J. Chabassier, R. Tournemenne, M. Ainsworth, and . Oden, A posteriori error estimation in finite 718 element anaysis, Comput. Methods in Appl, vol.717, p.716, 1997.

, Mech. Eng, vol.142, pp.1-88

. Amestoy, Multifrontal par-722 allel distributed symmetric and unsymmetric 723 solvers, Comput. Methods in Appl. Mech. Eng, vol.724, pp.501-520, 2000.

I. Babuska and M. R. Dorr, 726 "Error Estimates for the combined h and p ver-727 sions of the finite element method, 1981.

, Math, vol.37, pp.257-277

J. Backus, Input impedance 730 curves for the brass instruments, The Journal 731 of the Acoustical Society of America, vol.60, issue.2, p.480, 1976.

. Berjamin, , p.734, 2017.

C. Vergez and E. Cottanceau, Time-735 Domain numerical modeling of brass instruments 736 including nonlinear wave propagation, viscother-737 mal losses, and lips vibration, Acta Acust 738 united Ac, vol.103, issue.1, pp.117-131, 2017.

S. Bilbao, Direct simulation 740 of reed wind instruments, Computer Music 741 Journal, vol.33, issue.4, pp.43-55, 2009.

S. Bilbao and J. Chick, Finite difference time domain simula-744 tion for the brass instrument bore, J. Acoust, p.743, 2013.

, Soc. Am, vol.134, issue.5, pp.3860-3871

A. C. Braden, Bore opti-747 misation and impedance modelling of brass musi-748 cal instruments, 2007.

[. Braden, , p.751, 2009.

M. J. Campbell and D. M. , Trombone 752 bore optimization based on input impedance tar-753 gets, J. Acoust. Soc. Am, vol.125, issue.4, pp.2404-2412, 2009.

H. Brezis, Functional analy-755 sis, Sobolev spaces and partial differential equa-756 tions, 2011.

[. Buys, , p.758, 2017.

R. , Developing and evaluating a hybrid 759 wind instrument, Acta Acust united Ac, vol.103, issue.5, pp.830-846, 2017.

M. Campbell-;-campbell, Brass in-762 struments as we know them today, Acta Acust 763 united Ac, vol.90, issue.4, pp.600-610, 2004.

[. Caussé, Input impedance of brass 766 musical instruments-comparison between ex-767 periment and numerical models, J. Acoust. Soc. 768 Am, vol.75, issue.1, pp.241-254, 1984.

T. Chabassier, J. Chabassier, and R. Tournemenne, About the trans-771 fert matrix method in the context of acoustical 772 wave propagation in wind instruments, 2019.

A. Chaigne and J. Kergomard, Modern Acoustics and 776 Signal Processing "Acoustics of Musical Instru-777 ments, 2016.

C. Chandler-wilde and S. N. , The impedance boundary value prob-780 lem for the Helmholtz equation in a half-plane, p.781, 1997.

, Mathematical Methods in the Applied Sciences, vol.782, issue.20, pp.813-840

G. Cohen-;-cohen, Higher Order Nu-784 merical Methods for Transient Wave Equations, 2004.

G. Cohen-;-cohen and S. Fauqueux, Mixed finite elements with mass-lumping for the 788 transient wave equation, Journal of Computa-789 tional Acoustics, vol.787, issue.1, pp.171-188, 2000.

H. ;. Courant and R. Courant, , p.791, 1965.

D. Hilbert, Methods of mathemat-792 ical physics. partial differential equations, 1965.

. Dalmont, , p.795, 2001.

C. J. , J. , and N. , Radiation impedance 796 of tubes with different flanges: Numerical and ex-797 perimental investigations, Journal of Sound, vol.798, issue.3, pp.505-534, 2001.

[. Dauge, Exponential convergence of 801 hp-fem for maxwell's equations with weighted 802 regularization in polygonal domains, 2005.

