.28)] The fully discrete error analysis in Theorem 5.3 needs an unstated regularity assumption on the exact solution. In Eq. (5.28), the Taylor remainder terms R*_1 and R*_2 involve u_tttt, and the displayed bound contains ∫_{t_{n-1}}^{t_{n+1}} ‖u_tttt(s)‖^2 ds. The theorem statement, however, imposes no regularity hypotheses on u beyond the semidiscrete context. The same expansion is used in Theorem 5.4. Without an explicit assumption such as u_tttt ∈ L^2(0,T;L^2(Ω)), the right-hand side of (5.28) is not controlled, so the claimed O((Δt)^2) rates in (5.25), (5.26), (5.39), and (5.40) are not proved. This is not a cosmetic omission: second-order accuracy for this implicit midpoint-type scheme genuinely depends on this fourth-order time smoothness, and the theorem must state it.
[Section 6.2, proof of Eq. (6.5)] The proof of the O(h^{k+2}) superconvergence result (6.5) for the variant HDG method is only sketched. After the decomposition into Q1,...,Q8, several terms are bounded, but the text then says that the remaining terms are handled 'systematically' or 'similarly as done in subsection 4.2' without giving the actual estimates. In particular, the nonlinear terms Q4 and Q8 and the time-integrated φ terms require the local Lipschitz structure of f and are not straightforward copies of the earlier argument because the projection and error equations for the variant differ. Since the abstract explicitly advertises the error estimates for this variant, the proof needs to be completed rather than summarized.

minor comments (5)

[Eq. (5.37c)] In the non-conservative scheme, the flux equation (5.37c) writes the test function as v instead of v_h; this should be corrected.
[Proof of Theorem 5.1] In the proof of Theorem 5.1, the sentence 'By adding ... on both sides of (5.37c)' appears to refer to Eq. (5.5), not to (5.37c).
[Lemma 3.1] In the proof of Lemma 3.1, the sentence 'an application of Theorem 2.1 implies' is misleading; the argument used is a Cauchy-Schwarz/Hölder bound, not the energy conservation theorem.
[Theorem 6.1] The statement of Theorem 6.1 says 'u0, u1 ∈ L∞(0,T;H^{k+2})', but u0 and u1 are initial data, not time-dependent functions; the statement should read u0, u1 ∈ H^{k+2}.
[Notation, Section 5] The time-difference notation is overloaded: both ∂_t and overbar-∂_t are introduced in Section 5.1, and the distinction between ∂_tϕ^n and ∂_tϕ^n is easy to miss; a summary table or consistent use of different symbols would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the error and energy results are proved from external HDG projection theory and standard energy arguments; self-citations are minor and not load-bearing.

full rationale

The paper's central claims—semidiscrete O(h^{k+1}) convergence, postprocessing O(h^{k+2}), discrete energy conservation, and fully discrete O(h^{k+1}+(Δt)^2) error estimates—are not obtained by assuming the quantities they purport to prove. The error analysis uses the standard decomposition e_u=θ_u−ϑ_u, with projection errors ϑ controlled by the external HDG projection results of [11] (Lemma 2.1) and with θ estimated by energy arguments (Lemmas 3.3–3.5, Theorems 3.1–3.2). The fully discrete analysis of Theorem 5.3 proceeds from the same θ-error equations and Taylor expansions; the cited works involving the authors, [24,29,32], are used for standard technique (Baker-type estimates and Taylor remainder bounds) and are not themselves the target results. No parameter is fitted to data, and no quantity is defined in terms of the quantity being predicted. The discrete energy E(t) in Theorem 2.1 is defined from the scheme variables and shown to be conserved by direct differentiation of the HDG equations, not by invoking conservation. The concerns raised by the skeptic are correctness issues rather than circularity: Eq. (5.28) requires u_tttt∈L²(0,T;L²) although Theorem 5.3 states no explicit regularity hypotheses on u, and Lemma 5.1's claimed inequality appears scale-inconsistent and false as written (e.g., Φ1=Ψ1=M, Φ2=Ψ2=0). These may require repair or additional hypotheses, but they do not make the derivation circular. The low nonzero score reflects only the presence of minor self-citations that are not load-bearing.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central estimates rest on standard finite element machinery (HDG projection, elliptic regularity, duality) and on an unstated fourth-order time regularity assumption; no free parameters are fitted.

