pith. sign in

arxiv: 2604.03109 · v1 · submitted 2026-04-03 · 🧮 math.NA · cs.NA

An unconditionally stable space-time isogeometric method for a biharmonic wave equation

S. Chauhan , S. Chaudhary This is my paper

Pith reviewed 2026-05-13 18:55 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords biharmonic wave equationspace-time isogeometric analysisunconditional stabilitypenalty stabilizationB-spline discretizationvariational formulation
0
0 comments X

The pith

A non-consistent penalty term added to the space-time isogeometric discretization yields unconditional stability for the biharmonic wave equation.

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

The paper develops a space-time isogeometric method for solving the biharmonic wave equation, which models vibrations in thin plates and beams. It first shows that the continuous variational formulation has a unique solution. Using smooth B-spline basis functions to achieve the required H2 conformity, the discrete scheme is stable only when a CFL condition relating time step and mesh size is satisfied. By adding a non-consistent penalty term, the authors obtain unconditional stability, removing any restriction on the time step size. This allows efficient simulation of higher-order wave phenomena without the usual stability constraints.

Core claim

The central discovery is that incorporating a non-consistent penalty term into the space-time discrete formulation for the biharmonic wave equation produces unconditional stability, while the tensor-product structure of the discretization permits an efficient direct solver for the resulting linear systems.

What carries the argument

The non-consistent penalty term added to enforce unconditional stability in the space-time isogeometric discretization using globally C1 or higher continuous B-splines.

If this is right

  • The discrete problem becomes solvable without restrictions on the ratio of time step to spatial element size.
  • Convergence properties are preserved as shown by numerical experiments.
  • An efficient direct solver exploits the tensor product structure to solve the linear system.
  • Unique solvability is established for the continuous space-time variational formulation.

Where Pith is reading between the lines

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

  • Such penalty stabilization might extend to other fourth-order evolution equations in structural dynamics.
  • Removing the CFL constraint could significantly reduce computational cost for long-time simulations on fine spatial meshes.
  • Future work could analyze the consistency error introduced by the penalty term to bound its effect on accuracy.

Load-bearing premise

The non-consistent penalty term preserves the accuracy of the solution and does not introduce errors that prevent convergence to the true solution as the mesh is refined.

What would settle it

Numerical experiments showing that the error in the solution fails to converge to zero at the expected rate when the penalty parameter is active and the mesh is refined.

Figures

Figures reproduced from arXiv: 2604.03109 by S. Chaudhary, S. Chauhan.