, Models Methods Appl. Sci, vol.15, issue.4, pp.575-622

M. Fortin, An analysis of the 805 convergence of mixed finite element methods, 806 RAIRO. Analyse Numérique, vol.11, issue.4, pp.341-354, 1977.

T. Hélie, T. Hézard, R. Mignot, and D. Matignon, , 2013.

, One-dimensional acoustic models of horns and 823 comparison with measurements, Acta Acust 824 united Ac, vol.99, issue.6, pp.960-974

. Ihlenburg and F. Babu?ka-;-ihlenburg, , p.826, 1995.

I. Babu?ka, Dispersion analysis and er-827 ror estimation of galerkin finite element meth-828 ods for the Helmholtz equation, Int. J. Numer, 1995.

, Methods Engrg, vol.38, pp.3745-3774

W. Kausel, Optimization of 831 brasswind instruments and its application in bore 832 reconstruction, Journal of New Music Research, vol.833, issue.1, pp.69-82, 2001.

G. Kirchhoff, Ueber den 835 einfluss der wärmeleitung in einem gase auf die 836 schallbewegung, Annalen der Physik, vol.210, issue.6, pp.837-177, 1868.

. [le-roux, A new impedance 840 tube for large frequency band measurement 841 of absorbing materials, J. Acoust. Soc. Am, vol.842, issue.5, p.3119, 2008.

S. Lefebvre and A. Lefebvre, , p.844, 2012.

G. P. Scavone, Characterization of 845 woodwind instrument toneholes with the finite 846 element method, The Journal of the Acoustical 847 Society of America, vol.131, issue.4, pp.3153-3163, 2012.

D. Mapes-riordan-;-mapes-riordan, 849 "Horn modeling with conical and cylindrical 850 transmission-line elements, J. Audio Eng. Soc, vol.851, issue.6, pp.471-484, 1993.

G. R. Plitnik and W. J. Strong, Numerical method for 854 calculating input impedances of the oboe, The 855 Journal of the Acoustical Society of America, vol.856, issue.3, pp.816-825, 1979.

[. Quarteroni, Méthodes Numériques, p.859, 2007.

. Algorithmes,

. Rabiner and L. R. Schafer-;-rabiner, , p.862, 1978.

R. W. Schafer, Digital processing of 863 speech signals, vol.100, 1978.

S. Repin, A Posteriori Es-866 timates for Partial Differential Equations, p.867, 2008.

, Radon Series on Computational and Ap-868 plied Mathematics

. Gruyter,

S. W. Rienstra, Webster's 871 horn equation revisited, SIAM Journal on Ap-872 plied Mathematics, vol.65, issue.6, pp.1981-2004, 2005.

[. Sharp, A single microphone 875 capillary-based system for measuring the com-876 plex input impedance of musical wind instru-877 ments, Acta Acust united Ac, vol.97, issue.5, pp.819-829, 2011.

[. Silva, MoReeSC: A framework for the simulation and 881 analysis of sound production in reed and brass in-882 struments, Acta Acust united Ac, vol.880, issue.1, p.885, 2014.

J. Petiot, B. Talgorn, M. Kokkolaras, and J. Gilbert, Brass instruments 887 design using physics-based sound simulation 888 models and surrogate-assisted derivative-free 889 optimization, Journal of Mechanical Design, vol.890, issue.4, p.41401, 2017.

K. Van-den-doel and U. M. Ascher, Real-time numer-893 ical solution of webster's equation on a nonuni-894 form grid, IEEE Transactions on Audio, Speech, 895 and Language Processing, vol.16, issue.6, pp.1163-1172, 2008.

J. Virieux and O. , An overview of full-waveform in-899 version in exploration geophysics, Geophysics, vol.898, issue.6, pp.1-26, 2009.

J. C. Webster, An electri-902 cal method of measuring the intonation of cup-903 mouthpiece instruments, The Journal, p.904, 1947.

, Acoustical Society of America, vol.19, issue.5, pp.902-906

K. Zwikker, , p.906, 1949.

C. W. Kosten, Sound absorbing 907 materials, 1949.