Free Access
Issue
ESAIM: M2AN
Volume 55, 2021
Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
Page(s) S187 - S223
DOI https://doi.org/10.1051/m2an/2020026
Published online 26 February 2021
  1. 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. [Google Scholar]
  2. F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997) 267–279. [Google Scholar]
  3. A. Bensoussan, Some existence results for stochastic partial differential equations. In: Stochastic Partial Differential Equations and Applications (Trento, 1990). Longman Sci. Tech., Harlow (1992) 37–53. [Google Scholar]
  4. Y. Cao, R. Zhang and K. Zhang, Finite element and discontinuous Galerkin method for stochastic Helmholtz equation in two- and three-dimensions. J. Comput. Math. 26 (2008) 702–715. [Google Scholar]
  5. Y. Cao, R. Zhang and K. Zhang, Finite element method and discontinuous Galerkin method for stochastic scattering problem of Helmholtz type in ℝd (d = 2,3). Potential Anal. 28 (2008) 301–319. [Google Scholar]
  6. P. Castillo, B. Cockburn, D. Schotzau and C. Schwab, Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comput. 71 (2002) 455–478. [Google Scholar]
  7. T. Chen, B. Rozovskii and C.-W. Shu, Numerical solutions of stochastic PDEs driven by arbitrary type of noise. Stoch. Part. Differ. Equ.: Anal. Comput. 7 (2019) 1–39. [Google Scholar]
  8. P. Ciarlet, The Finite Element Method for Elliptic Problem. North Holland (1975). [Google Scholar]
  9. B. Cockburn and C.-W. Shu, The Runge-Kutta local projection P1-discontinuous Galerkin method for scalar conservation laws. MMNP 25 (1991) 337–361. [Google Scholar]
  10. B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52 (1998) 411–435. [Google Scholar]
  11. B. Cockburn and C.-W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141 (1998) 199–224. [Google Scholar]
  12. B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440–2463. [Google Scholar]
  13. B. Cockburn, S.-Y. Lin and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. Comput. Phys. 84 (1989) 90–113. [Google Scholar]
  14. B. Cockburn, S. Hou and C.-W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54 (1990) 545–581. [Google Scholar]
  15. Y.L. Dalecky and N.Y. Goncharuk, On a quasilinear stochastic differential equation of parabolic type. Stoch. Anal. App. 12 (1994) 103–129. [Google Scholar]
  16. A.M. Davie and J.G. Gaines, Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations. Math. Comput. 70 (2001) 121–134. [Google Scholar]
  17. A. Debussche and J. Printems, Weak order for the discretization of the stochastic heat equation. Math. Comput. 78 (2009) 845–863. [Google Scholar]
  18. K. Du and J. Liu, On the Cauchy problem for stochastic parabolic equations in Hölder spaces. Trans. Am. Math. Soc. 371 (2019) 2643–2664. [Google Scholar]
  19. Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by special additive noises. SIAM J. Numer. Anal. 40 (2002) 1421–1445. [Google Scholar]
  20. N.Y. Goncharuk, On a class of quasilinear stochastic differential equations of parabolic type: regular dependence of solutions on initial data. Stochastic Partial Differential Equations. Cambridge University Press, Cambridge (1995) 97–119. [Google Scholar]
  21. I. Gyöngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise I. Potential Anal. 9 (1998) 1–25. [CrossRef] [MathSciNet] [Google Scholar]
  22. I. Gyöngy and T. Martinez, On numerical solution of stochastic partial differential equations of elliptic type. Stochastics 78 (2006) 213–231. [Google Scholar]
  23. S. He, J. Wang and J. Yan, Semimartingale Theory and Stochastic Calculus. Science Press, New York, (1992). [Google Scholar]
  24. M. Hofmanová, Strong solutions of semilinear stochastic partial differential equations. Nonlinear Differ. Equ. App. 20 (2013) 757–778. [Google Scholar]
