A Systematic Study on Weak Galerkin Finite Element Method for Second Order Parabolic Problems

Deka, Bhupen; Kumar, Naresh

doi:10.48550/arxiv.2103.13669

Cited by 1 publication

(1 citation statement)

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“…Various PDEs arising from the mathematical modeling of practical problems in science and engineering are solved numerically via WG-FEMs. There exists vast literature on such PDEs; for example, elliptic equation [16][17][18][19], parabolic equation [20][21][22], hyperbolic equation [23,24], and so forth.…”

Section: Introductionmentioning

confidence: 99%

A stabilizer free weak Galerkin finite element method for second‐order Sobolev equation

Кумар

Deka

2022

Numerical Methods Partial

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In this study, we consider a stabilizer free weak Galerkin (SFWG) finite element method to solve a second‐order Sobolev equation. The SFWG method has various assets, including the support for higher order of accuracy and fewer coefficients. We have analyzed error estimates for both semidiscrete and fully discrete schemes using a non‐standard projection operator. The convergence rate of numerical error estimates of 0.3emO0.3em()hk+2+τ2$$ \kern0.3em O\kern0.3em \left({h}^{k+2}+{\tau}^2\right) $$ in L∞()H1$$ {L}^{\infty}\left({H}^1\right) $$ norm and 0.3emO0.3em()hk+3+τ2$$ \kern0.3em O\kern0.3em \left({h}^{k+3}+{\tau}^2\right) $$ in L∞()L2$$ {L}^{\infty}\left({L}^2\right) $$ norm are obtained, which are two order higher than the optimal order associated with WG finite element space ()ℙk(K),ℙk+1(∂K),[]ℙk+1(K)2.$$ \left({\mathrm{\mathbb{P}}}_k(K),{\mathrm{\mathbb{P}}}_{k+1}\left(\partial K\right),{\left[{\mathrm{\mathbb{P}}}_{k+1}(K)\right]}^2\right). $$ At last, several numerical examples are reported to validate the supercloseness property and efficiency of the proposed SFWG method. These experiments confirm the accuracy of the theoretical findings.

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