Hybrid Nodal Integral/Finite Element Method (NI-FEM) for Time-Dependent Convection Diffusion Equation

Nodal integral methods (NIM) are a class of efficient coarse mesh method that use transverse averaging to reduce the governing partial differential equation(s) (PDE) into a set of ordinary differential equations (ODE), and these ODEs or their approximations are analytically solved. Since this method depends on transverse averaging, the standard application of this approach gets restricted to domains that have boundaries that are parallel to one of the coordinate axes (2D) or coordinate planes (3D). The hybrid nodal-integral/finite-element method (NI-FEM) has been developed to extend the application of NIM to arbitrary domains. NI-FEM is based on the idea that the interior region and the regions with boundaries parallel to the coordinate axes (2D) or coordinate planes (3D) can be solved using NIM and the rest of the domain can be solved using FEM. The crux of the hybrid NI-FEM is in developing interfacial conditions at the common interfaces between the regions solved by the NIM and the FEM. Since the discrete variables in the two numerical approaches are different, this requires special treatment of the discrete quantities on the interface between the two different types of discretized elements. We here report the development of hybrid NI-FEM in a parallel framework in Fortran using PETSc for the time-dependent convection-diffusion equation (CDE) in arbitrary domains. Numerical solutions are compared with exact solutions, and the scheme is shown to be second order accurate in both space and time. The order of approximations used for the development of the scheme are also shown to be second order. The hybrid method is efficient compared to standalone conventional numerical schemes like FEM.