TY - JOUR

T1 - Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions

AU - Li, Dan

AU - Nie, Yufeng

AU - Wang, Chunmei

N1 - Funding Information:
The research of Dan Li was supported in part by National Natural Science Foundation of China grant number 11471262.The research of Yufeng Nie was supported in part by National Natural Science Foundation of China grants 11471262.The research of Chunmei Wang was partially supported by National Science Foundation Awards DMS-1849483 and DMS-1905195. The authors would like to gratefully acknowledge Dr. Junping Wang in NSF for his invaluable discussion and suggestion for this paper.
Publisher Copyright:
© 2019 Elsevier Ltd

PY - 2019/8/1

Y1 - 2019/8/1

N2 - A superconvergence error estimate for the gradient approximation of the second order elliptic problem in three dimensions is analyzed by using weak Galerkin finite element scheme on the uniform and non-uniform cubic partitions. Due to the loss of the symmetric property from two dimensions to three dimensions, this superconvergence result in three dimensions is not a trivial extension of the recent superconvergence result in two dimensions Li et al. (0000) from rectangular partitions to cubic partitions. The error estimate for the numerical gradient in the L2-norm arrives at a superconvergence order of O(hr)(1.5≤r≤2) when the lowest order weak Galerkin finite elements consisting of piecewise linear polynomials in the interior of the elements and piecewise constants on the faces of the elements are employed. A series of numerical experiments are illustrated to confirm the established superconvergence theory in three dimensions.

AB - A superconvergence error estimate for the gradient approximation of the second order elliptic problem in three dimensions is analyzed by using weak Galerkin finite element scheme on the uniform and non-uniform cubic partitions. Due to the loss of the symmetric property from two dimensions to three dimensions, this superconvergence result in three dimensions is not a trivial extension of the recent superconvergence result in two dimensions Li et al. (0000) from rectangular partitions to cubic partitions. The error estimate for the numerical gradient in the L2-norm arrives at a superconvergence order of O(hr)(1.5≤r≤2) when the lowest order weak Galerkin finite elements consisting of piecewise linear polynomials in the interior of the elements and piecewise constants on the faces of the elements are employed. A series of numerical experiments are illustrated to confirm the established superconvergence theory in three dimensions.

KW - Non-uniform cubic partitions

KW - Second order elliptic problem

KW - Superconvergence

KW - Three dimensions

KW - Weak Galerkin finite element method

UR - http://www.scopus.com/inward/record.url?scp=85063548373&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2019.03.010

DO - 10.1016/j.camwa.2019.03.010

M3 - Article

AN - SCOPUS:85063548373

VL - 78

SP - 905

EP - 928

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 3

ER -