  25. A. Jentzen and P.E. Kloeden, Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise. Proc. R. Soc. A: Math. Phys. Eng. Sci. 465 (2009) 649–667. [Google Scholar]
  26. G. Jiang and C.-W. Shu, On a cell entropy inequality for discontinuous Galerkin methods. Math. Comput. 62 (1994) 531–538. [Google Scholar]
  27. P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, 3rd ed. In: Vol. 23 of Applications in Mathematics, Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin (1999). [Google Scholar]
  28. M. Kovacs, S. Larsson and F. Saedpanah, Finite element approximation of the linear stochastic wave equation with additive noise. SIAM J. Numer. Anal. 48 (2010) 408–427. [Google Scholar]
  29. R.J. LeVeque, Numerical Methods for Conservation Laws. In: Lectures in Mathematics. Birkhauser, Basel (1992). [Google Scholar]
  30. Y. Li, C.-W. Shu and S. Tang, A discontinuous Galerkin method for stochastic conservation laws. SIAM J. Sci. Comput. 42 (2020) A54–A86. [Google Scholar]
  31. G.J. Lord and T. Shardlow, Postprocessing for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 45 (2007) 870–889. [Google Scholar]
  32. X. Mao, Stochastic Differential Equations and Applications, 2nd ed. Horwood, Chichester (2008). [Google Scholar]
  33. A. Millet and P.L. Morien, On implicit and explicit discretization schemes for parabolic SPDEs in any dimension. Stoch. Process. App. 115 (2005) 1073–1106. [Google Scholar]
  34. T. Müller-Gronbach and K. Ritter, Lower bounds and nonuniform time discretization for approximation of stochastic heat equations. Found. Comput. Math. 7 (2007) 135–181. [Google Scholar]
  35. T. Müller-Gronbach, K. Ritter and T. Wagner, Optimal pointwise approximation of infinite-dimensional Ornstein-Uhlenbeck processes. Stoch. Dyn. 8 (2008) 519–541. [Google Scholar]
  36. É. Pardoux, Stochastic Partial Differential Equations. Lecture notes for the course given at Fudan University, Shanghai, (2007). [Google Scholar]
  37. É. Pardoux and S. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Relat. Fields 98 (1994) 209–227. [Google Scholar]
  38. W. Pazner, N. Trask and P.J. Atzberger, Stochastic discontinuous Galerkin methods (SDGM) based on fluctuation-dissipation balance. Results Appl. Math. 4 (2019) 100068. [Google Scholar]
  39. C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations. In: Vol. 1905 of Lecture Notes in Mathematics. Springer, Berlin (2007). [Google Scholar]
  40. P. Protter, Stochastic Integration and Differential Equations, 2nd ed. Springer-Verlag, New York, (2004). [Google Scholar]
  41. M. Roozbahani, H. Aminikhah and M. Tahmasebi, Numerical solution of nonlinear SPDEs using a multi-scale method. Comput. Methods Diff. Equ. 6 (2018) 157–175. [Google Scholar]
  42. C. Roth, A combination of finite difference and Wong-Zakai methods for hyperbolic stochastic partial differential equations. Stoch. Anal. App. 24 (2006) 221–240. [Google Scholar]
  43. P.E. Souganidis, Fully nonlinear first- and second-order stochastic partial differential equations. In: CIME Lecture Notes (2016) 1–37. [Google Scholar]
  44. J.B. Walsh, Finite element methods for parabolic stochastic PDEs. Potential Anal. 23 (2005) 1–43. [Google Scholar]
  45. J.B. Walsh, On numerical solutions of the stochastic wave equation. Illinois J. Math. 50 (2006) 991–1018. [CrossRef] [Google Scholar]
  46. H.-J. Wang and Q. Zhang, Error estimate on a fully discrete local discontinuous Galerkin method for linear convection-diffusion problem. J. Comput. Math. 31 (2013) 283–307. [Google Scholar]
  47. Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 43 (2005) 1363–1384. [Google Scholar]

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