A two-dimensional finite element recursion relation for the transport equation using nine-diagonal solvers

Abstract

A Galerkin-based finite element recursion relation is used to solve the heat transport equation in two-dimensions. The finite element method (FEM) is a powerful technique that is commonly used for solving complex engineering problems. However, the implementation of the FEM in multi-dimensional problems can be computationally expensive. A finite element recursion algorithm based on bilinear triangular, bilinear quadrilateral and quadratic Lagrangian approximations are employed to discretize the 2-D advection-diffusion equation. This algorithm is an extension of the 1-D Chapeau (linear element) technique, which employed a tridiagonal recursion expression common to the classical central finite-difference approach. The global matrix is nine-diagonal (for 2-D) and is solved using a modified strongly implicit procedure and a left-to-right sweep method.