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Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes
Author(s) -
Mu Lin,
Wang Junping,
Ye Xiu
Publication year - 2014
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21855
Subject(s) - biharmonic equation , mathematics , finite element method , discontinuous galerkin method , piecewise , norm (philosophy) , polygon mesh , mixed finite element method , polyhedron , galerkin method , convergence (economics) , partial differential equation , mathematical analysis , extended finite element method , geometry , boundary value problem , physics , political science , law , economics , thermodynamics , economic growth
A new weak Galerkin (WG) finite element method is introduced and analyzed in this article for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter‐free. Optimal order error estimates in a discrete H 2 norm is established for the corresponding WG finite element solutions. Error estimates in the usual L 2 norm are also derived, yielding a suboptimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1003–1029, 2014