A Review of Collocation Approximations to Solutions of Differential Equations

This review considers piecewise polynomial functions, that have long been known to be a useful and versatile tool in numerical analysis, for solving problems which have solutions with irregular features, such as steep gradients and oscillatory behaviour. Examples of piecewise polynomial functions used include splines, in particular B-splines, and Hermite functions. Spline functions are useful for obtaining global approximations whilst Hermite functions are useful for approximation over finite elements. Our aim in this review is to study quintic Hermite functions and develop a numerical collocation scheme for solving ODEs and PDEs. This choice of basis functions is further motivated by the fact that we are interested in solving problems having solutions with steep gradients and oscillatory properties, for which this approximation basis seems to be a suitable choice. We derive the quintic Hermite basis and use it to formulate the orthogonal collocation on finite element (OCFE) method. We present the error analysis for third order ODEs and derive both global and nodal error bounds to illustrate the super-convergence property at the nodes. Numerical simulations using the Julia programming language are performed for both ODEs and PDEs and enhance the theoretical results.