Figure 1
Figure 1. Figure 1: Computational domains. Example 1. Unconditional stability: In this example, we consider the biharmonic wave equation (1.1) on the unit square domain Ω = (0, 1) × (0, 1), as illustrated in Figure 1a. The data f in equation (1.1) is chosen such that the manufactured solution of (1.1) is u(x, y, t) = t 2 sin2 (πx) sin2 (πy). We apply the proposed IgA-stab formulation (4.10), FEM￾stab formulation (4.11) and th… view at source ↗
Figure 2
Figure 2. Figure 2: Relative errors in (a) L 2 (L 2 (Ω)) norm, (b) L 2 (H1 0 (Ω)) ∩ H1 (L 2 (Ω)) norm and (c) X norm for the IgA stabilization method with splines of maximum regularity in both space and time direction for Square domain. Example 2. Performance of solver: To evaluate the performance of solver proposed in Section 5 for solving the linear system that raised from the IgA-stab formulation (4.10), here we perform on… view at source ↗
Figure 3
Figure 3. Figure 3: Relative errors in (a) L 2 (L 2 (Ω)) norm, (b) L 2 (H1 0 (Ω)) ∩ H1 (L 2 (Ω)) norm and (c) X norm for the IgA stabilization method with splines of maximum regularity in space and C p−2 regularity in time direction for Square domain. 10-2 10-1 100 h 10-20 100 1020 1040 1060 1080 10100 10120 10140 ||u-u h|| L 2(L 2 ( )/||u|| L 2(L 2 ( ) p=2 O(h2 ) p=3 O(h4 ) p=4 O(h5 ) (a) 10-2 10-1 100 h 100 1050 10100 10150… view at source ↗
Figure 4
Figure 4. Figure 4: Relative errors in (a) L 2 (L 2 (Ω)) norm, (b) L 2 (H1 0 (Ω)) ∩ H1 (L 2 (Ω)) norm and (c) X norm for the IgA stabilization method with splines of maximum regularity in space and C 0 regularity in time direction for Square domain. grows with factors 11.47, 14.02 and 13.52 for polynomial degrees 2, 3 and 4 respectively, which are smaller than the factor 16 resulting from the complexity O(N 4/3 dof ). The res… view at source ↗
Figure 5
Figure 5. Figure 5: Relative errors in (a) L 2 (L 2 (Ω)) norm, (b) L 2 (H1 0 (Ω)) ∩ H1 (L 2 (Ω)) norm and (c) X norm for the FEM stabilization method with splines of maximum regularity in both space and time direction for Square domain. 10-2 10-1 100 h 10-10 10-5 100 105 1010 1015 1020 1025 ||u-u h|| L 2(L 2 ( )/||u|| L 2(L 2 ( ) p=2 O(h2 ) p=3 O(h4 ) p=4 O(h5 ) (a) 10-2 10-1 100 h 10-10 10-5 100 105 1010 1015 1020 1025 1030 … view at source ↗
Figure 6
Figure 6. Figure 6: Relative errors in (a) L 2 (L 2 (Ω)) norm, (b) L 2 (H1 0 (Ω)) ∩ H1 (L 2 (Ω)) norm and (c) X norm for the FEM stabilization method with splines of maximum regularity in space and C p−2 regularity in time direction for Square domain. Example 3. Accuracy of the IgA-stab formulation: In this final example, we com￾pare the performance of the proposed IgA-stab formulation (4.10) with the FEM-stab for￾mulation (4… view at source ↗
Figure 7
Figure 7. Figure 7: Relative errors in (a) L 2 (L 2 (Ω)) norm, (b) L 2 (H1 0 (Ω)) ∩ H1 (L 2 (Ω)) norm and (c) X norm for the FEM stabilization method with splines of maximum regularity in space and C 0 regularity in time direction for Square domain. 10-2 10-1 100 h 10-10 10-5 100 105 1010 1015 1020 1025 ||u-u h|| L 2(L 2 ( )/||u|| L 2(L 2 ( ) p=2 O(h2 ) p=3 O(h4 ) p=4 O(h5 ) (a) 10-2 10-1 100 h 10-10 10-5 100 105 1010 1015 10… view at source ↗
Figure 8
Figure 8. Figure 8: Relative errors in (a) L 2 (L 2 (Ω)) norm, (b) L 2 (H1 0 (Ω)) ∩ H1 (L 2 (Ω)) norm and (c) X norm for the without stabilization method with splines of maxi￾mum regularity in both space and time direction for Square domain. We obtain the numerical solution of (1.1) using the IgA-stab formulation (4.10) for splines of maximal regularity and FEM-stab formulation (4.11) for splines of C 1 regularity in space an… view at source ↗
Figure 9
Figure 9. Figure 9: Relative errors in (a) L 2 (L 2 (Ω)) norm, (b) L 2 (H1 0 (Ω)) ∩ H1 (L 2 (Ω)) norm and (c) X norm for the without stabilization method with splines of maxi￾mum regularity in space and C p−2 regularity in time direction for Square domain. 10-2 10-1 100 h 10-10 100 1010 1020 1030 1040 1050 ||u-u h|| L 2(L 2 ( )/||u|| L 2(L 2 ( ) p=2 O(h2 ) p=3 O(h4 ) p=4 O(h5 ) (a) 10-2 10-1 100 h 10-10 100 1010 1020 1030 104… view at source ↗
Figure 10
Figure 10. Figure 10: Relative errors in (a) L 2 (L 2 (Ω)) norm, (b) L 2 (H1 0 (Ω)) ∩ H1 (L 2 (Ω)) norm and (c) X norm for the without stabilization method with splines of maxi￾mum regularity in space and C 0 regularity in time direction for Square domain. H1 0 (Ω) and H2 0 (Ω) norm obtained at final-time T. From [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Relative errors in (a) L 2 (Ω) norm, (b) H1 0 (Ω) norm and (c) H2 0 (Ω) norm for the IgA stabilization and FEM stabilization. 7. Conclusions In this work, we have analyzed the space-time isogeometric method for biharmonic wave equation (1.1). We proved the well-posedness of the space-time variational formulation when f ∈ L 2 (L 2 (Ω)) by exploiting the Fourier expansion of the trial and test functions. By… view at source ↗
read the original abstract

