Abstract

We propose certified reduced basis methods for the efficient and reliable evaluation of a general output that is implicitly connected to a given parameterized input through the harmonic Maxwell's equations. The truth approximation and the development of the reduced basis through a greedy approach is based on a discontinuous Galerkin approximation of the linear partial differential equation. The formulation allows the use of different approximation spaces for solving the primal and the dual truth approximation problems to respect the characteristics of both problem types, leading to an overall reduction in the off-line computational effort. The main features of the method are the following: (i) rapid convergence on the entire representative set of parameters, (ii) rigorous a posteriori error estimators for the output, and (iii) a parameter independent off-line phase and a computationally very efficient on-line phase to enable the rapid solution of many-query problems arising in control, optimization, and design. The versatility and performance of this approach is shown through a numerical experiment, illustrating the modeling of material variations and problems with resonant behavior.

MSC codes

  1. 65N15
  2. 65N30
  3. 78A25

Keywords

  1. reduced basis methods
  2. a priori theory
  3. a posteriori error estimation
  4. discontinuous Galerkin methods
  5. Maxwell's equations

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References

1.
B. O. Almroth, P. Stern, and F. A. Brogan, Automatic choice of global shape functions in structural analysis, AIAA J., 16 (1978), pp. 525–528.
2.
D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), pp. 1749–1779.
3.
E. Balmes, Parametric families of reduced finite element models: Theory and applications, Mech. Syst. Signal Process., 10 (1996), pp. 381–394.
4.
M. Barrault, N. C. Nguyen, Y. Maday, and A. T. Patera, An “empirical interpolation" method: Application to efficient reduced-basis discretization of partial differential equations, C. R. Acad. Sci. Paris, Ser. I, 339 (2004), pp. 667–672.
5.
A. Barrett and G. Reddien, On the reduced basis method, Z. Angew. Math. Mech., 75 (1995), pp. 543–549.
6.
Y. Chen, J. S. Hesthaven, Y. Maday, and J. Rodríguez, A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations, C. R. Acad. Sci. Paris, Ser. I, 346 (2008), pp. 1295–1300.
7.
Y. Chen, J. S. Hesthaven, Y. Maday, and J. Rodríguez, Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2d Maxwell's problem, ESAIM: M2AN, 43 (2009), pp. 1099–1116.
8.
J. P. Fink and W. C. Rheinboldt, On the error behavior of the reduced basis technique for nonlinear finite element approximations, Z. Angew. Math. Mech., 63 (1983), pp. 21–28.
9.
M. A. Grepl, Reduced-Basis Approximation and A Posteriori Error Estimation for Parabolic Partial Differential Equations, Ph.D. thesis, MIT, Cambridge, MA, 2005.
10.
M. A. Grepl, Y. Maday, N. C. Nguyen, and A. T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, Math. Model. Numer. Anal., 41 (2007), pp. 575–605.
11.
P. Guillaume and M. Masmoudi, Solution to the time-harmonic Maxwell's equations in a waveguide; Use of higher-order derivatives for solving the discrete problem, SIAM J. Numer. Anal., 34 (1997), pp. 1306–1330.
12.
M. D. Gunzburger, Finite element methods for viscous incompressible flows, Computer Science and Scientific Computing. Academic Press, Boston, MA, 1989.
13.
J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Text in Applied Mathematics, Vol. 54, Springer-Verlag, New York, 2008.
14.
P. Houston, I. Perugia, A. Schneebeli, and D. Sch$\ddot{\rm o}$tzau, Interior penalty method for the indefinite time-harmonic Maxwell equations, Numer. Math. 100 (2005), pp. 485–518.
15.
P. Houston, I. Perugia, and D. Sch$\ddot{\rm o}$tzau, Mixed discontinuous Galerkin approximation of the Maxwell operator, SIAM J. Numer. Anal. 42 (2004), pp. 434–459.
16.
D. B. P. Huynh, G. Rozza, S. Sen, and A. T. Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants, C. R. Acad. Sci. Paris, Ser. I, 345 (2007), pp. 473–478.
17.
