Space-time finite element methods stabilized using bubble function spaces

References

• Eriksson K, Johnson C, Thomeé V. Time discretization of parabolic problems by the discontinuous Galerkin method. RAIRO Modél Math Anal Numér. 1985;19:611–643. doi: 10.1051/m2an/1985190406111
   Web of Science ®Google Scholar
• Eriksson K, Johnson C. Error estimates and automatic time step control for nonlinear parabolic problems I. SIAM J Numer Anal. 1987;24:12–23. doi: 10.1137/0724002
   Web of Science ®Google Scholar
• Akrivis G, Makridakis C. Galerkin time-stepping methods for nonlinear parabolic equations. ESAIM: Math Model Numer Anal. 2004;38(2):261–289. doi: 10.1051/m2an:2004013
   Web of Science ®Google Scholar
• Thomeé V. Galerkin finite element methods for parabolic problems, Berlin, Heidelberg: Springer-Verlag; 2006. (Springer series in computational mathematics; vol. 25).
   Google Scholar
• Eriksson K, Johnson C. Adaptive finite element methods for parabolic problems IV: nonlinear problems. SIAM J Numer Anal. 1995;32(6):1729–1749. doi: 10.1137/0732078
   Web of Science ®Google Scholar
• Eriksson K, Johnson C. Adaptive finite element methods for parabolic problems V: long-time integration. SIAM J Numer Anal. 1995;32(6):1750–1763. doi: 10.1137/0732079
   Web of Science ®Google Scholar
• Schwab C, Stevenson R. Space-time adaptive wavelet methods for parabolic evolution problems. Math Comput. 2009;78:1293–1318. doi: 10.1090/S0025-5718-08-02205-9
   Web of Science ®Google Scholar
• Chegini N, Stevenson R. Adaptive wavelet schemes for parabolic problems: sparsematrices and numerical results. SIAM J Numer Anal. 2011;49:182–212. doi: 10.1137/100800555
   Web of Science ®Google Scholar
• Babuška I, Janik T. The h-p version of the finite element method for parabolic equations. Part I. The p-version in time. Numer Methods Partial Differ Equ. 1989;5(4):363–399. doi: 10.1002/num.1690050407
   Google Scholar
• Babuška I, Janik T. The h-p version of the finite element method for parabolic equations. II. The h-p version in time. Numer Methods Partial Differ Equ. 1990;6(4):343–369. doi: 10.1002/num.1690060406
   Google Scholar
• Hughes TJR, Hulbert GM. Space-time finite element methods for elastodynamics: formulations and error estimates. Comput Methods Appl Mech Engrg. 1988;66:339–363. doi: 10.1016/0045-7825(88)90006-0
   Web of Science ®Google Scholar
• Johnson C. Numerical solution of partial differential equations by the finite element method. Cambridge: Cambridge University Press; 1988.
   Google Scholar
• Hansbo P. Space-time oriented streamline diffusion methods for non-linear conservation laws in one dimension. Commun Numer Methods Eng. 1994;10(3):203–215. doi: 10.1002/cnm.1640100304
   Web of Science ®Google Scholar
• Hansbo P. The characteristic streamline diffusion method for the time-dependent incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng. 1992;99(2–3):171–186. doi: 10.1016/0045-7825(92)90039-M
   Web of Science ®Google Scholar
• Andreev R. Stability of sparse space-time finite element discretizations of linear parabolic evolution problems. IMA J Numer Anal. 2013;33(1):242–260. doi: 10.1093/imanum/drs014
   Web of Science ®Google Scholar
• Andreev R. On long time integration of the heat equation. Calcolo. 2016;53(1):19–34. doi: 10.1007/s10092-014-0133-9
   Web of Science ®Google Scholar
• Steinbach O. Space-time finite element methods for parabolic problems. Comput Methods Appl Math. 2015;15(4):551–556.
   Web of Science ®Google Scholar
• Langer U, Moore SE, Neumüller M. Space-time isogeometric analysis of parabolic evolution problems. Comput Methods Appl Mech Eng. 2016;306:342–363. doi: 10.1016/j.cma.2016.03.042
   Web of Science ®Google Scholar
• Langer U, Neumüller M, Toulopoulos I. Multipatch space-time isogeometric analysis of parabolic diffusion problems. In: Lirkov I, Margenov S, editors. Large-scale scientific computing (LSSC 2017). Cham: Springer; 2017. p. 21–32. (Lecture notes in computer science (LNCS); vol. 10665).
   Google Scholar
• Hofer C, Langer U, Neumüller M, Toulopoulos I. Time-multipatch discontinuous Galerkin space-time isogeometric analysis of parabolic evolution problems. Electron Trans Numer Anal. 2018;49:1–25. doi: 10.1553/etna_vol49s1
   Web of Science ®Google Scholar
• Antonio Di Pietro D, Ern A. Mathematical aspects of discontinuous Galerkin methods. Vol. 69. Heidelberg: Springer-Verlag; 2012.
   Google Scholar
• Brezzi F, Bristeau M-O, Franca LP, Mallet M, Rogé G. A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Comput Methods Appl Mech Eng. 1992;96(1):117–129. doi: 10.1016/0045-7825(92)90102-P
   Web of Science ®Google Scholar
• Baiocchi C, Brezzi F, Franca LP. Virtual bubbles and Galerkin-least-squares type methods (Ga.L.S.). Comput Methods Appl Mech Eng. 1993;105(1):125–141. doi: 10.1016/0045-7825(93)90119-I
   Web of Science ®Google Scholar
• Quarteroni A, Valli A. Numerical approximation of partial differential equations. Vol. 23. Berlin, Heidelberg: Springer-Verlag; 1994.
   Google Scholar
• Guermond JL. Stabilization of Galerkin approximations of transport equations by subgrid modeling. ESAIM: M2AN. 1999;33(6):1293–1316. doi: 10.1051/m2an:1999145
   Web of Science ®Google Scholar
• Guermond JL. Subgrid stabilization of Galerkin approximations of linear monotone operators. IMA J Numer Anal. 2001;21(1):165–197. doi: 10.1093/imanum/21.1.165
   Web of Science ®Google Scholar
• Adams R, Fournier J. Sobolev spaces. 2nd ed. Oxford: Academic Press-imprint Elsevier Science; 2003. (Pure and applied mathematics; vol. 140).
   Google Scholar
• Ladyzhenskaya OA. The boundary value problems of mathematical physics. New York: Springer; 1985. (Applied mathematical sciences series; vol. 49).
   Google Scholar
• Zeidler E. Nonlinear functional analysis and its applications, II/A: linear monotone operators. New York: Springer-Verlag; 1990.
   Google Scholar
• Evans LC. Partial differential equestions. 1st ed. Providence (RI): American Mathematical Society; 1998. (Graduate studies in mathematics; vol. 19).
   Google Scholar
• Brener SC, Scott LR. The mathematical theory of finite element methods. 3rd ed. New York: Springer-Verlag; 2008. (Texts in applied mathematics; vol. 15).
   Google Scholar
• Ciarlet PG. The finite element method for elliptic problems. New York (NY): North Holland Publishing Company; 1978. (Studies in mathematics and its applications).
   Google Scholar
• Ern A, Guermond J-L. Theory and practice of finite elements. New York: Springer-Verlag; 2004. (Applied mathematical sciences; vol. 159).
   Google Scholar
• Apel T. Interpolation of non-smooth functions on anisotropic finite element meshes. ESAIM: Math Model Numer Anal. 1999;33(6):1149–1185. doi: 10.1051/m2an:1999139
   Web of Science ®Google Scholar