February  2016, 9(1): xi-xvii. doi: 10.3934/dcdss.2016.9.1xi

The research of Alberto Valli

1. 

Dipartimento di Matematica, Universita degli Studi di Trento, Via Sommarive, 14, I-38050 POVO

2. 

Department of Mathematics, Pisa University, Via F.Buonarroti, 1, 56127-Pisa

3. 

EPFL, SB, SMA, MATHICSE, CMCS, Av. Piccard, Station 8, CH-1015 Lausanne, Switzerland

Published  December 2015

The scientific activity of Professor Alberto Valli has been mainly devoted to three different subjects: theoretical analysis of partial differential equations in fluid dynamics; domain decomposition methods; numerical approximation of problems arising in low-frequency electromagnetism.

For more information please click the “Full Text” above.
Citation: Ana Alonso Rodríguez, Hugo Beirão da Veiga, Alfio Quarteroni. The research of Alberto Valli. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : xi-xvii. doi: 10.3934/dcdss.2016.9.1xi
References:
[1]

L. Carbone and A. Valli, Filtrazione di un fluido in un mezzo non omogeneo tridimensionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 61 (1976), 161-164.

[2]

A. Valli, L'equazione di Eulero dei fluidi bidimensionali in domini con frontiera variabile, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 61 (1976), 1-5.

[3]

L. Carbone and A. Valli, Free boundary enclosure in a three-dimensional filtration problem, Appl. Math. Optim., 4 (1977), 1-14. doi: 10.1007/BF01442128.

[4]

A. Valli, Soluzioni classiche dell'equazione di Eulero dei fluidi bidimensionali in domini con frontiera variabile, Ricerche Mat., 26 (1977), 301-333.

[5]

L. Carbone and A. Valli, Asymptotic behaviour of the free boundary in a filtration problem, Boll. Un. Mat. Ital. B (5), 15 (1978), 217-224.

[6]

L. Carbone and A. Valli, Filtration through a porous non-homogeneous medium with variable cross-section, J. Analyse Math., 33 (1978), 191-221. doi: 10.1007/BF02790173.

[7]

H. Beirão da Veiga and A. Valli, On the motion of a non-homogeneous ideal incompressible fluid in an external force field, Rend. Sem. Mat. Univ. Padova, 59 (1978), 117-145.

[8]

H. Beirão da Veiga and A. Valli, Existence of $C^\infty$ solutions of the Euler equations for non-homogeneous fluids, Comm. Partial Differential Equations, 5 (1980), 95-107. doi: 10.1080/03605308008820134.

[9]

H. Beirão da Veiga and A. Valli, On the Euler equations for non-homogeneous fluids (I), Rend. Sem. Mat. Univ. Padova, 63 (1980), 151-168.

[10]

H. Beirão da Veiga and A. Valli, On the Euler equations for non-homogeneous fluids (II), J. Math. Anal. Appl., 73 (1980), 338-350. doi: 10.1016/0022-247X(80)90282-6.

[11]

A. Valli, Uniqueness theorems for compressible viscous fluids, especially when the Stokes relation holds, Boll. Un. Mat. Ital. C (5), 18 (1981), 317-325.

[12]

H. Beirão da Veiga, R. Serapioni and A. Valli, On the motion of non-homogeneous fluids in the presence of diffusion, J. Math. Anal. Appl., 85 (1982), 179-191. doi: 10.1016/0022-247X(82)90033-6.

[13]

A. Valli, A correction to the paper: "An existence theorem for compressible viscous fluids'', Ann. Mat. Pura Appl. (4), 132 (1982), 399-400. doi: 10.1007/BF01760990.

[14]

A. Valli, An existence theorem for compressible viscous fluids, Ann. Mat. Pura Appl. (4), 130 (1982), 197-213. doi: 10.1007/BF01761495.

[15]

P. Secchi and A. Valli, A free boundary problem for compressible viscous fluids, J. Reine Angew. Math., 341 (1983), 1-31. doi: 10.1515/crll.1983.341.1.

[16]

A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 10 (1983), 607-647.

[17]

A. Valli, Free boundary problems for compressible viscous fluids, in Fluid Dynamics (Varenna, 1982), Lecture Notes in Math., 1047, Springer, Berlin, 1984, 175-187. doi: 10.1007/BFb0072331.

