A study of the finite element method employing three-node elements in a one-dimensional domain

Three-node isoparametric elements are widely used for solving various one-dimensional problems with finite element discretization. However, it is not well known that the value of the Jacobian at either end of an element can be zero or negative when the interior node is placed at particular positions inside the element. In this study, we propose a simple remedy to fix this problem, and the proposed method is applied to the discretization of the one-dimensional advection-diffusion equation. With the use of this proposed scheme, accurate numerical results are obtained on two types of unequally spaced meshes, whereas the numerical solution obtained using isoparametric elements diverges from the analytical solution on one of the meshes, whose nodes are more concentrated in the vicinity of the edges of the domain. This proves the robustness of the proposed method on any unequally divided mesh.