Abstract
We investigate the approximation by space and time discretization of quasi linear evolution equations driven by nuclear or space time white noise. An error bound for the implicit Euler, the explicit Euler, and the Crank–Nicholson scheme is given and the stability of the schemes are considered. Lastly we give some examples of different space approximation, i.e., we consider approximation by eigenfunction, finite differences and wavelets.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bergh, J. and Löfström, J.: Interpolation Spaces: An Introduction, Grundlehren Math. Wiss. 223, Springer-Verlag, 1976.
Besov, O.V.: 'On some families of functional spaces. Imbedding and continuation theorems',Dokl. Akad. Nauk SSSR 126 (1959), 1163-1165.
Besov, O.V.: 'Investigation of a family of function spaces in connection with theorems of imbedding and extension',Amer. Math. Soc. Transl. Ser. II 40 (1964), 85-126.
Brenner, S.C. and Scott, L.R.:The Mathematical Theory of Finite Element Methods,Texts Appl. Math. 15, Springer, 1994.
Butzer, P.L. and Berens, H.:Semi-Groups of Operators and Approximation, Grundlehren Math. Wiss. 145,Springer-Verlag, 1967.
Chernoff, P.R.: 'Note on product formulas for operator semigroups',J. Funct. Anal. 2 (1968), 238-242.
Dahmen, W.: 'Wavelet and multiscale methods for operator equations', Acta Numerica 6 (1997), 55-228.
Dahmen, W.: 'Wavelet methods for PDEs — some recent developments',J. Comput. Appl. Math. 128 (2001), 133-185.
Dahmen, W., Kunoth, A. and Urban, K.: 'Biorthogonal spline wavelets on the interval — stability and moment conditions',Appl. Comput. Harmon. Anal. 6 (1999), 132-196.
Daubechies, I.:Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math. 61, CBMS, 1992.
Da Prato, G. and Zabczyk, J.:Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 44, Cambridge University Press, 1992.
Davie, A.M. and Gaines, J.G.: 'Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations', Math. Comput. 70(233) (2001), 121-134.
Flandoli, F.: 'On the semigroup approach to stochastic evolution equations', Stochastic Anal. Appl. 10 (1992), 181-203.
Flandoli, F.:Regularity Theory and Stochastic Flows for Parabolic SPDEs, Stochastics Monographs 9, Gordon and Breach, London, 1995.
Friedman, A.: Partial Differential Equations, Holt, Rinehart & Winston, Inc., New York, 1969.
Greksch, W. and Kloeden, P.: 'Time-discretised Galerkin approximations of parabolic stochastic PDEs', Bull. Austral. Math. Soc. 54 (1996), 79-85.
Gyöngy, I.: 'Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise: I', Potential Anal. 9 (1998), 1-25.
Gyöngy, I.: 'Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise: II', Potential Anal. 11 (1999), 1-37.
Harrington, R.F.: Field Computation by Moments Method, IEEE Press, New York, 1993.
Kato, T.: Perturbation Theory for Linear Operators, Reprint of the corr. print of the 2nd edn 1980, Vol.XXI, Springer-Verlag, 1959.
Kato, T.: 'Fractional powers of dissipative operators. II', J. Math. Soc. Japan 14 (1962), 242-248.
Kloeden, P.E. and Shott, S.: 'Linear-implicit strong schemes for Itô—Galerkin approximations of stochastic PDEs',J. Appl. Math. Stochastic Anal. 14 (2001), 47-53. Special issue: Advances in Applied Stochastics.
Meyer, Y.: Wavelets and Operators, Cambridge Stud. Adv. Math. 37, Cambridge University Press, 1992.
Miklavcic, M.:Applied Functional Analysis and Partial Differential Equations, World Scientific, Singapore, 1998.
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44,Springer-Verlag, New York, 1983.
Runst, T. and Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations, de Gruyter Ser. Nonlinear Anal. Appl. 3, de Gruyter, Berlin, 1996.
Seidler, J.: 'Da Prato—Zabczyk's maximal inequality revisited',Math. Bohem. 118 (1993), 67-106.
Shardlow, T.: 'Numerical methods for stochastic parabolic PDEs',Numer. Fund. Anal. Optim. 20 (1999), 121-145.
Stevenson, R.: 'Piecewise linear (pre-)wavelets on non-uniform meshes', in W. Hackbusch et al. (eds),Multigrid Methods V. Proceedings of the 5th European Multigrid Conference, Lecture Notes Comput. Sci. Eng., Vol.3,Springer, Berlin, 1998, pp. 306-319.
Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems, Springer Ser. Comput. Math. 25, Springer, 1997.
Trotter, H.F.: 'Approximation of semi-groups of operators',Pacific J. Math. 8 (1958), 887-919.
Trotter, H.F.: 'On the product of semi-groups of operators',Proc. Amer. Math. Soc. 10 (1959),545-551.
Wojtaszczyk, P.:A Mathematical Introduction to Wavelets, London Math. Soc. Stud. Texts 37, Cambridge University Press, 1997.
Yagi, A.: 'Coincidence entre des espaces d'interpolation et des domaines de puissances fractionnaires d'opérateurs', C.R. Acad. Sci. Paris, Ser I 299 (1984), 173-176.
Yoo, H.: 'Semi-discretization of stochastic partial differential equations on r1 by a finite difference method',Math. Comput. 69 (2000), 653-666.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hausenblas, E. Approximation for Semilinear Stochastic Evolution Equations. Potential Analysis 18, 141–186 (2003). https://doi.org/10.1023/A:1020552804087
Issue Date:
DOI: https://doi.org/10.1023/A:1020552804087