[18]

P. Marcati and A. Valli, Almost-periodic solutions to the Navier-Stokes equations for compressible fluids, Boll. Un. Mat. Ital. B (6), 4 (1985), 969-986.

[19]

A. Valli, Global existence theorems for compressible viscous fluids, in Nonlinear Variational Problems (Isola d'Elba, 1983) (eds. A. Marino, L. Modica, S. Spagnolo and M. Degiovanni), Res. Notes in Math., 127, Pitman, Boston, MA, 1985, 120-122.

[20]

A. Valli, On the integral representation of the solution to the Stokes system, Rend. Sem. Mat. Univ. Padova, 74 (1985), 85-114.

[21]

A. Valli, Navier-Stokes equations for compressible fluids: Global estimates and periodic solutions, in Nonlinear Functional Analysis and its Applications, Part 2 (Berkeley, Calif., 1983) (ed. F. E. Browder), Proc. Sympos. Pure Math., 45, Part 2, Amer. Math. Soc., Providence, RI, 1986, 467-476.

[22]

A. Valli, Qualitative properties of the solutions to the Navier-Stokes equations for compressible fluids, in Equadiff 6 (Brno, 1985) (eds. J. Vosmanský and M. Zlámal), Lecture Notes in Math., 1192, Springer, Berlin, 1986, 259-264. doi: 10.1007/BFb0076079.

[23]

A. Valli, Stationary solutions to the Navier-Stokes equations for compressible fluids, in BAIL IV (Novosibirsk, 1986) (eds. S. K. Godunov, J. J. H. Miller and V. A. Novikov), Boole, Dún Laoghaire, 1986, 417-422.

[24]

A. Valli and W. Zajączkowski, Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case, Comm. Math. Phys., 103 (1986), 259-296. doi: 10.1007/BF01206939.

[25]

A. Valli, On the existence of stationary solutions to compressible Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 99-113.

[26]

I. Straškraba and A. Valli, Asymptotic behaviour of the density for one-dimensional Navier-Stokes equations, Manuscripta Math., 62 (1988), 401-416. doi: 10.1007/BF01357718.

[27]

A. Valli and W. Zajączkowski, About the motion of non-homogeneous ideal incompressible fluids, Nonlinear Anal., 12 (1988), 43-50. doi: 10.1016/0362-546X(88)90011-9.

[28]

A. Valli, An existence theorem for non-homogeneous inviscid incompressible fluids, in Differential Equations (Xanthi, 1987) (eds. C. M. Dafermos, G. Ladas and G. Papanicolaou), Dekker, New York (NY), 1989, 691-698.

[29]

V. Lovicar, I. Straškraba and A. Valli, On bounded solutions of one-dimensional compressible Navier-Stokes equations, Rend. Sem. Mat. Univ. Padova, 83 (1990), 81-95.

[30]

A. Quarteroni and A. Valli, Domain decomposition for a generalized Stokes problem, in, Proceedings of the Third European Conference on Mathematics in Industry (Glasgow, 1988) (eds. J. Manley, S. McKee, and D. Owens), Teubner, Stuttgart, 1990, 59-74.

[31]

A. Valli, On the one-dimensional Navier-Stokes equations for compressible fluids, in The Navier-Stokes Equations (Oberwolfach, 1988) (eds. J. G. Heywood, K. Masuda, R. Rautmann and V. A. Solonnikov), Springer, Berlin, 1990, 173-179. doi: 10.1007/BFb0086068.

[32]

A. Quarteroni, G. Sacchi Landriani and A. Valli, Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements, Numer. Math., 59 (1991), 831-859. doi: 10.1007/BF01385813.

[33]

A. Quarteroni and A. Valli, Theory and applications of Steklov-Poincaré for boundary value problems: the heterogeneous operator case, in Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Moscow, 1990) (eds. R. Glowinski, Y. A. Kuznetsov, G. Meurant, J. Périaux and O. B. Widlund), SIAM, Philadelphia (PA), 1991, 58-81.

[34]

A. Quarteroni and A. Valli, Theory and applications of Steklov-Poincaré operators for boundary value problems, in Applied and Industrial Mathematics (Venice, 1989) (ed. R. Spigler), Kluwer Acad. Publ., Dordrecht, 1991, 179-203.

