PROGETTO PRIN 2012: Metodologie innovative nella modellistica differenziale numerica





Riferimenti bibliografici relativi al Punto 11. Stato dell’arte

Bibliographical references concerning Sect. 11. State of the Art



Nel seguito, elenchiamo alcuni significativi contributi in letteratura riguardanti ciascuna delle linee di ricerca presentate al Punto 10 del Progetto

Herafter,  we list some relevant contributions in the literature concerning each of the research lines presented in Sect. 10 of the Project.




Linea di ricerca / Research line S.1a+1b)


[1] T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194:39–41 (2005), 4135-4195.


[2] Y. Bazilevs, V.M. Calo, J.A. Cottrell, J.A. Evans, T.J.R. Hughes, S. Lipton, M.A. Scott, T.W. Sederberg, Isogeometric analysis using T-splines, Computer Methods in Applied Mechanics and Engineering, 199:5-8 (2010), 229-263.


[3] Y. Bazilevs, L. Beirčo da Veiga, J. A. Cottrell, T. J. R. Hughes, and G. Sangalli, Isogeometric analysis: approximation, stability and error estimates for h-refined meshes, Mathematical Models and Methods in Applied Sciencs 16:07 (2006), 1031-1090.


[4] F. Auricchio, L. Beirčo da Veiga, A. Buffa, C. Lovadina, A. Reali, G. Sangalli, A fully “locking-free” isogeometric approach for plane linear elasticity problems: A stream function formulation, Computer Methods in Applied Mechanics and Engineering 197:1-4 (2007), 160-172.


[5] F. Auricchio, L. Beirčo da Veiga, T. J. R. Hughes, A. Reali, G. Sangalli, Isogeometric collocation methods, Mathematical Models and Methods in Applied Sciences 20:11 (2010), 2075-2107.


[6] A. Buffa, G. Sangalli, R. Vázquez, Isogeometric analysis in electromagnetics: B-splines approximation, Computer Methods in Applied Mechanics and Engineering, Vol. 199 (17-20), pp. 1143-1152, 2010


[7] A. Buffa, C. de Falco, G. Sangalli, Isogeometric Analysis: new stable elements for the 2D Stokes equation, International Journal for Numerical Methods in Fluids Vol. 65 (11-12), pp.1407-1422, 2011


[8] L. Beirčo da Veiga, A. Buffa, J. Rivas, G. Sangalli, Some estimates for h–p–k-refinement in Isogeometric Analysis, Numerische Mathematik 118:2 (2011), 271-305.


[9] Beirao da Veiga, L.,Brezzi, F., Cangiani, A., Manzini, G., Marini, L. D. and Russo, A., Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23, 2013, pp. 119-214.


[10] F.Brezzi, L.D.Marini. Virtual element method for plate bending problems, Comput. Methods Appl. Mech. Engrg.,253:455-462 (2012)


[11] Beirao da Veiga, F.Brezzi, L.D.Marini. Virtual Elements for linear elasticity problems, SIAM J. Numer. Anal., 51:794-812 (2013)


[12] Brezzi, F., Lipnikov, K. and Shashkov, M., Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes, SIAM J. Numer. Anal.,  43(5), 2005, pp. 1872-1896.


[13] Brezzi, F., Lipnikov, K. and Simoncini, V., A family of mimetic finite difference methods on polygonal and polyhedral meshes, Math. Models Methods Appl. Sci., 15(10), 2005, pp. 1533-1551.


[14] L. Beirao da Veiga, K. Lipnikov e G. Manzini, Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes, SIAM J. Numer. Anal., 49: 1737-1760 (2011)


[15] K. Lipnikov, G. Manzini, M. Shashkov. Mimetic Finite Difference Method, submitted to Journal of Computational Physics




Linea di ricerca / Research line S.1c)


[16] I. Herrera, Trefftz method: A general theory, Numer. Methods Partial Differential Eq., 16 (2000), 561–580.


[17] O. Cessenat and B. Despres, Application of an Ultra Weak Variational Formulation of Elliptic PDEs to the Two-Dimensional Helmholtz Problem, SIAM Journal on Numerical Analysis, 35:1 (1998), 255-299.


[18] O. Cessenat and B. Despres, Using Plane Waves as Base Functions for Solving Time Harmonic Equations with the Ultra Weak Variational Formulation, Journal of Computational Acoustics, 11:02 (2003), 227-238.