assumptions (5)

standard math The HDG projection (Pi_V, Pi_W) from [11] is well-defined and has the approximation properties in Lemma 2.1.
Invoked throughout Section 3 without proof; the paper cites Appendix A of [11].
domain assumption On convex polygonal domains, the Poisson problem with L^q data has W^{2,q} regularity.
Used in Lemma 3.1 to derive L^p bounds for the discrete solution; this requires Omega convex and is standard but unproved.
domain assumption The exact solution u is smooth enough to lie in H^{k+2} in space and W^{3,1} or W^{2,2} in time, and f(u) = u^3 - u is locally Lipschitz with u bounded in L^infty.
Stated in the semidiscrete theorems and used for the mean-value estimates and the dual problem; the bounds fail if u is only H^{k+1}.
ad hoc to paper For the fully discrete scheme, u_tttt lies in L^2(0,T;L^2(Omega)) although Theorem 5.3 does not state this.
The Taylor remainders R*_1 and R*_2 in Eq (5.28) are controlled with fourth time derivatives; without this hypothesis the (Delta t)^2 error bound is not proved.
domain assumption Proposition 4.1's H^2 bound for the time-integrated dual solution holds with constant independent of h.
Assumed in the postprocessing superconvergence analysis; only a sketch of the energy argument is given.

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Cite this review

Pith. "Pith review of On Two Conservative HDG Schemes for Nonlinear Klein-Gordon Equation." pith.science (2026). https://pith.science/paper/XY7OBPUP

@misc{pith2026241115572,
  author       = {Pith},
  title        = {Pith review of: On Two Conservative HDG Schemes for Nonlinear Klein-Gordon Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XY7OBPUP}},
  note         = {Machine review of arXiv:2411.15572}
}

abstract

In this article, a hybridizable discontinuous Galerkin (HDG) method is proposed and analyzed for the Klein-Gordon equation with local Lipschitz-type non-linearity. {\it A priori} error estimates are derived, and it is proved that approximations of the flux and the displacement converge with order $O(h^{k+1}),$ where $h$ is the discretizing parameter and $k$ is the degree of the piecewise polynomials to approximate both flux and displacement variables. After post-processing of the semi-discrete solution, it is shown that the post-processed solution converges with order $O(h^{k+2})$ for $k \geq 1.$ Moreover, a second-order conservative finite difference scheme is applied to discretize in time %second-order convergence in time. and it is proved that the discrete energy is conserved with optimal error estimates for the completely discrete method. %Since at each time step, one has to solve a nonlinear system of algebraic equations, To avoid solving a nonlinear system of algebraic equations at each time step, a non-conservative scheme is proposed, and its error analysis is also briefly established. Moreover, another variant of the HDG scheme is analyzed, and error estimates are established. Finally, some numerical experiments are conducted to confirm our theoretical findings.

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Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

T. Achouri. Finite diﬀerence schemes for the two-dimensional se milinear wave equation. Numerical Methods for Partial Diﬀerential Equations , 35(1):200–221, 2019. 29

work page 2019
[2]

Baccouch

M. Baccouch. Optimal error estimates of the local discontinuou s Galerkin method for the two- dimensional sine-Gordon equation on cartesian grids. International Journal of Numerical Analysis & Modeling, 16(3):436–462, 2019

work page 2019
[3]

G. A. Baker. Error estimates for ﬁnite element methods for sec ond order hyperbolic equations. SIAM journal on numerical analysis , 13(4):564–576, 1976

work page 1976
[4]

Brezzi, J

F. Brezzi, J. Douglas, and L. D. Marini. Two families of mixed ﬁnite ele ments for second order elliptic problems. Numerische Mathematik , 47:217–235, 1985

work page 1985
[5]

G. Chen, B. Cockburn, J. Singler, and Y. Zhang. Superconverg ent interpolatory HDG methods for reaction diﬀusion equations. Journal of Scientiﬁc Computing , 81(3):2188–2212, 2019

work page 2019
[6]