This work presents a space-time isogeometric analysis of biharmonic wave problem, in contrast to the more common application of space-time methods to second order wave equations. We first establish the unique solvability of the continuous space-time variational formulation. In order to obtain $H^2$- conforming discretization of the biharmonic wave equation, we consider globally smooth B-spline functions having continuity higher than $C^0$. We prove that the resulting space-time discrete formulation is stable under a Courant-Friedrichs-Lewy (CFL) condition. Furthermore, we propose a stabilized formulation, achieved by adding a non-consistent penalty term, which yields unconditional stability. Exploiting the tensor product structure, an efficient direct solver is also provided for solving the linear system arising from the discrete formulation. A few numerical experiments are presented to demonstrate the stability and convergence properties of the proposed scheme as well as the efficiency of the proposed solver.

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 / 1 minor

Summary. The paper develops a space-time isogeometric analysis (IGA) scheme for the biharmonic wave equation using globally C^k B-spline basis functions. It first proves unique solvability of the continuous space-time variational formulation, then shows that the standard discrete IGA formulation is stable only under a CFL condition. A non-consistent penalty term is added to obtain unconditional stability, an efficient direct solver exploiting the tensor-product structure is derived, and numerical experiments are presented to illustrate stability and convergence.

Significance. If the consistency error introduced by the non-consistent penalty can be rigorously controlled without degrading optimal convergence rates, the method would provide a useful unconditionally stable space-time discretization for fourth-order wave problems, where high-order continuity requirements make standard explicit schemes restrictive. The tensor-product direct solver is a practical contribution for IGA implementations.

major comments (2)
  1. [Abstract / stabilized formulation section] Abstract and the section introducing the stabilized formulation: the claim that the non-consistent penalty yields unconditional stability while preserving convergence is load-bearing, yet no explicit bound is given on the consistency error term that arises from breaking Galerkin orthogonality. The penalty parameter must be shown to scale with mesh size or time step so that this term vanishes at the design order; otherwise the headline convergence claim fails.
  2. [Discrete stability section / numerical experiments] The CFL-stability proof for the unpenalized discrete form (likely §4 or equivalent) and the subsequent unconditional-stability proof for the penalized form: both proofs need to be checked for whether the penalty term is treated consistently in the energy estimates, and whether the numerical experiments report observed rates for multiple penalty values to confirm that accuracy is not lost.
minor comments (1)
  1. [Stabilized formulation] Notation for the penalty parameter and its dependence on h and Δt should be introduced clearly when the stabilized formulation is first written.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will incorporate the suggested clarifications and additions in the revised version.

read point-by-point responses

  1. Referee: [Abstract / stabilized formulation section] Abstract and the section introducing the stabilized formulation: the claim that the non-consistent penalty yields unconditional stability while preserving convergence is load-bearing, yet no explicit bound is given on the consistency error term that arises from breaking Galerkin orthogonality. The penalty parameter must be shown to scale with mesh size or time step so that this term vanishes at the design order; otherwise the headline convergence claim fails.

    Authors: We agree that an explicit bound on the consistency error is required to rigorously justify the convergence claim. In the revised manuscript we will add a detailed a priori error analysis for the stabilized formulation. Specifically, we will derive an estimate showing that the consistency error term is bounded by a quantity that vanishes at the optimal rate when the penalty parameter is chosen to scale as O(h^{-2}) (or an analogous power of the mesh size), thereby preserving the design-order convergence. This analysis will be inserted into the section on the stabilized formulation. revision: yes

  2. Referee: [Discrete stability section / numerical experiments] The CFL-stability proof for the unpenalized discrete form (likely §4 or equivalent) and the subsequent unconditional-stability proof for the penalized form: both proofs need to be checked for whether the penalty term is treated consistently in the energy estimates, and whether the numerical experiments report observed rates for multiple penalty values to confirm that accuracy is not lost.

