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CONVERGENCE ANALYSIS OF A QUADRATURE FINITE ELEMENT GALERKIN SCHEME FOR A BIHARMONIC PROBLEM
 

Summary: CONVERGENCE ANALYSIS OF A QUADRATURE FINITE
ELEMENT GALERKIN SCHEME FOR A BIHARMONIC PROBLEM
RAKHIM AITBAYEV
Abstract. A quadrature finite element Galerkin scheme for a Dirichlet boundary value problem
for the biharmonic equation is analyzed for a solution existence, uniqueness, and convergence. Con-
forming finite element space of Bogner-Fox-Schmit rectangles and an integration rule based on the
two-point Gaussian quadrature are used to formulate the discrete problem. An H2-norm error esti-
mate is obtained for the solution of the original finite element problem consistent with the solution
regularity. A standard quadrature error analysis gives a suboptimal order error estimate. Optimal
order error estimates under sufficient regularity assumptions are obtained using an alternative ap-
proach based on the equivalence of the quadrature problem with an orthogonal spline collocation
problem.
Key words: biharmonic problem, finite elements, Galerkin method, Gaussian
quadrature, orthogonal spline collocation
AMS subject classification. 65N12, 65N15, 65N30, 65N35
1. Introduction. In this article, we analyze existence, uniqueness, and conver-
gence of a quadrature finite element Galerkin approximation of a Dirichlet boundary
value problem (BVP) with the biharmonic equation on a rectangular polygonal region.
Problems with the biharmonic equation arise in many areas of applied mathematics,
for example, plate problems in plane elasticity and problems for the stream function

  

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

 

Collections: Mathematics