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Consistency and Stability of a Milstein-Galerkin Finite Element Scheme for Semilinear SPDE

by R. Kruse

(Report number 2013-20)

Abstract
We present an abstract concept for the error analysis of numerical schemes for semilinear stochastic partial differential equations (SPDEs) and demonstrate its usefulness by proving the strong convergence of a Milstein-Galerkin finite element scheme. By a suitable generalization of the notion of bistability from Beyn & Kruse (DCDS B, 2010) to the semigroup framework in Hilbert spaces, our main result includes a two-sided error estimate of the spatio-temporal discretization. In an additional section we derive an analogous result for a Milstein-Galerkin finite element scheme with truncated noise.

Keywords: semilinear SPDE, Milstein, Galerkin finite element methods, strong convergence, spatio-temporal discretization, Spijker norm, bistability, consistency, two-sided error estimate

@Techreport{K13_517,
  author = {R. Kruse},
  title = {Consistency and Stability of a Milstein-Galerkin Finite Element Scheme for Semilinear SPDE},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-20},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-20.pdf },
  year = {2013}
}

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