[19] T. Huttunen, P. Monk, F. Collino, and J. P. Kaipio, The Ultra-Weak Variational Formulation for Elastic Wave Problems. SIAM Journal on Scientific Computing 25:5 (2004), 1717-1742.


[20] D. Wang, R. Tezaur, J. Toivanen and C. Farhat, Overview of the discontinuous enrichment method, the ultra-weak variational formulation, and the partition of unity method for acoustic scattering in the medium frequency regime and performance comparisons, Int. J. Numer. Meth. Engng., 89 (2012), 403–417.


[21] R. Hiptmair, A. Moiola, and I. Perugia, Plane Wave Discontinuous Galerkin Methods for the 2D Helmholtz Equation: Analysis of the p-Version, SIAM Journal on Numerical Analysis, 49:1 (2011), 264-284.


[22] R. Hiptmair, A. Moiola and I. Perugia, Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations, Math. Comp., 82 (2013), 247-268.


[23] A. Moiola, R. Hiptmair and I. Perugia, Plane wave approximation of homogeneous Helmholtz solutions, Z. Angew. Math. Phys., 62 (2011), 809-837.

[24] Ben Abdallah N., Pinaud O.: Multiscale solution of transport in an open quantum system: Resonances and WKB interpolation, J. Comp. Phys. (2006) 213: pp. 288-310.

[25] Negulescu C.: Numerical analysis of a multiscale finite element scheme for the resolution of the stationary Schrödinger equation, Numer. Math. (2008) 108: pp. 625-652.

[26] Wang W., Shu C.W.: The WKB Local Discontinuous Galerkin Method for the Simulation of Schrödinger Equation in a Resonant Tunneling Diode, J Sci Comput (2009) 40: pp. 360-374.





Linea di ricerca / Research line S.2a)


[27] A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet methods for elliptic operator equations – convergence rates. Math. Comp, 70:27–75, 1998.


[28] P. Binev, W. Dahmen, and R. DeVore. Adaptive finite element methods with convergence rates. Numer. Math., 97(2):219–268, 2004.


[29] P. Morin, K. G. Siebert, A. Veeser, A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18 (2008), 707-737.


[30] R. Stevenson. Adaptive wavelet methods for solving operator equations: an overview. In Multiscale, nonlinear and adaptive approximation (A. Kunoth and R.A. DeVore, eds.), pages 543–597. Springer, Berlin, 2009.


[31] R. H. Nochetto, K. G. Siebert, A. Veeser,  Theory of adaptive finite element methods: an introduction. Multiscale, nonlinear and adaptive approximation, R. DeVore, A. Kunoth (Ed.), Springer Verlag Heidelberg, 409-542, 2009.


[32] R. H. Nochetto, A. Veeser, Primer of adaptive finite element methods. Multiscale and adaptivity: modeling, numerics and applications, G. Naldi, G. Russo (Ed.), Springer Verlag, Heidelberg, 125-226, 2012.


[33] R. A. De Vore,  Nonlinear Approximation, Acta Numerica, 7(1998), Cambridge University Press, pp. 51-150.


[34] R. A. DeVore, Optimal Computation, Proceedings of the International Congress of Mathematicians, Madrid, 2006.


[35] C. Canuto, R.H. Nochetto and M. Verani, Adaptive Fourier-Galerkin Methods, arXiv:1201.5648, accepted for publication in Math. Comput.


[36] C. Canuto, R.H. Nochetto, and M. Verani. Contraction and optimality properties of adaptive Legendre-Galerkin methods: the 1-dimensional case. arXiv:1206.5524v1, accepted for publication in Computers and Mathematics with Applications.


[37] P.F. Antonietti, L. Beirao da Veiga, C. Lovadina and M. Verani, Hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems, Accepted for publication in SIAM J. Numer. Anal.


[38] R.H. Nochetto, A. Veeser and M. Verani, A Safeguarded Dual Weighted Residual Method, IMA J. Numer. Anal. 29 (2009), no. 1, 126--140.


[39] A. Veeser and R. Verfuerth, Explicit upper bounds for dual norms of residuals. SIAM J. Numer. Anal. 47 (2009), 2387-2405.


[40] Q. Zou, A. Veeser, C. Gruesser and R. Kornhuber, Hierarchical error estimates for the energy functional in obstacle problems. Numer. Math. 117 (2011), 653-677.


[41] A. Veeser and R. Verfuerth, Poincare` constants for finite element stars. IMA J Numer Anal (2012) 32 (1): 30-47.




