Basic Concept and Application of Differential Operator in Time and Space by Orthogonal Collocation Finite Elements Method

Takaki Ohkubo

Professor emeritus, Department of civil and environmental engineering, Hakodate national college of technology（〒042-8501,14-1 Tokura Hakodate Hokkaido Japan）e-mail:ohkubo@hakodate-ct.ac.jp

Abstract: The aim of this paper is to spread the basic concept and the application of orthogonal collocation finite elements method (OCFEM) to the field of engineering and science. OCFEM is based on Weierstraß’s polynomial approximation theorem. In the formulation of PDE as strong form by OCFEM, the differential operator represented by matrix in time and space is used for the translation of PDE to algebraic equation. This paper explains the concept of matrix representation method of the differential operator in time and space, the coordinate transformation matrix based on the partial derivative of composite function and integration of element (for example, Consumption rate) or element boundary (for example, Flux) by using Gauss Legendre integration. On the other hand, this paper presents the formulation of advection-diffusion-reaction equation (PDE) as initial boundary value problem, which is Biofilm model, by differential operator of OCFEM.

Keywords: Orthogonal Collocation Finite Elements Method, Differential Operator, Time and Space, Coordinate Transformation Matrix, Biofilm Model, Advective Diffusive Equation, PDE, Initial Boundary Value Problem.

Pages: 37 – 69 | Full PDF Paper