Convergence Analysis of Orthogonal Spline Collocation for Elliptic Boundary Value Problems

Abstract

Existence, uniqueness, and optimal order H2 , H1 , and L2 error bounds are established for the orthogonal spline collocation solution of a Dirichlet boundary value problem on the unit square. The linear, elliptic, nonself-adjoint, partial differential equation is given in nondivergence form. The approximate solution, which is a tensor product of continuously differentiable piecewise polynomials of degree $r\geq 3$, is determined by satisfying the partial differential equation at the nodes of a composite Gauss quadrature.