Hybrid Nodal Integral/Finite Element Method for Time-Dependent Convection Diffusion Equation

Abstract

Nodal integral methods (NIM) are a class of efficient coarse mesh methods that use transverse averaging to reduce the governing partial differential equation(s) (PDE) into a set of ordinary differential equations (ODE). These ODEs or their approximations are analytically solved. These analytical solutions are used to then develop the numerical scheme. Transverse averaging is an essential step in the development of NIM, and hence the standard application of this approach gets restricted to domains that have boundaries parallel to one of the coordinate axes (in 2D) or coordinate planes (in 3D). The hybrid nodal-integral/finite element method (NI-FEM) reported here 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 discretized and solved using FEM. The crux of the hybrid NI-FEM is in developing interfacial conditions at the common interfaces between the NIM regions and FEM regions. 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 for the time-dependent convection-diffusion equation (CDE) in arbitrary domains. The resulting hybrid numerical scheme is implemented in a parallel framework in fortran and solved using portable, extensible toolkit for scientific computation (PETSc). The preliminary approach to domain decomposition is also discussed. 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 is also shown to be second order. The hybrid method is more efficient compared to standalone conventional numerical schemes like FEM.