# American Institute of Mathematical Sciences

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.

Citation: Ana Alonso Rodríguez, Hugo Beirão da Veiga, Alfio Quarteroni. The research of Alberto Valli. Discrete & 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. Google Scholar [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. Google Scholar [3] L. Carbone and A. Valli, Free boundary enclosure in a three-dimensional filtration problem,, Appl. Math. Optim., 4 (1977), 1. doi: 10.1007/BF01442128. Google Scholar [4] A. Valli, Soluzioni classiche dell'equazione di Eulero dei fluidi bidimensionali in domini con frontiera variabile,, Ricerche Mat., 26 (1977), 301. Google Scholar [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. Google Scholar [6] L. Carbone and A. Valli, Filtration through a porous non-homogeneous medium with variable cross-section,, J. Analyse Math., 33 (1978), 191. doi: 10.1007/BF02790173. Google Scholar [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. Google Scholar [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. doi: 10.1080/03605308008820134. Google Scholar [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. Google Scholar [10] H. Beirão da Veiga and A. Valli, On the Euler equations for non-homogeneous fluids (II),, J. Math. Anal. Appl., 73 (1980), 338. doi: 10.1016/0022-247X(80)90282-6. Google Scholar [11] A. Valli, Uniqueness theorems for compressible viscous fluids, especially when the Stokes relation holds,, Boll. Un. Mat. Ital. C (5), 18 (1981), 317. Google Scholar [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. doi: 10.1016/0022-247X(82)90033-6. Google Scholar [13] A. Valli, A correction to the paper: "An existence theorem for compressible viscous fluids'',, Ann. Mat. Pura Appl. (4), 132 (1982), 399. doi: 10.1007/BF01760990. Google Scholar [14] A. Valli, An existence theorem for compressible viscous fluids,, Ann. Mat. Pura Appl. (4), 130 (1982), 197. doi: 10.1007/BF01761495. Google Scholar [15] P. Secchi and A. Valli, A free boundary problem for compressible viscous fluids,, J. Reine Angew. Math., 341 (1983), 1. doi: 10.1515/crll.1983.341.1. Google Scholar [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. Google Scholar [17] A. Valli, Free boundary problems for compressible viscous fluids,, in Fluid Dynamics (Varenna, (1982), 175. doi: 10.1007/BFb0072331. Google Scholar [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. Google Scholar [19] A. Valli, Global existence theorems for compressible viscous fluids,, in Nonlinear Variational Problems (Isola d'Elba, (1983), 120. Google Scholar [20] A. Valli, On the integral representation of the solution to the Stokes system,, Rend. Sem. Mat. Univ. Padova, 74 (1985), 85. Google Scholar [21] A. Valli, Navier-Stokes equations for compressible fluids: Global estimates and periodic solutions,, in Nonlinear Functional Analysis and its Applications, (1983), 467. Google Scholar [22] A. Valli, Qualitative properties of the solutions to the Navier-Stokes equations for compressible fluids,, in Equadiff 6 (Brno, (1985), 259. doi: 10.1007/BFb0076079. Google Scholar [23] A. Valli, Stationary solutions to the Navier-Stokes equations for compressible fluids,, in BAIL IV (Novosibirsk, (1986), 417. Google Scholar [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. doi: 10.1007/BF01206939. Google Scholar [25] A. Valli, On the existence of stationary solutions to compressible Navier-Stokes equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 99. Google Scholar [26] I. Straškraba and A. Valli, Asymptotic behaviour of the density for one-dimensional Navier-Stokes equations,, Manuscripta Math., 62 (1988), 401. doi: 10.1007/BF01357718. Google Scholar [27] A. Valli and W. Zajączkowski, About the motion of non-homogeneous ideal incompressible fluids,, Nonlinear Anal., 12 (1988), 43. doi: 10.1016/0362-546X(88)90011-9. Google Scholar [28] A. Valli, An existence theorem for non-homogeneous inviscid incompressible fluids,, in Differential Equations (Xanthi, (1987), 691. Google Scholar [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. Google Scholar [30] A. Quarteroni and A. Valli, Domain decomposition for a generalized Stokes problem,, in, (1988), 59. Google Scholar [31] A. Valli, On the one-dimensional Navier-Stokes equations for compressible fluids,, in The Navier-Stokes Equations (Oberwolfach, (1988), 173. doi: 10.1007/BFb0086068. Google