A systematic study on weak Galerkin finite element method for second‐order parabolic problems

In the present work, we have described a systematic numerical study on weak Galerkin (WG) finite element method for second‐order linear parabolic problems by allowing polynomial approximations with various degrees for each local element. Convergence of both semidiscrete and fully discrete WG solutio...

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Published in:Numerical methods for partial differential equations Vol. 39; no. 3; pp. 2444 - 2474
Main Authors: Deka, Bhupen, Kumar, Naresh
Format: Journal Article
Language:English
Published: Hoboken, USA John Wiley & Sons, Inc 01.05.2023
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Subjects:
convergence analysis
discrete weak gradient
Finite element method
Galerkin method
Norms
parabolic problems
Polynomials
Robustness (mathematics)
semidiscrete and fully discrete schemes
weak Galerkin finite element method
ISSN:0749-159X, 1098-2426
Online Access:Get full text
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Summary:In the present work, we have described a systematic numerical study on weak Galerkin (WG) finite element method for second‐order linear parabolic problems by allowing polynomial approximations with various degrees for each local element. Convergence of both semidiscrete and fully discrete WG solutions are established in L∞L2$$ {L}^{\infty}\left({L}^2\right) $$ and L∞H1$$ {L}^{\infty}\left({H}^1\right) $$ norms for a general WG element 𝒫k(K),𝒫j(∂K),𝒫l(K)2, where k≥1$$ k\ge 1 $$, j≥0$$ j\ge 0 $$ and l≥0$$ l\ge 0 $$ are arbitrary integers. The fully discrete space–time discretization is based on a first order in time Euler scheme. Numerical experiments are reported to justify the robustness, reliability and accuracy of the WG finite element method.
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ISSN:0749-159X
1098-2426
DOI:10.1002/num.22973