[35]

C. Carlenzoli, A. Quarteroni and A. Valli, Spectral domain decomposition methods for compressible Navier-Stokes equations, in Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Norfolk, VA, 1991) (eds. D. E. Keyes, T. F. Chan, G. Meurant, J. S. Scroggs and R. G. Voigt), SIAM, Philadelphia (PA), 1992, 441-450.

[36]

A. Quarteroni, F. Pasquarelli and A. Valli, Heterogeneous domain decomposition: principles, algorithms, applications, in Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Norfolk, {VA, 1991)} (eds. D. E. Keyes, T. F. Chan, G. Meurant, J. S. Scroggs and R. G. Voigt), SIAM, Philadelphia (PA), 1992, 129-150.

[37]

A. Valli, Mathematical results for compressible flows, in Mathematical Topics in Fluid Mechanics (Lisbon, 1991) (eds. J. F. Rodrigues and A. Sequeira), Longman Sci. Tech., Harlow, 1992, 193-229.

[38]

A. Quarteroni and A. Valli, Mathematical modelling and numerical approximation of fluid flow, in Methods and Techniques in Computational Chemistry: METECC-94. Volume C: Structure and Dynamics (ed. E. Clementi), STEF, Cagliari, 1993, 247-298.

[39]

C. Carlenzoli, A. Quarteroni and A. Valli, Numerical solution of the Navier-Stokes equations for viscous compressible flows, in Applied Mathematics in Aerospace Science and Engineering (Erice, 1991) (eds. A. Miele and A. Salvetti), Plenum, New York (NY), 1994, 81-111.

[40]

A. Alonso and A. Valli, A new approach to the coupling of viscous and inviscid Stokes equations, East-West J. Numer. Math., 3 (1995), 29-41.

[41]

A. Alonso and A. Valli, Some remarks on the characterization of the space of tangential traces of $H(rot;\Omega)$ and the construction of an extension operator, Manuscripta Math., 89 (1996), 159-178. doi: 10.1007/BF02567511.

[42]

A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations, in 27th Computational Fluid Dynamics (ed. H. Deconinck), Von Karman Institute for Fluid Dynamics, Rhode-Saint-Genèse, 1996, 1-90.

[43]

A. Alonso and A. Valli, Domain decomposition algorithms for low-frequency time-harmonic Maxwell equations, in Numerical Modelling in Continuum Mechanics (Prague, 1997) (eds. M. Feistauer, R. Rannacher and K. Kozel), Matfyzpress, Prague, 1997, 3-17.

[44]

A. Alonso and A. Valli, A domain decomposition approach for heterogeneous time-harmonic Maxwell equations, Comput. Methods Appl. Mech. Engrg., 143 (1997), 97-112. doi: 10.1016/S0045-7825(96)01144-9.

[45]

A. Alonso, R. L. Trotta and A. Valli, Coercive domain decomposition algorithms for advection-diffusion equations and systems, J. Comput. Appl. Math., 96 (1998), 51-76. doi: 10.1016/S0377-0427(98)00091-0.

[46]

A. Alonso and A. Valli, Finite element approximation of heterogeneous time-harmonic Maxwell equations via a domain decomposition approach, in International Conference on Differential Equations (Lisboa, 1995) (eds. L. Magalhães, C. Rocha and L. Sanchez), World Sci. Publ., River Edge, NJ, 1998, 227-232.

[47]

A. Alonso and A. Valli, Unique solvability for high-frequency heterogeneous time-harmonic Maxwell equations via Fredholm alternative theory, Math. Methods Appl. Sci., 21 (1998), 463-477. doi: 10.1002/(SICI)1099-1476(199804)21:6<463::AID-MMA947>3.0.CO;2-U.

[48]

A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comp., 68 (1999), 607-631. doi: 10.1090/S0025-5718-99-01013-3.

[49]

A. Quarteroni and A. Valli, Domain decomposition methods for compressible flows, in Error Control and Adaptivity in Scientific Computing (Antalya, 1998) (eds. H. Bulgak and C. Zenger), Kluwer Acad. Publ., Dordrecht, 1999, 221-245.

[50]

A. Alonso Rodríguez and A. Valli, Domain decomposition algorithms for time-harmonic Maxwell equations with damping, M2AN Math. Model. Numer. Anal., 35 (2001), 825-848. doi: 10.1051/m2an:2001137.

