Journal of the Korean Society for Industrial and Applied Mathematics

  • 제19권4호
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  • Pages.387-407
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  • 2015
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  • 1226-9433(pISSN)
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  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
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THE h × p FINITE ELEMENT METHOD FOR OPTIMAL CONTROL PROBLEMS CONSTRAINED BY STOCHASTIC ELLIPTIC PDES

  • LEE, HYUNG-CHUN (DEPARTMENT OF MATHEMATICS, AJOU UNIVERSITY) ;
  • LEE, GWOON (DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MARY WASHINGTON)
  • 투고 : 2015.09.26
  • 심사 : 2015.12.13
  • 발행 : 2015.12.25
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초록

This paper analyzes the $h{\times}p$ version of the finite element method for optimal control problems constrained by elliptic partial differential equations with random inputs. The main result is that the $h{\times}p$ error bound for the control problems subject to stochastic partial differential equations leads to an exponential rate of convergence with respect to p as for the corresponding direct problems. Numerical examples are used to confirm the theoretical results.

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참고문헌

  1. E. H. Georgoulis and E. Suli, Optimal error estimates for the hp-version interior penalty discontinous Galerkin finite element method, IMA J. Numer. Anal., 25 (2005), 205-220. https://doi.org/10.1093/imanum/drh014
  2. P. Houston, C. Schwab, and E. Suli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 39 (2002), 2133-2163. https://doi.org/10.1137/S0036142900374111
  3. I. Mozolevski and E. Suli, A priori error analysis for the hp-version of the discontinuous Galerkin finite element method for the biharmonic equation, Comput. Meth. Appl. Math., 3 (2003), 1-12.
  4. I. Mozolevski, E. Suli, and P. R. Bosing, hp-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation, J. Sci. Comput., 30 (2007), 465-491. https://doi.org/10.1007/s10915-006-9100-1
  5. D. Schotzau and C. Schwab, Time discretization of parabolic problems by the hp-version of the discontinous Galerkin finite element method, SIAM J. Numer. Anal., 38 (2000), 837-875. https://doi.org/10.1137/S0036142999352394
  6. I. Babuska, R. Tempone, and G. E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal., 42 (2004), 800-825. https://doi.org/10.1137/S0036142902418680
  7. I. Babuska, R. Tempone, and G. E. Zouraris, Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation, Comput. Method Appl. Mech. Engrg., 194 (2005), 1251-1294. https://doi.org/10.1016/j.cma.2004.02.026
  8. P. Frauenfelder, C. Schwab, and R. A. Todor, Finite elements for elliptic problems with stochastic coefficients, Comput. Method Appl. Mech. Engrg., 194 (2005), 205-228. https://doi.org/10.1016/j.cma.2004.04.008
  9. I. Babuska and P. Chatzipantelidis, On solving elliptic stochastic partial differential equations, Comput. Method Appl. Mech. Engrg., 191 (2002), 4093-4122. https://doi.org/10.1016/S0045-7825(02)00354-7
  10. I. Babuska, K. Liu, and R. Tempone, Solving stochastic partial differential equations based on the experimental data, Math. Models Methods Appl. Sci., 13 (2003), 415-444. https://doi.org/10.1142/S021820250300257X
  11. M. K. Deb, I. Babuska, and J. T. Oden, Solution of stochastic partial differential equations using Galerkin finite element techniques, Comput. Method Appl. Mech. Engrg., 190 (2001), 6359-6372. https://doi.org/10.1016/S0045-7825(01)00237-7
  12. M. D. Gunzburger, H.-C. Lee, and J. Lee, Error estimates of stochastic optimal Neumann boundary control problems, SIAM J. Numer. Anal., 49 (2011), 1532-1552. https://doi.org/10.1137/100801731
  13. H.-C. Lee and J. Lee, A Stochastic Galerkin Method for Stochastic Control Problems, Commun. Comput. Phys., 14 (2013), 77-106. https://doi.org/10.4208/cicp.241011.150612a
  14. L. S. Hou, J. Lee, and H. Manouzi Finite Element Approximations of Stochastic Optimal Control Problems Constrained by Stochastic Elliptic PDEs, J. Math. Anal. Appl. Vol., 384 (2011), 87-103. https://doi.org/10.1016/j.jmaa.2010.07.036
  15. C. Schwab and R. A. Todor, Sparse finite elements for elliptic problems with stochastic loading, Numer. Math., (2003), 707-734. https://doi.org/10.1007/s00211-003-0455-z
  16. K. Chrysafinos, Moving mesh finite element methods for an optimal control problem for the advection-diffusion equation, J. Sci. Comput., 25 (2005), 401-421. https://doi.org/10.1007/s10915-004-4804-6
  17. M. D. Gunzburger, L. S. Hou, and T. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls, Math. Comp. 57 (1991), 123-151. https://doi.org/10.1090/S0025-5718-1991-1079020-5
