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

• 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; Wiley Subscription Services, Inc
• 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
• DOI: 10.1002/num.22973

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.