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We apply orthogonal spline collocation with splines of degree r ≥ 3 to solve, on the unit square, Poisson’s equation with Neumann boundary conditions. We show that the H¹ norm error is of order r and explain how to compute efficiently the approximate solution using a matrix decomposition algorithm involving the solution of a symmetric generalized eigenvalue problem. [ABSTRACT FROM AUTHOR]

For the heat equation on a rectangle and nonzero Dirichlet boundary conditions, we consider an ADI orthogonal spline collocation method without perturbation terms, to specify boundary values of intermediate solutions at half time levels on the vertical sides of the rectangle. We show that, at each time level, the method has optimal convergence rate in the L2 norm in space. Numerical results for splines of orders 4; 5; 6 confirm our theoretical convergence rates and demonstrate suboptimal convergence rates in the H1 norm. We also demonstrate numerically that the scheme without the perturbation terms is applicable to variable coefficient problems yielding the same convergence rates obtained for the heat equation. [ABSTRACT FROM AUTHOR]

We use extrapolated alternating direction implicit Crank–Nicolson orthogonal spline collocation with splines of order r to solve the coupled Burgers' equations in two space variables and two unknown functions. The scheme is initialized with an alternating direction implicit predictor–corrector method. We show theoretically that the H 1 norm of the error at each time level is of order r in space and of order 2 in time. Our numerical results confirm these theoretical orders. [ABSTRACT FROM AUTHOR]

We solve the Dirichlet and mixed Dirichlet-Neumann boundary value problems for the variable coefficient Cauchy-Navier equations of elasticity in a square using a Legendre spectral Galerkin method. The resulting linear system is solved by the preconditioned conjugate gradient (PCG) method with a preconditioner which is shown to be spectrally equivalent to the matrix of the resulting linear system. Numerical tests demonstrating the convergence properties of the scheme and PCG are presented. [ABSTRACT FROM AUTHOR]