TY - JOUR
T1 - A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity
AU - Wang , Yiying
AU - Zou , Yongkui
AU - Liu , Xuan
AU - Zhou , Chenguang
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 2
SP - 514
EP - 533
PY - 2024
DA - 2024/05
SN - 17
DO - http://doi.org/10.4208/nmtma.OA-2023-0163
UR - https://global-sci.org/intro/article_detail/nmtma/23110.html
KW - Stabilizer free weak Galerkin FEM, weak gradient, error estimate, lower regularity, second-order elliptic equation.
AB - <p style="text-align: justify;">This paper presents error analysis of a stabilizer free weak Galerkin finite
element method (SFWG-FEM) for second-order elliptic equations with low regularity solutions. The standard error analysis of SFWG-FEM requires additional regularity on solutions, such as $H^2$-regularity for the second-order convergence. However,
if the solutions are in $H^{1+s}$ with $0 &lt; s &lt; 1,$ numerical experiments show that the
SFWG-FEM is also effective and stable with the $(1+s)$-order convergence rate, so
we develop a theoretical analysis for it. We introduce a standard $H^2$ finite element
approximation for the elliptic problem, and then we apply the SFWG-FEM to approach this smooth approximating finite element solution. Finally, we establish the
error analysis for SFWG-FEM with low regularity in both discrete $H^1$-norm and standard $L^2$-norm. The $(P_k(T ), P_{k−1}(e), [P_{k+1}(T)]^d)$ elements with dimensions of space $d = 2, 3$ are employed and the numerical examples are tested to confirm the theory.</p>