## Preprint 2013-07

## Ricardo Oyarzúa, Dominik Schötzau:

### An exactly divergence-free finite element method for a generalized Boussinesq problem

### Abstract:

We propose and analyze a mixed finite element method with exactly divergence-free velocities for the numerical simulation of a generalized Boussinesq problem, describing the motion of a non-isothermal incompressible fluid subject to a heat source. The method is based on using divergence-conforming elements of order k for the velocities, discontinuous elements of order k - 1 for the pressure, and standard continuous elements of order k for the discretization of the temperature. The H1-conformity of the velocities is enforced by a discontinuous Galerkin approach. The resulting numerical scheme yields exactly divergence free velocity approximations; thus, it is probably energy-stable without the need to modify the underlying differential equations. We prove the existence and stability of discrete solutions, and derive optimal error estimates in the mesh size for small and smooth solutions.

This preprint gave rise to the following definitive publication(s):

**Ricardo OYARZúA, Tong QIN, Dominik SCHöTZAU: ***An exactly divergence-free finite element method for a generalized Boussinesq problem*. IMA Journal of Numerical Analysis, vol. 34, 3, pp. 1104-1135, (2014).