Linea di ricerca / Research line S.2b)


[42] D. Givoli, T. Hagstrom, I. Patlashenko, Finite element formulation with high-order absorbing conditions for time-dependent waves, Comput. Methods Appl. Mech. Engrg. 195 (2006), 3666-3690.


[43] D. Baffet, D. Givoli, On the stability of the high order Higdon absorbing boundary conditions, Appl. Numer. Math. 61 (2011), 768-784.


[44] M.J. Grote, I. Sim, Local nonreflecting boundary condition for time-dependent multiple scattering, J. Comput. Phys. 230 (2011), 3135-3154.


[45] X. Wu, J. Zhang, High order local absorbing boundary conditions for heat equation in unbounded domains, J. Comput. Math. 29 (2011), 74-90.


[46] S. Falletta, G. Monegato, L. Scuderi, Space-time BIE methods for non homogeneous exterior wave equation problems. The Dirichlet case. In: IMA J. Numer. Anal. 32 (2012), 202-226.


[47] S. Falletta, G. Monegato (2013), An exact non-reflecting boundary condition for 2D  time-dependent  wave equation problems,  Wave Motion, to appear.


[48] C. S. Peskin. The immersed boundary method, in Acta Numerica, 2002. Cambridge University Press, 2002. 


[49] D. Boffi, L. Gastaldi, L. Heltai and C.S. Peskin, A note on the hyper-elastic formulation of the immersed boundary method Comp. Meth. Appl. Mech. Eng., 197 (2008) 2210-2231 


[50] D. Boffi, N. Cavallini and L. Gastaldi, Finite element approach to immersed boundary method with different fluid and solid densities M3AS Math. Models Methods Appl. Sci., 21 (2011) 2523-2550 doi:10.1142/S0218202511005829


[51] D. Boffi, N. Cavallini, F. Gardini and L. Gastaldi, Local mass conservation of Stokes finite elements, J. Sci. Comput., 52 (2012) 383-400 doi:10.1007/s10915-011-9549-4


[52] M. Discacciati and A. Quarteroni, Navier–Stokes/Darcy coupling: modeling, analysis, and numerical approximation. Rev. Mat. Complut. 22(2), 315–426  (2009).


[53] M. Discacciati, P. Gervasio and A. Quarteroni, Heterogeneous  mathematical models in fluid dynamics and associated solution  algorithms, in Multiscale and Adaptivity: Modeling, Numerics and Applications, G. Naldi and G. Russo, Eds. Series: Lecture Notes in Mathematics,  Vol. 2040. January, 2012.






Linea di ricerca / Research line S.2c)


[54] P. M. Adler, Fractures and Fracture Networks, Kluwer Academic,  Dordrecht, 1999. 


[55] M. C. Cacas, E. Ledoux, G. de Marsily, B. Tillie, A. Barbreau, E.  Durand, B. Feuga, and P. Peaudecerf, Modeling fracture flow with a stochastic discrete  fracture network: calibration and validation: 1. the flow model, Water Resour. Res., 26  (1990), pp. 479–489. 


[56] W. S. Dershowitz and H. H. Einstein, Characterizing rock joint geometry  with joint system models, Rock Mechanics and Rock Engineering, 1 (1988), pp. 21–51.


[57] H. Mustapha and K. Mustapha, A new approach to simulating flow in  discrete fracture networks with an optimized mesh, SIAM J. Sci. Comput., 29 (2007), pp.  1439–1459. 


[58] G. Pichot, J. Erhel, and J.-R. de Dreuzy, A generalized mixed hybrid  mortar method for solving flow in stochastic discrete fracture networks, SIAM Journal on  scientific computing, 34 (2012), pp. B86–B105. 


[59] J.-R. de Dreuzy, G. Pichot, B. Poirriez, and J. Erhel, Synthetic  benchmark for modeling flow in 3D fractured media, Computers and Geosciences, 50 (2013), pp. 59–71.


[60] Berrone S., Pieraccini S., Scialo` S. (2012), A PDE-constrained optimization formulation for discrete fracture  network flows. <>, SIAM J. Sci. Comp. 35(2) (2013), pp. B487-B510.


[61] Berrone S., Pieraccini S., Scialo' S. (2012), On simulations of discrete fracture network flows with an  optimization-based extended finite element method. <>, SIAM J. Sci. Comp. 35(2) (2013), pp. A908-A935.