An hp-Discontinuous Galerkin Method for the Optimal Control Problem of Laser Surface Hardening of Steel

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1. V. Arnăutu, D. Hömberg and J. Sokołowski, Convergence results for a nonlinear parabolic control problem. Numer. Funct. Anal. Optim.20 (1999) 805–824.  
2. D.N. Arnold, An interior penalty method for discontinuous elements. SIAM J. Numer. Anal.19 (1982) 742–760.  
3. 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) 1749–1779.  
4. I. Babuška, The finite element method with penalty. Math. Comput.27 (1973) 221–228.  
5. C. Bernardi, M. Dauge and Y. Maday, Polynomials in the Sobolev world. Preprint of the Laboratoire Jacques-Louis LionsR03038 (2003).  
6. P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978).  
7. M. Crouzeix, V. Thomee, L.B. Wahlbin, Error estimates for spatial discrete approximation of semilinear parabolic equation with initial data of low regularity. Math. Comput.53 187 (1989) 25–41.  
8. J. Douglas, Jr. and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Computing Methods in Applied Sciences, Lecture Notes in Phys.58. Springer-Verlag, Berlin (1976).  
9. K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems I: a linear model problem. SIAM J. Numer. Anal.28 (1991) 43–77.  
10. K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems II: optimal error estimates in $L_{\infty }{L}_2$ and $L_{\infty }{L}_{\infty }$. SIAM J. Numer. Anal.32 (1995) 706–740.  
11. L.C. Evans, Partial Differential Equations. American Mathematics Society, Providence, Rhode Island (1998).  
12. T. Gudi, N. Nataraj and A.K. Pani, Discontinuous Galerkin methods for quasi-linear elliptic problems of nonmonotone type. SIAM J. Numer. Anal.45 (2007) 163–192.  
13. T. Gudi, N. Nataraj and A.K. Pani, hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems. Numer. Math.109 (2008) 233–268.  
14. T. Gudi, N. Nataraj and A.K. Pani, An hp-local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type. Math. Comput.77 (2008) 731–756.  
15. N. Gupta, N. Nataraj and A.K. Pani, An optimal control problem of laser surface hardening of steel. Int. J. Numer. Anal. Model.7 (2010).  
16. N. Gupta, N. Nataraj and A.K. Pani, A priori error estimates for the optimal control of laser surface hardening of steel. Paper communicated.  
17. D. Hömberg, A mathematical model for the phase transitions in eutectoid carbon steel. IMA J. Appl. Math.54 (1995) 31–57.  
18. D. Hömberg, Irreversible phase transitions in steel. Math. Methods Appl. Sci.20 (1997) 59–77.  
19. D. Hömberg and J. Fuhrmann, Numerical simulation of surface hardening of steel. Int. J. Numer. Meth. Heat Fluid Flow9 (1999) 705–724.  
20. D. Hömberg and J. Sokolowski, Optimal control of laser hardening. Adv. Math. Sci.8 (1998) 911–928.  
21. D. Hömberg and S. Volkwein, Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition. Math. Comput. Model.37 (2003) 1003–1028.  
22. D. Hömberg and W. Weiss, PID-control of laser surface hardening of steel. IEEE Trans. Control Syst. Technol.14 (2006) 896–904.  
23. P. Houston, C. Schwab and E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal.39 (2002) 2133–2163.  
24. A. Lasis and E. Süli, hp-version discontinuous Galerkin finite element methods for semilinear parabolic problems. Report No. 03/11, Oxford university computing laboratory (2003).  
25. A. Lasis and E. Süli, Poincaré-type inequalities for broken Sobolev spaces. Isaac Newton Institute for Mathematical Sciences, Preprint No. NI03067-CPD (2003).  
26. J.B. Leblond and J. Devaux, A new kinetic model for anisothermal metallurgical transformations in steels including effect of austenite grain size. Acta Metall.32 (1984) 137–146.  
27. V.I. Mazhukin and A.A. Samarskii, Mathematical modelling in the technology of laser treatments of materials. Surveys Math. Indust.4 (1994) 85-149.  
28. D. Meidner, and B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control problems part I: problems without control constraints. SIAM J. Control Optim.47 (2007) 1150–1177.  
29. J.A. Nitsche, Über ein Variationprinzip zur Lösung Dirichlet-Problemen bei Verwen-dung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9–15.  
30. J.T. Oden, I. Babuška and C.E. Baumann, A discontinuous hp finite element method for diffusion problems. J. Comput. Phys.146 (1998) 491–519.  
31. S. Prudhomme, F. Pascal and J.T. Oden, Review of error estimation for discontinuous Galerkin method. TICAM-report 00-27 (2000).  
32. B. Rivière, Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. Frontiers in Mathematics35. SIAM 2008. ISBN: 978-0-898716-56-6.  
33. B. Rivière and M.F. Wheeler, A discontinuous Galerkin method applied to nonlinear parabolic equations. The Center for Substance Modeling, TICAM, The University of Texas, Austin TX 78712, USA.  
34. B. Rivière, M.F. Wheeler and V. Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal.39 (2001) 902–931.  
35. V. Thomée, Galerkin finite element methods for parabolic problems. Springer (1997).  
36. S. Volkwein, Non-linear conjugate gradient method for the optimal control of laser surface hardening, Optim. Methods Softw.19 (2004) 179–199.  
37. M.F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal.15 (1978) 152–161.