Cockburn

B. Cockburn. The hybridizable discontinuous Galerkin methods. I n Proceedings of the International Congress of Mathematicians 2010, (volume 4) Vol. I: Plenary Lectures and Ceremonies Vols. II–IV: Invited Lectures, pages 2749–2775. World Scientiﬁc, 2010

work page 2010
[7]

Cockburn

B. Cockburn. Hybridizable discontinuous Galerkin methods for se cond-order elliptic problems: overview, a new result and open problems. Japan Journal of Industrial and Applied Mathematics , 40(3):1637–1676, 2023

work page 2023
[8]

Cockburn, Z

B. Cockburn, Z. Fu, A. Hungria, L. Ji, M. A. Sanchez, and F.J. Sa yas. Stormer-Numerov HDG methods for acoustic waves. Journal of Scientiﬁc Computing , 75(2):597–624, 2018

work page 2018

Show all 32 references

[9]

Cockburn and J

B. Cockburn and J. Gopalakrishnan. The derivation of hybridizab le discontinuous Galerkin methods for stokes ﬂow. SIAM Journal on Numerical Analysis , 47(2):1092–1125, 2009

2009
[10]

Cockburn, J

B. Cockburn, J. Gopalakrishnan, and R. Lazarov. Uniﬁed hybr idization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic pro blems. SIAM Journal on Numerical Analysis, 47(2):1319–1365, 2009

2009
[11]

Cockburn, J

B. Cockburn, J. Gopalakrishnan, and F.-J. Sayas. A projectio n-based error analysis of HDG meth- ods. Mathematics of Computation , 79(271):1351–1367, 2010

2010
[12]

Cockburn, W

B. Cockburn, W. Qiu, and K. Shi. Conditions for superconverge nce of HDG methods for second- order elliptic problems. Mathematics of Computation , 81(279):1327–1353, 2012

2012
[13]

Cockburn and V

B. Cockburn and V. Quenneville-B´ elair. Uniform-in-time superc onvergence of the HDG methods for the acoustic wave equation. Mathematics of Computation , 83(285):65–85, 2014

2014
[14]

Cockburn and K

B. Cockburn and K. Shi. Conditions for superconvergence of H DG methods for Stokes ﬂow. Math- ematics of Computation , 82(282):651–671, 2013

2013
[15]

Cockburn, J

B. Cockburn, J. R. Singler, and Y. Zhang. Interpolatory HDG m ethod for parabolic semilinear PDEs. Journal of Scientiﬁc Computing , 79(3):1777–1800, 2019

2019
[16]

J. F. Currie, J. A. Krumhansl, A. R. Bishop, and S. E. Trullinger. Statistical mechanics of one- dimensional solitary-wave-bearing scalar ﬁelds: Exact results and ideal-gas phenomenology. Physical Review B , 22(2):477, 1980

1980
[17]

D. B. Duncan. Sympletic ﬁnite diﬀerence approximations of the n onlinear Klein-Gordon equation. SIAM Journal on numerical analysis , 34(5):1742–1760, 1997

1997
[18]

Jim´ enez and L

S. Jim´ enez and L. V´ azquez. Analysis of four numerical schem es for a nonlinear Klein-Gordon equation. Applied Mathematics and Computation , 35(1):61–94, 1990

1990
[19]

Johnson and V

C. Johnson and V. Thom´ ee. Error estimates for some mixed ﬁn ite element methods for parabolic type problems. RAIRO. Analyse num´ erique, 15(1):41–78, 1981

1981
[20]

Kapitanski

L. Kapitanski. Global and unique weak solutions of nonlinear wave equations. Mathematical Research Letters, 1(2):211–223, 1994

1994
[21]

S. Kesavan. Topics in Functional Analysis and Applications . New Age International Pvt Limited Publishers, 2015. 30

2015
[22]

R. M. Kirby, S. J. Sherwin, and B. Cockburn. To HDG or to CG: A c omparative study. Journal of Scientiﬁc Computing , 51(1):183–212, 2012

2012
[23]

H. Kragh. Equation with the many fathers. The Klein–Gordon eq uation in 1926. American Journal of Physics , 52(11):1024–1033, 1984

