FEniCS framework in geoscientific applications

Abstract

The FEniCS Project is an open source finite element framework for solving PDEs. The advantage of this framework is a tight connection between the mathematical model, the associated variational problem and the numerical algorithm. This connection is achieved by the form language UFL and the user interface DOLFIN. Using the Python version of the DOLFIN interface, we implement numerical schemes for solving quasistatic Stokes equations with Newtonian and non-Newtonian viscosity accompanied by the heat and the compositional field transer equations in two dimensions. We use a mixed finite element formulation for the incompressible Stokes equations with an independent approximation of velocity and pressure. The advection-diffusion equations for the heat and the composition transfer, with similar mathematical properties, are discretized using an upwinded discontinuous Galerkin formulation, because of the discontinuities and sharp gradients found in these equations. To minimize numerical dispersion error for the compositional field, the discontinuous method is reinforced by a filtering algorithm. We consider two numerical examples: firstly an isothermal and isoviscous gravitational instability; and secondly thermal convection with a non-Newtonian viscosity. Gravitational instability problem has applications in the modelling of the sinking of solidified magma leading to the formation of basalt layers, while a non-Newtonian power law viscosity is used in studies of mantle convection. Results presented demonstrate functionality of the FEniCS project and its most impressive feature: comparatively easy way of experimenting with mathematical description of natural processes.