Scholar [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. doi: 10.1007/BF01385813. Google Scholar [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), 58. Google Scholar [34] A. Quarteroni and A. Valli, Theory and applications of Steklov-Poincaré operators for boundary value problems,, in Applied and Industrial Mathematics (Venice, (1989), 179. Google Scholar [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, (1991), 441. Google Scholar [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, (1991), 129. Google Scholar [37] A. Valli, Mathematical results for compressible flows,, in Mathematical Topics in Fluid Mechanics (Lisbon, (1991), 193. Google Scholar [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), (1993), 247. Google Scholar [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), 81. Google Scholar [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. Google Scholar [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. doi: 10.1007/BF02567511. Google Scholar [42] A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations,, in 27th Computational Fluid Dynamics (ed. H. Deconinck), (1996), 1. Google Scholar [43] A. Alonso and A. Valli, Domain decomposition algorithms for low-frequency time-harmonic Maxwell equations,, in Numerical Modelling in Continuum Mechanics (Prague, (1997), 3. Google Scholar [44] A. Alonso and A. Valli, A domain decomposition approach for heterogeneous time-harmonic Maxwell equations,, Comput. Methods Appl. Mech. Engrg., 143 (1997), 97. doi: 10.1016/S0045-7825(96)01144-9. Google Scholar [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. doi: 10.1016/S0377-0427(98)00091-0. Google Scholar [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), 227. Google Scholar [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. doi: 10.1002/(SICI)1099-1476(199804)21:6<463::AID-MMA947>3.0.CO;2-U. Google Scholar [48] A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations,, Math. Comp., 68 (1999), 607. doi: 10.1090/S0025-5718-99-01013-3. Google Scholar [49] A. Quarteroni and A. Valli, Domain decomposition methods for compressible flows,, in Error Control and Adaptivity in Scientific Computing (Antalya, (1998), 221. Google Scholar [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. doi: 10.1051/m2an:2001137. Google Scholar [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), 157. doi: 10.1007/978-3-642-56118-4_10. Google Scholar [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), 143. doi: 10.1007/978-3-642-55745-3_10. Google Scholar [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. doi: 10.1017/S0956792503005151. Google Scholar [54] A. Alonso Rodríguez, R. Hiptmair and A. Valli, Mixed finite element approximation of eddy current problems,, IMA J. Numer. Anal., 24 (2004), 255. doi: 10.1093/imanum/24.2.255. Google Scholar [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), 1. Google Scholar [56] A. Alonso Rodríguez, R. Hiptmair and A. Valli, A hybrid formulation of eddy current problems,, Numer. Methods Partial Differential Equations, 21 (2005), 742. doi: 10.1002/num.20060. Google Scholar [57] A. Quarteroni, M. Sala and A. Valli, An interface-strip domain decomposition preconditioner,, SIAM J. Sci. Comput., 28 (2006), 498. doi: 10.1137/04061057X. Google Scholar [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. doi: 10.1016/j.cma.2006.10.008. Google Scholar [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. doi: 10.1137/06065091X. Google Scholar [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. doi: 10.1002/mma.900. Google Scholar [61] A. Alonso Rodríguez and A. Valli, Voltage and current excitation for time-harmonic eddy-current problems,, SIAM J. Appl. Math., 68 (2008), 1477. doi: 10.1137/070697677. Google Scholar [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. doi: 10.1016/j.apnum.2008.12.002. Google Scholar [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. doi: 10.1007/s00211-009-0241-7. Google Scholar [64] A. Alonso Rodríguez, J. Camaño and A. Valli, Inverse source problems for eddy current equations,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/1/015006. Google Scholar [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. doi: 10.1016/j.aml.2011.12.013. Google Scholar [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. doi: 10.1137/120890648. Google Scholar [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. doi: 10.1016/j.camwa.2014.06.017. Google Scholar [68] A. Alonso Rodríguez and A. Valli, Finite element potentials,, Appl. Numer. Math., 95 (2015), 2. doi: 10.1016/j.apnum.2014.05.014. Google Scholar [69] A. Alonso Rodríguez and A. Valli, Eddy Current Approximation of Maxwell Equations,, Springer Italia, (2010). doi: 10.1007/978-88-470-1506-7. Google Scholar [70] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations,, Oxford University Press, (1999). Google Scholar [71] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations,, Springer, (1994). Google Scholar

