 
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 BognerFoxSchmit rectangles and an integration rule based on the
twopoint Gaussian quadrature are used to formulate the discrete problem. An H2norm 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
