TY - JOUR

T1 - An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes

AU - Wang, Chunmei

AU - Wang, Junping

N1 - Funding Information:
The research of Chunmei Wang was partially supported by The Project of Graduate Education Innovation of Jiangsu Province (CXZZ13_0387). The research of Junping Wang was supported by the NSF IR/D program, while working at National Science Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

PY - 2014/12/1

Y1 - 2014/12/1

N2 - This paper presents a new and efficient numerical algorithm for the biharmonic equation by using weak Galerkin (WG) finite element methods. The WG finite element scheme is based on a variational form of the biharmonic equation that is equivalent to the usual H2-semi norm. Weak partial derivatives and their approximations, called discrete weak partial derivatives, are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. The discrete weak partial derivatives serve as building blocks for the WG finite element method. The resulting matrix from the WG method is symmetric, positive definite, and parameter free. An error estimate of optimal order is derived in an H2-equivalent norm for the WG finite element solutions. Error estimates in the usual L2 norm are established, yielding optimal order of convergence for all the WG finite element algorithms except the one corresponding to the lowest order (i.e., piecewise quadratic elements). Some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme.

AB - This paper presents a new and efficient numerical algorithm for the biharmonic equation by using weak Galerkin (WG) finite element methods. The WG finite element scheme is based on a variational form of the biharmonic equation that is equivalent to the usual H2-semi norm. Weak partial derivatives and their approximations, called discrete weak partial derivatives, are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. The discrete weak partial derivatives serve as building blocks for the WG finite element method. The resulting matrix from the WG method is symmetric, positive definite, and parameter free. An error estimate of optimal order is derived in an H2-equivalent norm for the WG finite element solutions. Error estimates in the usual L2 norm are established, yielding optimal order of convergence for all the WG finite element algorithms except the one corresponding to the lowest order (i.e., piecewise quadratic elements). Some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme.

KW - Biharmonic equation

KW - Finite element methods

KW - Polyhedral meshes

KW - Weak Galerkin

KW - Weak partial derivatives

UR - http://www.scopus.com/inward/record.url?scp=84919427269&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2014.03.021

DO - 10.1016/j.camwa.2014.03.021

M3 - Article

AN - SCOPUS:84919427269

VL - 68

SP - 2314

EP - 2330

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 12

ER -