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##### 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. Google Scholar [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. Google Scholar [3] L. Carbone and A. Valli, Free boundary enclosure in a three-dimensional filtration problem,, Appl. Math. Optim., 4 (1977), 1. doi: 10.1007/BF01442128. Google Scholar [4] A. Valli, Soluzioni classiche dell'equazione di Eulero dei fluidi bidimensionali in domini con frontiera variabile,, Ricerche Mat., 26 (1977), 301. Google Scholar [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. Google Scholar [6] L. Carbone and A. Valli, Filtration through a porous non-homogeneous medium with variable cross-section,, J. Analyse Math., 33 (1978), 191. doi: 10.1007/BF02790173. Google Scholar [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. Google Scholar [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. doi: 10.1080/03605308008820134. Google Scholar [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. Google Scholar [10] H. Beirão da Veiga and A. Valli, On the Euler equations for non-homogeneous fluids (II),, J. Math. Anal. Appl., 73 (1980), 338. doi: 10.1016/0022-247X(80)90282-6. Google Scholar [11] A. Valli, Uniqueness theorems for compressible viscous fluids, especially when the Stokes relation holds,, Boll. Un. Mat. Ital. C (5), 18 (1981), 317. Google Scholar [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. doi: 10.1016/0022-247X(82)90033-6. Google Scholar [13] A. Valli, A correction to the paper: "An existence theorem for compressible viscous fluids'',, Ann. Mat. Pura Appl. (4), 132 (1982), 399. doi: 10.1007/BF01760990. Google Scholar [14] A. Valli, An existence theorem for compressible viscous fluids,, Ann. Mat. Pura Appl. (4), 130 (1982), 197. doi: 10.1007/BF01761495. Google Scholar [15] P. Secchi and A. Valli, A free boundary problem for compressible viscous fluids,, J. Reine Angew. Math., 341 (1983), 1. doi: 10.1515/crll.1983.341.1. Google Scholar [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. Google Scholar [17] A. Valli, Free boundary problems for compressible viscous fluids,, in Fluid Dynamics (Varenna, (1982), 175. doi: 10.1007/BFb0072331. Google Scholar [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. Google Scholar [19] A. Valli, Global existence theorems for compressible viscous fluids,, in Nonlinear Variational Problems (Isola d'Elba, (1983), 120. Google Scholar [20] A. Valli, On the integral representation of the solution to the Stokes system,, Rend. Sem. Mat. Univ. Padova, 74 (1985), 85. Google Scholar [21] A. Valli, Navier-Stokes equations for compressible fluids: Global estimates and periodic solutions,, in Nonlinear Functional Analysis and its Applications, (1983), 467. Google Scholar [22] A. Valli, Qualitative properties of the solutions to the Navier-Stokes equations for compressible fluids,, in Equadiff 6 (Brno, (1985), 259. doi: 10.1007/BFb0076079. Google Scholar [23] A. Valli, Stationary solutions to the Navier-Stokes equations for compressible fluids,, in BAIL IV (Novosibirsk, (1986), 417. Google Scholar [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. doi: 10.1007/BF01206939. Google Scholar [25] A. Valli, On the existence of stationary solutions to compressible Navier-Stokes equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 99. Google Scholar [26] I. Straškraba and A. Valli, Asymptotic behaviour of the density for one-dimensional Navier-Stokes equations,, Manuscripta Math., 62 (1988), 401. doi: 10.1007/BF01357718. Google Scholar [27] A. Valli and W. Zajączkowski, About the motion of non-homogeneous ideal incompressible fluids,, Nonlinear Anal., 12 (1988), 43. doi: 10.1016/0362-546X(88)90011-9. Google Scholar [28] A. Valli, An existence theorem for non-homogeneous inviscid incompressible