☆ 4.5 Article

Strong and weak convergence order of finite element methods for stochastic PDEs with spatial white noise

NUMERISCHE MATHEMATIK (2016)

Journal

NUMERISCHE MATHEMATIK

Volume 134, Issue 1, Pages 61-89

Publisher

SPRINGER HEIDELBERG

DOI: 10.1007/s00211-015-0768-8

Keywords

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Categories

Mathematics, Applied

Funding

OSD/AFOSR MURI [FA9550-09-1-0613]
NSF/DMS [DMS-1216437, DMS-1148284]
WPI
ARO [W911NF-13-1-0012]

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Abstract

We investigate the strong and weak convergence order of piecewise linear finite element methods for a class of semilinear elliptic equations with additive spatial white noise using a spectral truncation of white noise. Taking the number of modes for spectral approximation of the white noise at the order of , we show that the strong convergence order of the finite element approximation is , where h is the element size and is the dimension. Moreover, we show that the weak convergence order is . We then consider a fourth-order equation and an advection-reaction equation and show that a spectral approximation of the white noise can lead to higher convergence order, in both strong and weak sense, when the solutions are smooth. Numerical results confirm our prediction for one- and two-dimensional elliptic problems.

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