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BIT Numerical Mathematics

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22-03-2019

Numerically computable a posteriori-bounds for the stochastic Allen–Cahn equation

Authors: Dirk Blömker, Minoo Kamrani

Published in: BIT Numerical Mathematics | Issue 3/2019

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Abstract

The aim of this paper is the derivation of an a-posteriori error estimate for the numerical method based on an exponential scheme in time and spectral Galerkin methods in space. We obtain analytically a rigorous bound on the conditional mean square error, which is conditioned to the given realization of the data calculated by a numerical method. This bound is explicitly computable and uses only the computed numerical approximation. Thus one can check a-posteriori the error for a given numerical computation for a fixed discretization without relying on an asymptotic result. All estimates are only based on the numerical data and the structure of the equation, but they do not use any a-priori information of the solution, which makes the approach applicable to equations where global existence and uniqueness of solutions is not known. For simplicity of presentation, we develop the method here in a relatively simple situation of a stable one-dimensional Allen-Cahn equation with additive forcing.

previous article Weak backward error analysis for stochastic Hamiltonian Systems

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Title: Numerically computable a posteriori-bounds for the stochastic Allen–Cahn equation
Authors: Dirk Blömker
Minoo Kamrani
Publication date: 22-03-2019
Publisher: Springer Netherlands
Published in: BIT Numerical Mathematics / Issue 3/2019
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-019-00745-8

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