1926
[24]

A. K. Pani and J. Y. Yuan. Mixed ﬁnite element method for a stro ngly damped wave equation. Numerical Methods for Partial Diﬀerential Equations: An In ternational Journal , 17(2):105–119, 2001

2001
[25]

W. Qiu, J. Shen, and K. Shi. An HDG method for linear elasticity with strong symmetric stresses. Mathematics of Computation , 87(309):69–93, 2018

2018
[26]

Qiu and K

W. Qiu and K. Shi. An HDG method for convection diﬀusion equation . Journal of Scientiﬁc Computing, 66(1):346–357, 2016

2016
[27]

S´ anchez, T

N. S´ anchez, T. S´ anchez-Vizuet, and M. Solano. Error analy sis of an unﬁtted HDG method for a class of non-linear elliptic problems. Journal of Scientiﬁc Computing , 90(3):1–28, 2022

2022
[28]

S´ anchez, T

N. S´ anchez, T. S´ anchez-Vizuet, and M. E. Solano. A priori a nd a posteriori error analysis of an unﬁtted HDG method for semi-linear elliptic problems. Numerische Mathematik , 148(4):919–958, 2021

2021
[29]

R. K. Sinha and A. K. Pani. The eﬀect of spatial quadrature on ﬁ nite element galerkin approxima- tions to hyperbolic integro-diﬀerential equations. Numerical functional analysis and optimization , 19(9-10):1129–1153, 1998

1998
[30]

Tourigny

Y. Tourigny. Product approximation for nonlinear Klein-Gordon equations. IMA journal of Nu- merical Analysis, 10(3):449–462, 1990

1990
[31]

Y. Wong, Q. Chang, and L. Gong. An initial-boundary value proble m of a nonlinear Klein-Gordon equation. Applied Mathematics and Computation , 84(1):77–93, 1997

1997
[32]

Yadav and A

S. Yadav and A. K. Pani. Superconvergent discontinuous Galer kin methods for nonlinear parabolic initial and boundary value problems. Journal of Numerical Mathematics , 27(3):183–202, 2019. 31

2019

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

T. Achouri. Finite diﬀerence schemes for the two-dimensional se milinear wave equation. Numerical Methods for Partial Diﬀerential Equations , 35(1):200–221, 2019. 29

work page 2019

[2] [2]

Baccouch

M. Baccouch. Optimal error estimates of the local discontinuou s Galerkin method for the two- dimensional sine-Gordon equation on cartesian grids. International Journal of Numerical Analysis & Modeling, 16(3):436–462, 2019

work page 2019

[3] [3]

G. A. Baker. Error estimates for ﬁnite element methods for sec ond order hyperbolic equations. SIAM journal on numerical analysis , 13(4):564–576, 1976

work page 1976

[4] [4]

Brezzi, J

F. Brezzi, J. Douglas, and L. D. Marini. Two families of mixed ﬁnite ele ments for second order elliptic problems. Numerische Mathematik , 47:217–235, 1985

work page 1985

[5] [5]

G. Chen, B. Cockburn, J. Singler, and Y. Zhang. Superconverg ent interpolatory HDG methods for reaction diﬀusion equations. Journal of Scientiﬁc Computing , 81(3):2188–2212, 2019

work page 2019

[6] [6]

Cockburn

B. Cockburn. The hybridizable discontinuous Galerkin methods. I n Proceedings of the International Congress of Mathematicians 2010, (volume 4) Vol. I: Plenary Lectures and Ceremonies Vols. II–IV: Invited Lectures, pages 2749–2775. World Scientiﬁc, 2010

work page 2010

[7] [7]

Cockburn

B. Cockburn. Hybridizable discontinuous Galerkin methods for se cond-order elliptic problems: overview, a new result and open problems. Japan Journal of Industrial and Applied Mathematics , 40(3):1637–1676, 2023

work page 2023

[8] [8]

Cockburn, Z

B. Cockburn, Z. Fu, A. Hungria, L. Ji, M. A. Sanchez, and F.J. Sa yas. Stormer-Numerov HDG methods for acoustic waves. Journal of Scientiﬁc Computing , 75(2):597–624, 2018

work page 2018