Volume 38, Issue 2
A Balanced Oversampling Finite Element Method for Elliptic Problems with Observational Boundary Data

J. Comp. Math., 38 (2020), pp. 355-374.

Published online: 2020-02

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• Abstract

In this paper we propose a finite element method for solving elliptic equations with observational Dirichlet boundary data which may subject to random noises. The method is based on the weak formulation of Lagrangian multiplier and requires balanced oversampling of the measurements of the boundary data to control the random noises. We show the convergence of the random finite element error in expectation and, when the noise is sub-Gaussian, in the Orlicz $\psi_2$-norm which implies the probability that the finite element error estimates are violated decays exponentially. Numerical examples are included.

• Keywords

Observational boundary data, Elliptic equation, Sub-Gaussian random variable.

65N30, 65D10, 41A15

zmchen@lsec.cc.ac.cn (Zhiming Chen)

ruituo@tamu.edu (Rui Tuo)

zhangwl@sustc.edu.cn (Wenlong Zhang)

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• TXT
@Article{JCM-38-355, author = {Chen , Zhiming and Tuo , Rui and Zhang , Wenlong }, title = {A Balanced Oversampling Finite Element Method for Elliptic Problems with Observational Boundary Data}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {38}, number = {2}, pages = {355--374}, abstract = {

In this paper we propose a finite element method for solving elliptic equations with observational Dirichlet boundary data which may subject to random noises. The method is based on the weak formulation of Lagrangian multiplier and requires balanced oversampling of the measurements of the boundary data to control the random noises. We show the convergence of the random finite element error in expectation and, when the noise is sub-Gaussian, in the Orlicz $\psi_2$-norm which implies the probability that the finite element error estimates are violated decays exponentially. Numerical examples are included.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1810-m2017-0168}, url = {http://global-sci.org/intro/article_detail/jcm/14521.html} }
TY - JOUR T1 - A Balanced Oversampling Finite Element Method for Elliptic Problems with Observational Boundary Data AU - Chen , Zhiming AU - Tuo , Rui AU - Zhang , Wenlong JO - Journal of Computational Mathematics VL - 2 SP - 355 EP - 374 PY - 2020 DA - 2020/02 SN - 38 DO - http://doi.org/10.4208/jcm.1810-m2017-0168 UR - https://global-sci.org/intro/article_detail/jcm/14521.html KW - Observational boundary data, Elliptic equation, Sub-Gaussian random variable. AB -

In this paper we propose a finite element method for solving elliptic equations with observational Dirichlet boundary data which may subject to random noises. The method is based on the weak formulation of Lagrangian multiplier and requires balanced oversampling of the measurements of the boundary data to control the random noises. We show the convergence of the random finite element error in expectation and, when the noise is sub-Gaussian, in the Orlicz $\psi_2$-norm which implies the probability that the finite element error estimates are violated decays exponentially. Numerical examples are included.

Zhiming Chen, Rui Tuo & Wenlong Zhang. (2020). A Balanced Oversampling Finite Element Method for Elliptic Problems with Observational Boundary Data. Journal of Computational Mathematics. 38 (2). 355-374. doi:10.4208/jcm.1810-m2017-0168
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