Convergence Analysis of Orthogonal Spline Collocation for Elliptic Boundary Value Problems

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.