[51]

A. Alonso Rodríguez and A. Valli, Domain decomposition methods for time-harmonic Maxwell equations: Numerical results}, in Recent Developments in Domain Decomposition Methods (Zürich, 2001) (eds. L. F. Pavarino and A. Toselli), Springer, Berlin, 2002, 157-171. doi: 10.1007/978-3-642-56118-4_10.

[52]

A. Alonso Rodríguez, P. Fernandes and A. Valli, The time-harmonic eddy-current problem in general domains: Solvability via scalar potentials, in Computational Electromagnetics (Kiel, 2001) (eds. C. Carstensen, S. Funken, W. Hackbusch, R. H. W. Hoppe and P. Monk), Springer, Berlin, 2003, 143-163. doi: 10.1007/978-3-642-55745-3_10.

[53]

A. Alonso Rodríguez, P. Fernandes and A. Valli, Weak and strong formulations for the time-harmonic eddy-current problem in general multi-connected domains, European J. Appl. Math., 14 (2003), 387-406. doi: 10.1017/S0956792503005151.

[54]

A. Alonso Rodríguez, R. Hiptmair and A. Valli, Mixed finite element approximation of eddy current problems, IMA J. Numer. Anal., 24 (2004), 255-271. doi: 10.1093/imanum/24.2.255.

[55]

A. Alonso Rodríguez and A. Valli, Mixed finite element approximation of eddy current problems based on the electric field, in ECCOMAS 2004: European Congress on Computational Methods in Applied Sciences and Engineering (Jyväskylä, 2004) (eds. P. Neittaanmäki, T. Rossi, K. Majava and O. Pironneau), volume 1, University of Jyväskylä. Department of Mathematics, Jyväskylä, 2004, 1-12.

[56]

A. Alonso Rodríguez, R. Hiptmair and A. Valli, A hybrid formulation of eddy current problems, Numer. Methods Partial Differential Equations, 21 (2005), 742-763. doi: 10.1002/num.20060.

[57]

A. Quarteroni, M. Sala and A. Valli, An interface-strip domain decomposition preconditioner, SIAM J. Sci. Comput., 28 (2006), 498-516. doi: 10.1137/04061057X.

[58]

O. Bíró and A. Valli, The Coulomb gauged vector potential formulation for the eddy-current problem in general geometry: well-posedness and numerical approximation, Comput. Methods Appl. Mech. Engrg., 196 (2007), 1890-1904. doi: 10.1016/j.cma.2006.10.008.

[59]

M. Discacciati, A. Quarteroni and A. Valli, Robin-Robin domain decomposition methods for the Stokes-Darcy coupling, SIAM J. Numer. Anal., 45 (2007), 1246-1268. doi: 10.1137/06065091X.

[60]

P. Fernandes and A. Valli, Lorenz-gauged vector potential formulations for the time-harmonic eddy-current problem with $L^\infty$-regularity of material properties, Math. Methods Appl. Sci., 31 (2008), 71-98. doi: 10.1002/mma.900.

[61]

A. Alonso Rodríguez and A. Valli, Voltage and current excitation for time-harmonic eddy-current problems, SIAM J. Appl. Math., 68 (2008), 1477-1494. doi: 10.1137/070697677.

[62]

A. Alonso Rodríguez and A. Valli, A FEM-BEM approach for electro-magnetostatics and time-harmonic eddy-current problems, Appl. Numer. Math., 59 (2009), 2036-2049. doi: 10.1016/j.apnum.2008.12.002.

[63]

A. Alonso Rodríguez, A. Valli and R. Vázquez Hernández, A formulation of the eddy current problem in the presence of electric ports, Numer. Math., 113 (2009), 643-672. doi: 10.1007/s00211-009-0241-7.

[64]

A. Alonso Rodríguez, J. Camaño and A. Valli, Inverse source problems for eddy current equations, Inverse Problems, 28 (2012), 015006, 15 pp. doi: 10.1088/0266-5611/28/1/015006.

[65]

A. Valli, Solving an electrostatics-like problem with a current dipole source by means of the duality method, Appl. Math. Lett., 25 (2012), 1410-1414. doi: 10.1016/j.aml.2011.12.013.

[66]

A. Alonso Rodríguez, E. Bertolazzi, R. Ghiloni and A. Valli, Construction of a finite element basis of the first de Rham cohomology group and numerical solution of 3D magnetostatic problems, SIAM J. Numer. Anal., 51 (2013), 2380-2402. doi: 10.1137/120890648.