fluids,, in Differential Equations (Xanthi, (1987), 691. Google Scholar [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. Google Scholar [30] A. Quarteroni and A. Valli, Domain decomposition for a generalized Stokes problem,, in, (1988), 59. Google Scholar [31] A. Valli, On the one-dimensional Navier-Stokes equations for compressible fluids,, in The Navier-Stokes Equations (Oberwolfach, (1988), 173. doi: 10.1007/BFb0086068. Google Scholar [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. doi: 10.1007/BF01385813. Google Scholar [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), 58. Google Scholar [34] A. Quarteroni and A. Valli, Theory and applications of Steklov-Poincaré operators for boundary value problems,, in Applied and Industrial Mathematics (Venice, (1989), 179. Google Scholar [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, (1991), 441. Google Scholar [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, (1991), 129. Google Scholar [37] A. Valli, Mathematical results for compressible flows,, in Mathematical Topics in Fluid Mechanics (Lisbon, (1991), 193. Google Scholar [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), (1993), 247. Google Scholar [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), 81. Google Scholar [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. Google Scholar [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. doi: 10.1007/BF02567511. Google Scholar [42] A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations,, in 27th Computational Fluid Dynamics (ed. H. Deconinck), (1996), 1. Google Scholar [43] A. Alonso and A. Valli, Domain decomposition algorithms for low-frequency time-harmonic Maxwell equations,, in Numerical Modelling in Continuum Mechanics (Prague, (1997), 3. Google Scholar [44] A. Alonso and A. Valli, A domain decomposition approach for heterogeneous time-harmonic Maxwell equations,, Comput. Methods Appl. Mech. Engrg., 143 (1997), 97. doi: 10.1016/S0045-7825(96)01144-9. Google Scholar [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. doi: 10.1016/S0377-0427(98)00091-0. Google Scholar [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), 227. Google Scholar [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. doi: 10.1002/(SICI)1099-1476(199804)21:6<463::AID-MMA947>3.0.CO;2-U. Google Scholar [48] A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations,, Math. Comp., 68 (1999), 607. doi: 10.1090/S0025-5718-99-01013-3. Google Scholar [49] A. Quarteroni and A. Valli, Domain decomposition methods for compressible flows,, in Error Control and Adaptivity in Scientific Computing (Antalya, (1998), 221. Google Scholar [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. doi: 10.1051/m2an:2001137. Google Scholar [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), 157. doi: 10.1007/978-3-642-56118-4_10. Google Scholar [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), 143. doi: 10.1007/978-3-642-55745-3_10. Google Scholar [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. doi: 10.1017/S0956792503005151. Google Scholar [54] A. Alonso Rodríguez, R. Hiptmair and A. Valli, Mixed finite element approximation of eddy current problems,, IMA J. Numer. Anal., 24 (2004), 255. doi: 10.1093/imanum/24.2.255. Google Scholar [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), 1. Google Scholar [56] A. Alonso Rodríguez, R. Hiptmair and A. Valli, A hybrid formulation of eddy current problems,, Numer. Methods Partial Differential Equations, 21 (2005), 742. doi: 10.1002/num.20060. Google Scholar [57] A. Quarteroni, M. Sala and A. Valli, An interface-strip domain decomposition preconditioner,, SIAM J. Sci. Comput., 28 (2006), 498. doi: 10.1137/04061057X. Google Scholar [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. doi: 10.1016/j.cma.2006.10.008. Google Scholar [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. doi: 10.1137/06065091X. Google Scholar [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. doi: 10.1002/mma.900. Google Scholar [61] A. Alonso Rodríguez and A. Valli, Voltage and current excitation for time-harmonic eddy-current problems,, SIAM J. Appl. Math., 68 (2008), 1477. doi: 10.1137/070697677. Google Scholar [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. doi: 10.1016/j.apnum.2008.12.002. Google Scholar [63] A. Alonso Rodríguez, A. Valli and R. 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