Talk by Debendra Kumar Swain
Posted on: 28 Jan 2026
Title: Finite Element Galerkin Approximations to the Boussinesq System of Equations
Speaker: Debendra Kumar Swain, IIT Goa
Date: 5 February 2026
Time: 4 PM IST
Venue: Zoom
Abstract
This work examines numerical methods applied to the Boussinesq system of equations, which is widely used in real–world applications to analyze fluid flow and heat transfer phenomena. The system consists of the Navier–Stokes system of equations coupled with the convection–diffusion equation for temperature. To begin with, a three–step two–grid continuous Galerkin (CG) finite element method (FEM) is proposed, combining Newton’s iteration for spatial discretization with the backward Euler (BE) method for temporal discretization.
The two–grid technique is an efficient approach for solving nonlinear problems, where the solution is first computed on a coarse mesh and then used to linearize the problem on a finer mesh. This method helps achieve accurate approximations while mitigating the high computational cost associated with the nonlinear nature of the model. Subsequently, a discontinuous Galerkin (DG) method is investigated, employing DG method for spatial discretization and BE method for time discretization. This is followed by a two–grid DG FEM approach with BE method for time discretization. The DG method is a powerful technique for solving partial differential equations, offering high accuracy, local conservation, and stability even on irregular meshes. It excels at capturing complex phenomena and supports varying polynomial degrees across elements.
Additionally, a local discontinuous Galerkin (LDG) method for spatial discretization combined with BE method for time discretization is explored. As a variant of the DG method, the LDG method improves stability and conservation by applying local penalty terms at element boundaries, ensuring continuity and local conservation. For all proposed numerical schemes, stability and optimal error estimates are derived for velocity, pressure, and temperature. Finally, numerical experiments are conducted to evaluate the performance of the proposed methods.