[67]

A. Alonso Rodríguez, J. Camaño, R. Rodríguez and A. Valli, A posteriori error estimates for the problem of electrostatics with a dipole source, Comput. Math. Appl., 68 (2014), 464-485. doi: 10.1016/j.camwa.2014.06.017.

[68]

A. Alonso Rodríguez and A. Valli, Finite element potentials, Appl. Numer. Math., 95 (2015), 2-14. doi: 10.1016/j.apnum.2014.05.014.

[69]

A. Alonso Rodríguez and A. Valli, Eddy Current Approximation of Maxwell Equations, Springer Italia, Milan, 2010. doi: 10.1007/978-88-470-1506-7.

[70]

A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, Oxford, 1999.

[71]

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, Berlin, 1994.

show all references

References:
[1]

L. Carbone and A. Valli, Filtrazione di un fluido in un mezzo non omogeneo tridimensionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 61 (1976), 161-164.

[2]

A. Valli, L'equazione di Eulero dei fluidi bidimensionali in domini con frontiera variabile, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 61 (1976), 1-5.

[3]

L. Carbone and A. Valli, Free boundary enclosure in a three-dimensional filtration problem, Appl. Math. Optim., 4 (1977), 1-14. doi: 10.1007/BF01442128.

[4]

A. Valli, Soluzioni classiche dell'equazione di Eulero dei fluidi bidimensionali in domini con frontiera variabile, Ricerche Mat., 26 (1977), 301-333.

[5]

L. Carbone and A. Valli, Asymptotic behaviour of the free boundary in a filtration problem, Boll. Un. Mat. Ital. B (5), 15 (1978), 217-224.

[6]

L. Carbone and A. Valli, Filtration through a porous non-homogeneous medium with variable cross-section, J. Analyse Math., 33 (1978), 191-221. doi: 10.1007/BF02790173.

[7]

H. Beirão da Veiga and A. Valli, On the motion of a non-homogeneous ideal incompressible fluid in an external force field, Rend. Sem. Mat. Univ. Padova, 59 (1978), 117-145.

[8]

H. Beirão da Veiga and A. Valli, Existence of $C^\infty$ solutions of the Euler equations for non-homogeneous fluids, Comm. Partial Differential Equations, 5 (1980), 95-107. doi: 10.1080/03605308008820134.

[9]

H. Beirão da Veiga and A. Valli, On the Euler equations for non-homogeneous fluids (I), Rend. Sem. Mat. Univ. Padova, 63 (1980), 151-168.

[10]

H. Beirão da Veiga and A. Valli, On the Euler equations for non-homogeneous fluids (II), J. Math. Anal. Appl., 73 (1980), 338-350. doi: 10.1016/0022-247X(80)90282-6.

[11]

A. Valli, Uniqueness theorems for compressible viscous fluids, especially when the Stokes relation holds, Boll. Un. Mat. Ital. C (5), 18 (1981), 317-325.

[12]

H. Beirão da Veiga, R. Serapioni and A. Valli, On the motion of non-homogeneous fluids in the presence of diffusion, J. Math. Anal. Appl., 85 (1982), 179-191. doi: 10.1016/0022-247X(82)90033-6.

[13]

A. Valli, A correction to the paper: "An existence theorem for compressible viscous fluids'', Ann. Mat. Pura Appl. (4), 132 (1982), 399-400. doi: 10.1007/BF01760990.

[14]

A. Valli, An existence theorem for compressible viscous fluids, Ann. Mat. Pura Appl. (4), 130 (1982), 197-213. doi: 10.1007/BF01761495.

[15]

P. Secchi and A. Valli, A free boundary problem for compressible viscous fluids, J. Reine Angew. Math., 341 (1983), 1-31. doi: 10.1515/crll.1983.341.1.

[16]

A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 10 (1983), 607-647.

[17]

A. Valli, Free boundary problems for compressible viscous fluids, in Fluid Dynamics (Varenna, 1982), Lecture Notes in Math., 1047, Springer, Berlin, 1984, 175-187. doi: 10.1007/BFb0072331.

[18]

P. Marcati and A. Valli, Almost-periodic solutions to the Navier-Stokes equations for compressible fluids, Boll. Un. Mat. Ital. B (6), 4 (1985), 969-986.

