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Journal of Scientific Computing

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01-08-2015

A Weak Galerkin Finite Element Scheme for the Biharmonic Equations by Using Polynomials of Reduced Order

Authors: Ran Zhang, Qilong Zhai

Published in: Journal of Scientific Computing | Issue 2/2015

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Abstract

A new weak Galerkin (WG) finite element method for solving the biharmonic equation in two or three dimensional spaces by using polynomials of reduced order is introduced and analyzed. The WG method is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the numerical scheme, yet without compromising the accuracy of the numerical approximation. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete \(H^2\) norm and the standard \(L^2\) norm. In addition, the paper also presents some numerical experiments to demonstrate the power of the WG method. The numerical results show a great promise of the robustness, reliability, flexibility and accuracy of the WG method.

previous article Guaranteed Energy Error Estimators for a Modified Robust Crouzeix–Raviart Stokes Element

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Title: A Weak Galerkin Finite Element Scheme for the Biharmonic Equations by Using Polynomials of Reduced Order
Authors: Ran Zhang
Qilong Zhai
Publication date: 01-08-2015
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 2/2015
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-014-9945-7

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