Numerical analysis of semilinear stochastic evolution equations in Banach spaces

• Published in: Journal of computational and applied mathematics Vol. 147; no. 2; pp. 485 - 516
• Main Author: Hausenblas, Erika
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.10.2002; Elsevier
• Subjects: Difference and functional equations, recurrence relations; Exact sciences and technology; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Partial differential equations; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Stochastic analysis; Stochastic delay equation; Stochastic evolution equations; Stochastic partial differential equations; 65M15; Stochastic delay equation; Numerical approximation; 35R60; Stochastic partial differential equations; 35A40; Stochastic evolution equations; 60H15; Transcendental equation; Numerical computation; Error estimation; Recurrence relation; Evolution equation; Galerkin method; Difference equation; Banach space; Euler scheme; Discretization method; Stochastic equation
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/S0377-0427(02)00483-1

The solution of stochastic evolution equations generally relies on numerical computation. Here, usually the main idea is to discretize the SPDE spatially obtaining a system of SDEs that can be solved by e.g., the Euler scheme. In this paper, we investigate the discretization error of semilinear stochastic evolution equations in L p -spaces, resp. Banach spaces. The space discretization may be done by Galerkin approximation, for the time discretization we consider the implicit Euler, the explicit Euler scheme and the Crank–Nicholson scheme. In the last section, we give some examples, i.e., we consider an SPDEs driven by nuclear Wiener noise approximated by wavelets and delay equation approximated by finite differences.