[19]

A. Valli, Global existence theorems for compressible viscous fluids, in Nonlinear Variational Problems (Isola d'Elba, 1983) (eds. A. Marino, L. Modica, S. Spagnolo and M. Degiovanni), Res. Notes in Math., 127, Pitman, Boston, MA, 1985, 120-122.

[20]

A. Valli, On the integral representation of the solution to the Stokes system, Rend. Sem. Mat. Univ. Padova, 74 (1985), 85-114.

[21]

A. Valli, Navier-Stokes equations for compressible fluids: Global estimates and periodic solutions, in Nonlinear Functional Analysis and its Applications, Part 2 (Berkeley, Calif., 1983) (ed. F. E. Browder), Proc. Sympos. Pure Math., 45, Part 2, Amer. Math. Soc., Providence, RI, 1986, 467-476.

[22]

A. Valli, Qualitative properties of the solutions to the Navier-Stokes equations for compressible fluids, in Equadiff 6 (Brno, 1985) (eds. J. Vosmanský and M. Zlámal), Lecture Notes in Math., 1192, Springer, Berlin, 1986, 259-264. doi: 10.1007/BFb0076079.

[23]

A. Valli, Stationary solutions to the Navier-Stokes equations for compressible fluids, in BAIL IV (Novosibirsk, 1986) (eds. S. K. Godunov, J. J. H. Miller and V. A. Novikov), Boole, Dún Laoghaire, 1986, 417-422.

[24]

A. Valli and W. Zajączkowski, Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case, Comm. Math. Phys., 103 (1986), 259-296. doi: 10.1007/BF01206939.

[25]

A. Valli, On the existence of stationary solutions to compressible Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 99-113.

[26]

I. Straškraba and A. Valli, Asymptotic behaviour of the density for one-dimensional Navier-Stokes equations, Manuscripta Math., 62 (1988), 401-416. doi: 10.1007/BF01357718.

[27]

A. Valli and W. Zajączkowski, About the motion of non-homogeneous ideal incompressible fluids, Nonlinear Anal., 12 (1988), 43-50. doi: 10.1016/0362-546X(88)90011-9.

[28]

A. Valli, An existence theorem for non-homogeneous inviscid incompressible fluids, in Differential Equations (Xanthi, 1987) (eds. C. M. Dafermos, G. Ladas and G. Papanicolaou), Dekker, New York (NY), 1989, 691-698.

[29]

V. Lovicar, I. Straškraba and A. Valli, On bounded solutions of one-dimensional compressible Navier-Stokes equations, Rend. Sem. Mat. Univ. Padova, 83 (1990), 81-95.

[30]

A. Quarteroni and A. Valli, Domain decomposition for a generalized Stokes problem, in, Proceedings of the Third European Conference on Mathematics in Industry (Glasgow, 1988) (eds. J. Manley, S. McKee, and D. Owens), Teubner, Stuttgart, 1990, 59-74.

[31]

A. Valli, On the one-dimensional Navier-Stokes equations for compressible fluids, in The Navier-Stokes Equations (Oberwolfach, 1988) (eds. J. G. Heywood, K. Masuda, R. Rautmann and V. A. Solonnikov), Springer, Berlin, 1990, 173-179. doi: 10.1007/BFb0086068.

[32]

A. Quarteroni, G. Sacchi Landriani and A. Valli, Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements, Numer. Math., 59 (1991), 831-859. doi: 10.1007/BF01385813.

[33]

A. Quarteroni and A. Valli, Theory and applications of Steklov-Poincaré for boundary value problems: the heterogeneous operator case, in Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Moscow, 1990) (eds. R. Glowinski, Y. A. Kuznetsov, G. Meurant, J. Périaux and O. B. Widlund), SIAM, Philadelphia (PA), 1991, 58-81.

[34]

A. Quarteroni and A. Valli, Theory and applications of Steklov-Poincaré operators for boundary value problems, in Applied and Industrial Mathematics (Venice, 1989) (ed. R. Spigler), Kluwer Acad. Publ., Dordrecht, 1991, 179-203.

[35]

C. Carlenzoli, A. Quarteroni and A. Valli, Spectral domain decomposition methods for compressible Navier-Stokes equations, in Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Norfolk, VA, 1991) (eds. D. E. Keyes, T. F. Chan, G. Meurant, J. S. Scroggs and R. G. Voigt), SIAM, Philadelphia (PA), 1992, 441-450.

