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

Abstract

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 𝒫kfalse(Kfalse),𝒫jfalse(∂Kfalse),[]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.

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

“…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 [16–19], parabolic equation [20–22], hyperbolic equation [23, 24], and so forth.…”