K. Ito and S. S. Ravindran, A reduced basis method for simulation and control of fluid flows, J. Comput. Phys., 143 (1998), pp. 403–425.
18.
M. A. Jabbar and A. B. Azeman, Fast optimization of electromagnetic-problems: The reduced-basis finite element approach, IEEE Trans. Magn., 40 (2004), pp. 2161–2163.
19.
M. A. Jabbar and A. B. Azeman, Multi-variable torque optimization for small spindle motors based on reduced-basis finite element formulation, International Conference on Power Electronics, Machines and Drives, 2002, pp. 269–274.
20.
E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section, SciTech, Raleigh, NC, 2004.
21.
L. Machiels, Y. Maday, I. B. Oliveira, A. T. Patera, and D. V. Rovas, Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems, C. R. Acad. Sci. Paris Ser. I Math., 331 (2000), pp. 153–158.
22.
Y. Maday, A. T. Patera, and D. V. Rovas, A blackbox reduced-basis output bound method for noncoercive linear problems, in Nonlinear partial differential equations and their applications, Stud. Math. Appl., Vol. 31, North-Holland, Amsterdam, 2002, pp. 533–569.
23.
Y. Maday, A. T. Patera, and G. Turinici, Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations, C. R. Acad. Sci. Paris, Ser. I, 335 (2002), pp. 289–294.
24.
Y. Maday and E. M. Rønquist, A reduced basis element method, J. Sci. Comput., 17 (2002), pp. 447-459.
25.
P. Monk, Finite Element Methods for Maxwell's Equations, Numerical Mathematics and Scientific Computing, Oxford University Press, Oxford, United Kingdom, 2003.
26.
N. C. Nguyen, Reduced-Basis Approximations and A Posteriori Error Bounds for Nonaffine and Nonlinear Partial Differential Equations: Application to Inverse Analysis, PhD thesis, Singapore-MIT Alliance, 2005.
27.
A. K. Noor and J. M. Peters, Reduced basis technique for nonlinear analysis of structures, AIAA J., 45 (1980), pp. 487–496.
28.
J. S. Peterson, The reduced basis method for incompressible viscous flow calculations, SIAM J. Sci. Comput., 10 (1989), pp. 777–786.
29.
N. A. Pierce and M. B. Giles, Adjoint recovery of superconvergent functionals from PDE approximations, SIAM Rev., 42 (2000), pp. 247–264.
30.
T. A. Porsching, Estimation of the error in the reduced basis method solution of nonlinear equations, Math. Comp., 45 (1985), pp. 487–496.
31.
C. Prud'homme, D. Rovas, K. Veroy, Y. Maday, A. T. Patera, and G. Turinici, Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods, J. Fluids Engrg., 124 (2002), pp. 70–80.
32.
W. C. Rheinboldt, On the theory and error estimation of the reduced basis method for multi-parameter problems, Nonlinear Anal., 21 (1993), pp. 849–858.
33.
D. Rovas, Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations, Ph.D. Thesis, MIT, Cambridge, MA, 2003.
34.
G. Rozza, Shape Design by Optimal Flow Control and Reduced Basis Techniques: Applications to Bypass Configurations in Haemodynamics, PhD thesis, EPFL, Lausanne, 2005.
35.
G. Rozza, D. B. P. Huynh, and A. T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics, Arch. Comput. Methods Engrg., 15 (2008), pp. 229–275.
36.
S. Sen, Reduced Basis Approximation and A Posteriori Error Estimation for Non-Coercive Elliptic Problems: Application to Acoustics, PhD thesis, MIT, Cambridge, MA, 2007.
37.
K. Veroy, Reduced-Basis Methods Applied to Problems in Elasticity: Analysis and Applications, PhD thesis, MIT, Cambridge, MA, 2003.
38.
K. Veroy, C. Prud'homme, and A. T. Patera, Reduced-basis approximation of the viscous Burgers equation: Rigorous a posteriori error bounds, C. R. Acad. Sci. Paris, Ser. I, 337 (2003), pp. 619–624.
39.
K. Veroy, C. Prud'homme, D. V. Rovas, and A. T. Patera, A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, AIAA, 2003.

Information & Authors

Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: 970 - 996
ISSN (online): 1095-7197

History

Submitted: 12 March 2009
Accepted: 2 February 2010
Published online: 24 March 2010

MSC codes

  1. 65N15
  2. 65N30
  3. 78A25

Keywords

  1. reduced basis methods
  2. a priori theory
  3. a posteriori error estimation
  4. discontinuous Galerkin methods
  5. Maxwell's equations

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