    Authors: We have re-checked the energy estimates. The penalty term is incorporated directly into the discrete energy norm used for both the CFL and unconditional stability proofs, so the estimates remain consistent. To make this transparent we will add a short clarifying paragraph in the stability sections that explicitly tracks the contribution of the penalty. In addition, we will augment the numerical experiments with convergence tables for at least three distinct penalty scalings (including values both above and below the recommended threshold) to demonstrate that optimal rates are retained whenever the scaling condition is satisfied. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with no circular reductions

full rationale

The paper first establishes unique solvability of the continuous space-time variational formulation for the biharmonic wave equation. It then proves CFL stability for the discrete IGA scheme using globally smooth B-splines. A non-consistent penalty term is added to remove the CFL restriction and obtain unconditional stability, with an efficient direct solver exploiting tensor-product structure. No load-bearing step reduces by construction to its own inputs: there are no self-definitional equivalences, no fitted parameters renamed as predictions, no uniqueness theorems imported from self-citations, and no ansatz smuggled via prior work. The claims rest on standard variational analysis and explicit penalty addition whose consistency effects are asserted to be controlled, without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Only the abstract is available, so concrete free parameters, background axioms, and invented entities cannot be audited in detail; the non-consistent penalty term functions as an ad-hoc stabilization device whose justification is deferred to the stability proof.

invented entities (1)
  • non-consistent penalty term no independent evidence
    purpose: to remove the CFL restriction and obtain unconditional stability
    Introduced explicitly in the stabilized formulation; no independent evidence outside the paper is supplied in the abstract.

pith-pipeline@v0.9.0 · 5459 in / 1246 out tokens · 52691 ms · 2026-05-13T18:55:03.069462+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

42 extracted references · 42 canonical work pages

  1. [1]

    Basu and N

    D. Basu and N. S. V. Kameswara Rao. Analytical solutions f or Euler–Bernoulli beam on visco-elastic foundation subjected to moving load. International Journal for Numerical and Analytical Methods in Geomechanics, 37(8):945–960, 2012. 2

    work page 2012

  2. [2]

    F. P. Bretherton. Resonant interactions between waves. The case of discrete oscillations. Journal of Fluid Mechanics, 20(3):457–479, 1964. 2

    work page 1964

  3. [3]

    Lazer and P.J

    A.C. Lazer and P.J. Mckenna. Large scale oscillatory beh aviour in loaded asymmetric systems. Annales de l’Institut Henri Poincaré C, Analyse non linéaire , 4(3):243–274, 1987. 2

    work page 1987

  4. [4]

    B. Li, G. Fairweather, and B. Bialecki. Discrete-time or thogonal spline collocation methods for vibra- tion problems. SIAM Journal on Numerical Analysis , 39(6):2045–2065, 2002. 2

    work page 2045

  5. [5]

    Danumjaya, A

    P. Danumjaya, A. K. Pany, and A. K. Pani. Morley FEM for the fourth-order nonlinear reaction- diffusion problems. Computers & Mathematics with Applications , 99:229–245, 2021. 2

    work page 2021

  6. [6]

    E. H. Georgoulis and J. M. Virtanen. Adaptive discontinu ous Galerkin approximations to fourth order parabolic problems. 84(295):2163–2190, 2015. 2

    work page 2015

  7. [7]

    Gudi and H

    T. Gudi and H. S. Gupta. A fully discrete C0 interior penalty Galerkin approximation of the extended Fisher-Kolmogorov equation. 247:1–16, 2013. 2

    work page 2013

  8. [8]

    A. Das, B. P. Lamichhane, and N. Nataraj. A unified mixed fin ite element method for fourth-order time-dependent problems using biorthogonal systems. Computers & Mathematics with Applications , 165:52–69, 2024. 2

    work page 2024

  9. [9]

    S. He, H. Li, and Y. Liu. Analysis of mixed finite element me thods for fourth-order wave equations. Computers & Mathematics with Applications , 65(1):1–16, 2013. 2

    work page 2013

  10. [10]

    Q. Tao, Y. Xu, and C. Shu. A discontinuous Galerkin metho d and its error estimate for nonlinear fourth-order wave equations. Journal of Computational and Applied Mathematics , 386:113230, 2021. 2 26 SHREYA CHAUHAN AND SUDHAKAR CHAUDHARY

    work page 2021

  11. [11]

    Bause, M

    M. Bause, M. Lymbery, and K. Osthues. C1-conforming variational discretization of the biharmonic wave equation. Computers & Mathematics with Applications , 119:208–219, 2022. 2

    work page 2022

  12. [12]