[36]

A. Quarteroni, F. Pasquarelli and A. Valli, Heterogeneous domain decomposition: principles, algorithms, applications, in Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Norfolk, {VA, 1991)} (eds. D. E. Keyes, T. F. Chan, G. Meurant, J. S. Scroggs and R. G. Voigt), SIAM, Philadelphia (PA), 1992, 129-150.

[37]

A. Valli, Mathematical results for compressible flows, in Mathematical Topics in Fluid Mechanics (Lisbon, 1991) (eds. J. F. Rodrigues and A. Sequeira), Longman Sci. Tech., Harlow, 1992, 193-229.

[38]

A. Quarteroni and A. Valli, Mathematical modelling and numerical approximation of fluid flow, in Methods and Techniques in Computational Chemistry: METECC-94. Volume C: Structure and Dynamics (ed. E. Clementi), STEF, Cagliari, 1993, 247-298.

[39]

C. Carlenzoli, A. Quarteroni and A. Valli, Numerical solution of the Navier-Stokes equations for viscous compressible flows, in Applied Mathematics in Aerospace Science and Engineering (Erice, 1991) (eds. A. Miele and A. Salvetti), Plenum, New York (NY), 1994, 81-111.

[40]

A. Alonso and A. Valli, A new approach to the coupling of viscous and inviscid Stokes equations, East-West J. Numer. Math., 3 (1995), 29-41.

[41]

A. Alonso and A. Valli, Some remarks on the characterization of the space of tangential traces of $H(rot;\Omega)$ and the construction of an extension operator, Manuscripta Math., 89 (1996), 159-178. doi: 10.1007/BF02567511.

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A. Alonso and A. Valli, Domain decomposition algorithms for low-frequency time-harmonic Maxwell equations, in Numerical Modelling in Continuum Mechanics (Prague, 1997) (eds. M. Feistauer, R. Rannacher and K. Kozel), Matfyzpress, Prague, 1997, 3-17.

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A. Alonso and A. Valli, A domain decomposition approach for heterogeneous time-harmonic Maxwell equations, Comput. Methods Appl. Mech. Engrg., 143 (1997), 97-112. doi: 10.1016/S0045-7825(96)01144-9.

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A. Alonso, R. L. Trotta and A. Valli, Coercive domain decomposition algorithms for advection-diffusion equations and systems, J. Comput. Appl. Math., 96 (1998), 51-76. doi: 10.1016/S0377-0427(98)00091-0.

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A. Alonso and A. Valli, Unique solvability for high-frequency heterogeneous time-harmonic Maxwell equations via Fredholm alternative theory, Math. Methods Appl. Sci., 21 (1998), 463-477. doi: 10.1002/(SICI)1099-1476(199804)21:6<463::AID-MMA947>3.0.CO;2-U.

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A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comp., 68 (1999), 607-631. doi: 10.1090/S0025-5718-99-01013-3.

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A. Quarteroni and A. Valli, Domain decomposition methods for compressible flows, in Error Control and Adaptivity in Scientific Computing (Antalya, 1998) (eds. H. Bulgak and C. Zenger), Kluwer Acad. Publ., Dordrecht, 1999, 221-245.

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A. Alonso Rodríguez and A. Valli, Domain decomposition algorithms for time-harmonic Maxwell equations with damping, M2AN Math. Model. Numer. Anal., 35 (2001), 825-848. doi: 10.1051/m2an:2001137.

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A. Alonso Rodríguez and A. Valli, Domain decomposition methods for time-harmonic Maxwell equations: Numerical results}, in Recent Developments in Domain Decomposition Methods (Zürich, 2001) (eds. L. F. Pavarino and A. Toselli), Springer, Berlin, 2002, 157-171. doi: 10.1007/978-3-642-56118-4_10.

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A. Alonso Rodríguez, P. Fernandes and A. Valli, The time-harmonic eddy-current problem in general domains: Solvability via scalar potentials, in Computational Electromagnetics (Kiel, 2001) (eds. C. Carstensen, S. Funken, W. Hackbusch, R. H. W. Hoppe and P. Monk), Springer, Berlin, 2003, 143-163. doi: 10.1007/978-3-642-55745-3_10.

