Error estimates for space–time discontinuous Galerkin formulation based on proper orthogonal decomposition

References

• Volker J, Novo J. Error analysis of the SUPG finite element discretization of evolutionary convection--diffusion--reaction equations. SIAM J. Numer. Anal. 2011;49:1149–1176.
   Web of Science ®Google Scholar
• Feistauer M, Švadlenka K. Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems. J. Numer. Math. 2004;12:97–117.
   Google Scholar
• Dolejší V, Feistauer M, Kučera V, et al. An optimal L∞(L2) -error estimate for the discontinuous Galerkin approximation of a nonlinear non-stationary convection-diffusion problem. IMA J. Numer. Anal. 2008;28:496–521. doi:10.1093/imanum/drm023.
   Web of Science ®Google Scholar
• Vlasák M, Dolejší V, Hájek J. A priori error estimates of an extrapolated space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Methods Partial Differ. Equ. 2011;27:1456–1482.
   Web of Science ®Google Scholar
• Dolejší V, Feistauer M. Error estimates of the discontinuous Galerkin method for nonlinear nonstationary convection-diffusion problems. Numer. Funct. Anal. Optim. 2005;26:349–383.
   Web of Science ®Google Scholar
• Eriksson K, Johnson C. Adaptive finite element methods for parabolic problems. II. Optimal error estimates in L∞L2 and L∞L∞. SIAM J. Numer. Anal. 1995;32:706–740.
   Web of Science ®Google Scholar
• Thomée V. Galerkin finite element methods for parabolic problems. 2nd ed. Vol. 25, Springer series in computational mathematics. Berlin: Springer-Verlag; 2006.
   Google Scholar
• Kunisch K, Volkwein S. Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 2001;90:117–148.
   Web of Science ®Google Scholar
• Volkwein S. Proper orthogonal decomposition: applications in optimization and control. Lecture notes. CEA-EDF-INRIA summer school numerical analysis summer school model reduction and reduced basis methods: application in optimization. Saint-Lambert-des-Bois, France, 2008 June 23--July 4. p. 74. Available from: http://www.math.uni-konstanz.de/numerik/personen/volkwein/teaching/Lecture-Notes-Volkwein.pdf.
   Google Scholar
• Singer MA, Green WH. Using adaptive proper orthogonal decomposition to solve the reaction--diffusion equation. Appl. Numer. Math. 2009;59:272–279.
   Web of Science ®Google Scholar
• Iliescu T, Wang Z. Variational multiscale proper orthogonal decomposition: convection-dominated convection--diffusion--reaction equations. Math. Comput. 2013;82:1357–1378. doi:10.1090/S0025-5718-2013-02683-X.
   Web of Science ®Google Scholar
• Giere S, Iliescu T, John V, et al. SUPG reduced order models for convection-dominated convection--diffusion--reaction equations. Comput. Methods Appl. Mech. Eng. 2015;289:454–474.
   Web of Science ®Google Scholar
• Arnold D, Brezzi F, Cockburn B, et al. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 2001/02;39:1749–1779.
   Web of Science ®Google Scholar
• Ayuso B, Marini LD. Discontinuous Galerkin methods for advection--diffusion--reaction problems. SIAM J. Numer. Anal. 2009;47:1391–1420.
   Web of Science ®Google Scholar
• Rivière B. Discontinuous Galerkin methods for solving elliptic and parabolic equations. Vol. 35, Frontiers in applied mathematics. Theory and implementation. Philadelphia (PA): Society for Industrial and Applied Mathematics (SIAM); 2008.
   Google Scholar
• Holmes P, Lumley JL, Berkooz G. Turbulence, coherent structures, dynamical systems and symmetry. Cambridge monographs on mechanics. Cambridge: Cambridge University Press; 1996.
   Google Scholar
• Chrysafinos K, Walkington NJ. Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 2006;44:349–366. electronic.
   Web of Science ®Google Scholar
• Akman T, Karasözen B. Variational time discretization methods for optimal control problems governed by diffusion-convection-reaction equations. J. Comput. Appl. Math. 2014;272:41–56.
   Web of Science ®Google Scholar
• Dolejší V, Feistauer M, Schwab C. A finite volume discontinuous Galerkin scheme for nonlinear convection--diffusion problems. Calcolo. 2002;39:1–40.
   Web of Science ®Google Scholar
• Singler JR. New POD error expressions, error bounds, and asymptotic results for reduced order models of parabolic PDEs. SIAM J. Numer. Anal. 2014;52:852–876.
   Web of Science ®Google Scholar
• Iliescu T, Wang Z. Variational multiscale proper orthogonal decomposition: Navier--Stokes equations. Numer. Methods Partial Differ. Equ. 2014;30:641–663. doi:10.1002/num.21835.
   Web of Science ®Google Scholar
• Kostova T, Oxberry G, Chand K, et al. Error bounds and analysis of proper orthogonal decomposition model reduction methods using snapshots from the solution and the time derivatives. 2015. arXiv preprint arXiv:150102004.
   Google Scholar
• Iliescu T, Wang Z. Are the snapshot difference quotients needed in the proper orthogonal decomposition? SIAM J. Sci. Comput. 2014;36:A1221–A1250. doi:10.1137/130925141.
   Web of Science ®Google Scholar
• Ciarlet PG. The finite element method for elliptic problems. Amsterdam: North-Holland; 1978.
   Google Scholar
• Nouy A. A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations. Comput. Methods Appl. Mech. Eng. 2010;199:1603–1626.
   Web of Science ®Google Scholar
• Borggaard J, Iliescu T, Wang Z. Artificial viscosity proper orthogonal decomposition. Math. Comput. Model. 2011;53:269–279. doi:10.1016/j.mcm.2010.08.015.
   Web of Science ®Google Scholar