A Systematic Study on Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

A Systematic Study on Weak Galerkin Finite Element Methods for Second Order Elliptic Problems This article provides a systematic study for the weak Galerkin (WG) finite element method for second order elliptic problems by exploring polynomial approximations with various degrees for each local element. A typical local WG element is of the form $$P_k(T)\times P_j(\partial T)\Vert P_\ell (T)^2$$ P k ( T ) × P j ( ∂ T ) ‖ P ℓ ( T ) 2 , where $$k\ge 1$$ k ≥ 1 is the degree of polynomials in the interior of the element T, $$j\ge 0$$ j ≥ 0 is the degree of polynomials on the boundary of T, and $$\ell \ge 0$$ ℓ ≥ 0 is the degree of polynomials employed in the computation of weak gradients or weak first order partial derivatives. A general framework of stability and error estimate is developed for the corresponding numerical solutions. Numerical results are presented to confirm the theoretical results. The work reveals some previously undiscovered strengths of the WG method for second order elliptic problems, and the results are expected to be generalizable to other type of partial differential equations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Scientific Computing Springer Journals

A Systematic Study on Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer Science+Business Media, LLC
Subject
Mathematics; Algorithms; Computational Mathematics and Numerical Analysis; Mathematical and Computational Engineering; Theoretical, Mathematical and Computational Physics
ISSN
0885-7474
eISSN
1573-7691
D.O.I.
10.1007/s10915-017-0496-6
Publisher site
See Article on Publisher Site

Abstract

This article provides a systematic study for the weak Galerkin (WG) finite element method for second order elliptic problems by exploring polynomial approximations with various degrees for each local element. A typical local WG element is of the form $$P_k(T)\times P_j(\partial T)\Vert P_\ell (T)^2$$ P k ( T ) × P j ( ∂ T ) ‖ P ℓ ( T ) 2 , where $$k\ge 1$$ k ≥ 1 is the degree of polynomials in the interior of the element T, $$j\ge 0$$ j ≥ 0 is the degree of polynomials on the boundary of T, and $$\ell \ge 0$$ ℓ ≥ 0 is the degree of polynomials employed in the computation of weak gradients or weak first order partial derivatives. A general framework of stability and error estimate is developed for the corresponding numerical solutions. Numerical results are presented to confirm the theoretical results. The work reveals some previously undiscovered strengths of the WG method for second order elliptic problems, and the results are expected to be generalizable to other type of partial differential equations.

Journal

Journal of Scientific ComputingSpringer Journals

Published: Jul 26, 2017

References

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