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A. Alonso Rodríguez, P. Fernandes and A. Valli, Weak and strong formulations for the time-harmonic eddy-current problem in general multi-connected domains, European J. Appl. Math., 14 (2003), 387-406. doi: 10.1017/S0956792503005151.

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A. Alonso Rodríguez, R. Hiptmair and A. Valli, Mixed finite element approximation of eddy current problems, IMA J. Numer. Anal., 24 (2004), 255-271. doi: 10.1093/imanum/24.2.255.

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A. Alonso Rodríguez and A. Valli, Mixed finite element approximation of eddy current problems based on the electric field, in ECCOMAS 2004: European Congress on Computational Methods in Applied Sciences and Engineering (Jyväskylä, 2004) (eds. P. Neittaanmäki, T. Rossi, K. Majava and O. Pironneau), volume 1, University of Jyväskylä. Department of Mathematics, Jyväskylä, 2004, 1-12.

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A. Alonso Rodríguez, R. Hiptmair and A. Valli, A hybrid formulation of eddy current problems, Numer. Methods Partial Differential Equations, 21 (2005), 742-763. doi: 10.1002/num.20060.

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A. Quarteroni, M. Sala and A. Valli, An interface-strip domain decomposition preconditioner, SIAM J. Sci. Comput., 28 (2006), 498-516. doi: 10.1137/04061057X.

[58]

O. Bíró and A. Valli, The Coulomb gauged vector potential formulation for the eddy-current problem in general geometry: well-posedness and numerical approximation, Comput. Methods Appl. Mech. Engrg., 196 (2007), 1890-1904. doi: 10.1016/j.cma.2006.10.008.

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M. Discacciati, A. Quarteroni and A. Valli, Robin-Robin domain decomposition methods for the Stokes-Darcy coupling, SIAM J. Numer. Anal., 45 (2007), 1246-1268. doi: 10.1137/06065091X.

[60]

P. Fernandes and A. Valli, Lorenz-gauged vector potential formulations for the time-harmonic eddy-current problem with $L^\infty$-regularity of material properties, Math. Methods Appl. Sci., 31 (2008), 71-98. doi: 10.1002/mma.900.

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A. Alonso Rodríguez and A. Valli, Voltage and current excitation for time-harmonic eddy-current problems, SIAM J. Appl. Math., 68 (2008), 1477-1494. doi: 10.1137/070697677.

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A. Alonso Rodríguez and A. Valli, A FEM-BEM approach for electro-magnetostatics and time-harmonic eddy-current problems, Appl. Numer. Math., 59 (2009), 2036-2049. doi: 10.1016/j.apnum.2008.12.002.

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A. Alonso Rodríguez, A. Valli and R. Vázquez Hernández, A formulation of the eddy current problem in the presence of electric ports, Numer. Math., 113 (2009), 643-672. doi: 10.1007/s00211-009-0241-7.

[64]

A. Alonso Rodríguez, J. Camaño and A. Valli, Inverse source problems for eddy current equations, Inverse Problems, 28 (2012), 015006, 15 pp. doi: 10.1088/0266-5611/28/1/015006.

[65]

A. Valli, Solving an electrostatics-like problem with a current dipole source by means of the duality method, Appl. Math. Lett., 25 (2012), 1410-1414. doi: 10.1016/j.aml.2011.12.013.

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A. Alonso Rodríguez, E. Bertolazzi, R. Ghiloni and A. Valli, Construction of a finite element basis of the first de Rham cohomology group and numerical solution of 3D magnetostatic problems, SIAM J. Numer. Anal., 51 (2013), 2380-2402. doi: 10.1137/120890648.

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A. Alonso Rodríguez, J. Camaño, R. Rodríguez and A. Valli, A posteriori error estimates for the problem of electrostatics with a dipole source, Comput. Math. Appl., 68 (2014), 464-485. doi: 10.1016/j.camwa.2014.06.017.

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A. Alonso Rodríguez and A. Valli, Finite element potentials, Appl. Numer. Math., 95 (2015), 2-14. doi: 10.1016/j.apnum.2014.05.014.

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A. Alonso Rodríguez and A. Valli, Eddy Current Approximation of Maxwell Equations, Springer Italia, Milan, 2010. doi: 10.1007/978-88-470-1506-7.

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