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A Quadrature Finite Element Galerkin Scheme for a Biharmonic Problem on a Rectangular Polygon
 

Summary: A Quadrature Finite Element Galerkin Scheme for
a Biharmonic Problem on a Rectangular Polygon
Rakhim Aitbayev
Department of Mathematics, New Mexico Institute of Mining and Technology,
Socorro, New Mexico
Received 7 November 2006; accepted 18 April 2007
Published online in Wiley InterScience (www.interscience.wiley.com).
DOI 10.1002/num.20278
A quadrature Galerkin scheme with the Bogner­Fox­Schmit element for a biharmonic problem on a rectan-
gular polygon is analyzed for existence, uniqueness, and convergence of the discrete solution. It is known
that a product Gaussian quadrature with at least three-points is required to guarantee optimal order conver-
gence in Sobolev norms. In this article, optimal order error estimates are proved for a scheme based on the
product two-point Gaussian quadrature by establishing a relation with an underdetermined orthogonal spline
collocation scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 00: 000­000, 2007
Keywords: biharmonic problem; finite element method; Gaussian quadrature; orthogonal spline colloca-
tion; rectangular element
I. INTRODUCTION
Let R2
be an open rectangular polygon with the boundary aligned with the coordinate
axes. In this article, we propose a two-point Gaussian quadrature finite-element Galerkin scheme

  

Source: Aitbayev, Rakhim - Department of Mathematics, New Mexico Institute of Mining and Technology

 

Collections: Mathematics