Convergence Analysis of a Quadrature Finite Element Galerkin Scheme for a Biharmonic Problem

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. Conforming 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 estimate 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 approach based on the equivalence of the quadrature problem with an orthogonal spline collocation problem.