    M. He, J. Tian, P. Sun, and Z. Zhang. An energy-conservin g finite element method for nonlinear fourth-order wave equations. Applied Numerical Mathematics , 183:333–354, 2023. 2

    work page 2023

  13. [13]

    Nataraj, R

    N. Nataraj, R. Ruiz-Baier, and A. Yousuf. Semi- and full y-discrete analysis of lowest-order nonstan- dard finite element methods for the biharmonic wave problem. Computational Methods in Applied Mathematics, 25(4):921–948, 2025. 2

    work page 2025

  14. [14]

    T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeom etric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering , 194(39):4135–4195, 2005. 3

    work page 2005

  15. [15]

    Tagliabue, L

    A. Tagliabue, L. Dedè, and A. Quarteroni. Isogeometric analysis and error estimates for high order partial differential equations in fluid dynamics. Computers & Fluids , 102:277–303, 2014. 3

    work page 2014

  16. [16]

    Sogn and S

    J. Sogn and S. Takacs. Multigrid solvers for isogeometr ic discretizations of the second biharmonic problem. Mathematical Models and Methods in Applied Sciences , 33(09):1803–1828, 2023. 3

    work page 2023

  17. [17]

    Manni, E

    C. Manni, E. Sande, and H. Speleers. Outlier-free splin e spaces for isogeometric discretizations of biharmonic and polyharmonic eigenvalue problems. Computer Methods in Applied Mechanics and Engineering, 417:116314, 2023. A Special Issue in Honor of the Lifetime A chievements of T. J. R. Hughes. 3

    work page 2023

  18. [18]

    X. Meng, Y. Qin, and G. Hu. The convergence analysis of a c lass of stabilized semi-implicit iso- geometric methods for the Cahn-Hilliard equation. Journal of Scientific Computing , 102:26, 2025. 3

    work page 2025

  19. [19]

    Steinbach and H

    O. Steinbach and H. Yang. Space-time finite element meth ods for parabolic evolution equations: dis- cretization, a posteriori error estimation, adaptivity an d solution. In Space-Time Methods:Applications to Partial Differential Equations , volume 25, pages 207–248. De Gruyter, Berlin, Boston, 2019 . 3

    work page 2019

  20. [20]

    Langer, S

    U. Langer, S. E. Moore, and M. Neumüller. Space–time iso geometric analysis of parabolic evolution problems. Computer Methods in Applied Mechanics and Engineering , 306:342–363, 2016. 3

    work page 2016

  21. [21]

    G. Loli, M. Montardini, G. Sangalli, and M. Tani. An effici ent solver for space–time isogeometric Galerkin methods for parabolic problems. Computers & Mathematics with Applications , 80(11):2586– 2603, 2020. 3

    work page 2020

  22. [22]

    Montardini, M

    M. Montardini, M. Negri, G. Sangalli, and M. Tani. Space -time least-squares isogeometric method and efficient solver for parabolic problems. Mathematics of Computation , 89(323):1193–2227, 2020. 3

    work page 2020

  23. [23]

    Chaudhary, S

    S. Chaudhary, S. Chauhan, and M. Montardini. Space-tim e isogeometric method for a nonlocal par- abolic problem. Advances in Computational Mathematics , 52:12, 2026. 3

    work page 2026

  24. [24]

    Henning, D

    J. Henning, D. Palitta, V. Simoncini, and K. Urban. An ul traweak space-time variational formulation for the wave equation: Analysis and efficient numerical solut ion. ESAIM: Mathematical Modelling and Numerical Analysis, 56(4):1173–1198, 2022. 3

    work page 2022

  25. [25]

    Steinbach and M

    O. Steinbach and M. Zank. Coercive space-time finite ele ment methods for initial boundary value problems. ETNA - Electronic Transactions on Numerical Analysis , 52:154–194, 2020. 3, 4, 6, 7, 8, 12, 14

    work page 2020

  26. [26]

    Steinbach and M

    O. Steinbach and M. Zank. A Stabilized Space–Time Finite Element Method for the Wave E quation, pages 341–370. Springer International Publishing, Cham, 2 019. 3, 4, 8, 15

    work page

  27. [27]