  18. M. D. Gunzburger, L. S. Hou, and T. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Cirichlet controls, RAIRO Model. Math. Anal. Numer., 25 (1991), 711-748. https://doi.org/10.1051/m2an/1991250607111
  19. M. D. Gunzburger and L. S. Hou, Finite-dimensional approximation of a class of constrained nonlinear optimal control problems, SIAM J. Control. Optim., 34 (1996), 1001-1043. https://doi.org/10.1137/S0363012994262361
  20. L. S. Hou and S. S. Lavindran, A penalized neumann control approach for solving an optimal dirichlet control problem for the Navier-Stokes equations, SIAM J. Control. Optim., 36 (1998), 1795-1814. https://doi.org/10.1137/S0363012996304870
  21. I. Babuska, B. A. Szabo, and I. N. Katz, The p-version of the finite element method, SIAM J. Numer. Anal., 18 (1981), 515-545. https://doi.org/10.1137/0718033
  22. W. Gui and I. Babuska, The h, p and h-p versions of the finite element method in 1 dimension. I. The error analysis of the p-version, Numer. Math., 49 (1986), 577-612. https://doi.org/10.1007/BF01389733
  23. W. Gui and I. Babuska, The h, p and h-p versions of the finite element method in 1 dimension. II. The error analysis of the h- and h-p versions, Numer. Math., 49 (1986), 613-657. https://doi.org/10.1007/BF01389734
  24. W. Gui and I. Babuska, The h, p and h-p versions of the finite element method in 1 dimension. III. The adaptive h-p version, Numer. Math., 49 (1986), 659-683. https://doi.org/10.1007/BF01389735
  25. B. Guo and I. Babuska, it The h-p version of the finite element method. I. The basic approximation results, Comput. Mech., 1 (1986), 21-41. https://doi.org/10.1007/BF00298636
  26. I. Babuska and B. Q. Guo, The h-p version of the finite element method for domains with curved boundaries, SIAM J. Numer. Anal., 25 (1988), 837-861. https://doi.org/10.1137/0725048
  27. B. Guo, The h-p version of the finite element method for elliptic equations of order 2m, Numer. Math., 53 (1988), 199-224. https://doi.org/10.1007/BF01395885
  28. I. Babuska and H.-S. Oh, The p-version of the finite element method for domains with corners and for infinite domains, Numer. Methods Partial Differential Eq., 6 (1990), 371-392. https://doi.org/10.1002/num.1690060407
  29. I. Babuska and M. Suri, The p and h-p versions of the finite element method, basic principles and properties, SIAM Rev., 36 (1994), 578-632. https://doi.org/10.1137/1036141
  30. I. Babuska and J. M. Melenk, The partition of unity finite element method, Technical Note BN-1185, Inst. Phys. Sci. Tech., (1995).
  31. I. Babuska and F. Ihlenburg, Finite element solution of the Helmholtz equation with high wave number part II: The h-p version of the FEM, SIAM J. Numer. Anal., 34 (1997), 315-358. https://doi.org/10.1137/S0036142994272337
  32. P. Chen and A. Quarteroni, Weighted Reduced Basis Method for Stochastic Optimal Control Problems with Elliptic PDE Constraint, SIAM/ASA J. Uncert. Quant., 2 (2014), 364-396. https://doi.org/10.1137/130940517
  33. E. Rosseel and G. N.Wells, Optimal control with stochastic PDE constraints and uncertain controls, Comput. Methods Appl. Mech. Engrg., (2012), 152-167.
  34. H. Tiesler, R. M. Kirby, D. Xiu, and T. Preusser, Stochastic collocation for optimal control problems with stochastic PDE constraints, SIAM J. Control Optim., 50 (2012), 2659-2682. https://doi.org/10.1137/110835438
  35. M. Eigel, C. J. Gittelson, C. Schwab, and E. Zander, A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes, ESAIM: M2AN, 49 (2015), 1367-1398. https://doi.org/10.1051/m2an/2015017
  36. R. Adams, Sobolev Spaces, Academic, New York, 1975.
  37. J. Galvis and M. Sarkis, Approximating infinity-dimensional stochastic Darcy's equations without uniform ellipticity, SIAM J. Numer. Anal., 47 (2009), 3624-3651. https://doi.org/10.1137/080717924
  38. S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Second Edition, Springer, 2002.
  39. R. G. Ghanem and P. D. Spanos, Stochastic finite elements: A spectral approach, Springer-Verlag, 1991.
  40. W. Luo, Wiener chaos expansion and numerical solutions of stochastic partial differential equations, Ph.D. thesis, California institute of technology, Pasadena, California 2006.
  41. F. Brezzi, J. Rappaz, and P. Raviart, Finite-dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions, Numer. Math., 36 (1980), 1-25. https://doi.org/10.1007/BF01395985
  42. M. Crouzeix and J. Rappaz, On numerical approximation in bifurcation theory, Masson, Parix, 1990.
  43. V. Girault and P. Raviart, Finite element methods for Navier-Stokes equations, Springer, Berlin, 1986.
  44. D. Xiu, Numerical methods for stochastic computationss, Princeton, 2010.