Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

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}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).