    M. Zank. Higher-order space-time continuous galerkin methods for the wave equation. In 14th WCCM- ECCOMAS Congress 2020 , volume 700 of WCCM-ECCOMAS Congress, pages 1–10. Scipedia, 2021. 3, 8, 15

    work page 2020

  28. [28]

    Fraschini, G

    S. Fraschini, G. Loli, A. Moiola, and G. Sangalli. An unc onditionally stable space–time isogeometric method for the acoustic wave equation. Computers & Mathematics with Applications , 169:205–222,

    work page

  29. [29]

    Ferrari and S

    M. Ferrari and S. Fraschini. Stability of conforming sp ace-time isogeometric methods for the wave equation. Mathematics of Computation , 95(358):683–719, 2025. 4, 7, 8, 13, 14, 15, 18 SPACE-TIME IGA FOR BIHARMONIC W A VE EQUATION 27

    work page 2025

  30. [30]

    Bignardi and A

    P. Bignardi and A. Moiola. A space–time continuous and c oercive formulation for the wave equation. Numerische Mathematik , 157(4):1211–1258, 2025. 4

    work page 2025

  31. [31]

    Löscher, O

    R. Löscher, O. Steinbach, and M. Zank. Numerical result s for an unconditionally stable space-time finite element method for the wave equation. In Domain decomposition methods in science and engi- neering XXVI, volume 145 of Lecture Notes in Computational Science and Engineering, pages 625–632. Springer, Cham, 2022. 4

    work page 2022

  32. [32]

    Ferrari, I

    M. Ferrari, I. Perugia, and E. Zampa. Inf-sup stable spa ce-time discretization of the wave equation based on a first-order-in-time variational formulation. arXiv preprint arXiv:2506.05886 , 2025. 4

    work page arXiv 2025

  33. [33]

    When all else fails, integrate by parts

    E. A. Spence. “When all else fails, integrate by parts”: An overview of new and old variational formu- lations for linear elliptic PDEs. In A. S. Fokas and B. Pellon i, editors, Unified Transform for Boundary Value Problems: Applications and Advances , Other Titles in Applied Mathematics, chapter 6, pages 93–159. Society for Industrial and Applied Mathemat...

    work page

  34. [34]

    A. Moiola. Scattering of time-harmonic acoustic waves : Helmholtz equation, boundary integral equa- tions and BEM. Classnotes, 2021. 7

    work page 2021

  35. [35]

    Fraschini

    S. Fraschini. Stability of space-time isogeometric me thods for wave propagation problems. arXiv preprint arXiv:2303.15460, 2021. Master’s thesis. Universit‘a degli Studi di Pavia. 7

    work page arXiv 2021

  36. [36]

    Schumaker

    L. Schumaker. Spline Functions: Basic Theory . Cambridge University Press, 2007. 10

    work page 2007

  37. [37]

    Takacs and T

    S. Takacs and T. Takacs. Approximation error estimates and inverse inequalities for B-splines of maximum smoothness. Mathematical Models and Methods in Applied Sciences , 26(07):1411–1445,

    work page

  38. [38]

    Bazilevs, L

    Y. Bazilevs, L. Beirão da Veiga, J. A. Cottrell, T. J. R. H ughes, and G. Sangalli. Isogeometric analysis: Approximation, stability and error estimates for h-refined meshes. Mathematical Models and Methods in Applied Sciences , 16(07):1031–1090, 2006. 14

    work page 2006

  39. [39]

    Loli and G

    G. Loli and G. Sangalli. Space–time isogeometric analy sis: a review with application to wave propa- gation. SeMA Journal , 82(4):633–644, 2025. 16

    work page 2025

  40. [40]

    Ferrari, S

    M. Ferrari, S. Fraschini, G. Loli, and I. Perugia. Uncon ditionally stable space–time isogeometric discretization for the wave equation in Hamiltonian formul ation. ESAIM: M2AN , 59(5):2447–2490,

    work page

  41. [41]

    R. Vázquez. A new design for the implementation of isoge ometric analysis in octave and matlab: GeoPDEs 3.0. Computers & Mathematics with Applications , 72(3):523–554, 2016. 17

    work page 2016

  42. [42]

    Fraschini, G

    S. Fraschini, G. Loli, A. Moiola, and G. Sangali. Xtiga- waves. https://github.com/XTIgA-Waves/XTIgA-Waves.git, 2023. 17

    work page 2023