Comparison of the 3D nodal integral method for arbitrary hexahedral elements and spectral element method

The main advantage of using Nodal integral methods (NIM) in solving partial differential equations (PDEs) is that they provide an accurate solution for relatively coarse meshes. Due to the use of the transverse integration step in the derivation, most of the NIMs developed for different PDEs were restricted to simple cuboid elements. However, in most practical applications, it is difficult to capture the desired geometries using such elements. More complicated shapes, such as tetrahedron and hexahedron shapes, are often needed to represent complex geometries. Recently, NIM was developed to solve the 3D steady-state convection-diffusion equation (CDE) for arbitrary hexagonal elements [1]. In this paper, the derivation is extended to the time-dependent CDE. The scheme developed here is applied to find the temperature distribution inside a hollow cylinder geometry. The results of the newly derived method and those obtained using the open source code (NEK5000) that uses the spectral element method are compared with the exact solution. Since both methods use hexahedral meshes and provide accurate solution for coarse meshes, the same meshes are used in both methods. The results show that the accuracy of NIM developed for arbitrary hexahedral elements is maintained even for coarse mesh domains and higher Peclet numbers. The order of the accuracy is between first and second order.