Educated bases for the HiMod reduction of advection-diffusion-reaction problems with general boundary conditions

37/2015

Educated bases for the HiMod reduction of advection-diffusion-reaction problems with general boundary conditions

Friday 10th July 2015

Aletti, M.; Perotto, S.; Veneziani, A.

Hierarchical Model (HiMod) reduction is a method introduced in \cite{perotto:2008} to effectively solve advection-diffusion-reaction (ADR) and fluid dynamics problems in pipes. The rationale of the method is to regard the solution as a mainstream axial dynamics added by transverse components. The mainstream component is approximated by finite elements as often done in classical 1D models (like the popular Euler equations for gasdynamics). However, the HiMod formulation includes also the transverse dynamics by a spectral expansion. A few modes are expected to capture the transverse (somehow secondary) dynamics with a good level of approximation. This drastically reduces the size of the discrete problem, yet preserving accuracy. The method is ``hierarchical'' since the selection of the number of transverse modes can be hierarchically and adaptively performed~\cite{perotto:2013}. We have previously considered only Dirichlet boundary conditions for the lateral walls of the pipe and the procedure was tested only in 2D domains. With an appropriate selection of the spectral basis functions, here we extend our formulation to 3D problems with general boundary conditions, still pursuing an essential approach. The modal basis functions fulfill by construction the (homogeneous) boundary conditions associated with the solution. This is achieved by solving a Sturm-Liouville eigenpair problem. We analyze this approach and provide a convergence analysis for the numerical error in the case of a linear ADR problem in rectangles (2D) and slabs (3D). Numerical results confirm the theory.