On orthogonal collocation solutions of partial differential equations

In contrast to the h-version most frequently used, a p-version of the Orthogonal Collocation Method as applied to differential equations in two-dimensional domains is examined. For superior accuracy and convergence, the collocation points are chosen to be the zeros of a Legendre polynomial plus the two endpoints. Hence the method is called the Legendre Collocation Method. The approximate solution in an element is written as a Lagrange interpolation polynomial. This form of the approximate solution makes it possible to fully automate the method on a personal computer using conventional memory. The Legendre Collocation Method provides a formula for the derivatives in terms of the values of the function in matrix form. The governing differential equation and boundary conditions are satisfied by matrix equations at the collocation points. The resulting set of simultaneous equations is then solved for the values of the solution function using LU decomposition and back substitution. The Legendre Collocation Method is applied further to the problems containing singularities. To obtain an accurate approximation in a neighborhood of the singularity, an eigenfunction solution is specifically formulated to the given problem, and its coefficients are determined by least-squares or minimax approximation techniques utilizing the results previously obtained by the Le Legendre Collocation Method. This combined method gives accurate results for the values of the solution function and its derivatives in a neighborhood of the singularity. All results of a selected number of example problems are compared with the available solutions. Numerical experiments confirm the superior accuracy in the computed values of the solution function at the collocation points