Mantle Convection Simulations using the Hybridized Discontinuous Galerkin Method

Abstract:

The equations that describe mantle convection, under the assumption of incompressibility and the Boussinesq approximation, are the Stokes equations for the velocity and pressure fields, and a nonlinear advection-diffusion equation for the temperature field. We discuss the simulation of the mantle convection equations using the Hybridized Discontinuous Galerkin (HDG) method.

The discontinuous Galerkin method (DG) is a high-order method that has been used extensively in the last few years. DG methods combine the advantages of continuous Galerkin finite element methods (CG) and finite volume methods (FVM), being able to handle complex geometries like CG and maintaining local conservative properties like FVM. However, the DG method suffers from the problem of large memory requirements. This is due to the number of unknowns in the system increasing as O((p+1)^d N) where p is the order of the local polynomial, d is the spatial dimension, and N is the number of elements in the triangulation. For d>1, high-order polynomials will lead to a rapid increase in memory requirements. In HDG, the globally coupled unknowns are the degrees of freedom belonging to the unknown functions on the edges (or faces) of the elements instead of the interior of the elements. The number of unknowns now increases as O((p+1)^(d-1)E), where E is the number of edges in the triangulation.

In HDG the approximate variables and the corresponding approximate fluxes are expressed in terms of the approximate traces of the variables on the element faces. Unique values of these traces are defined by enforcing the continuity of the normal component of the fluxes across element faces. We present implementations of the HDG method for the Stokes system and the unsteady nonlinear advection diffusion equation. We discuss the convergence properties of the various HDG implementations presented and technqiues that have been developed for improving these convergence properties. We also present results from running the code against various benchmarks, and discuss the performance of the HDG method in addressing the overshoot phenomenon observed in advection dominated problems with steep gradients. Finally we present scaling results for our implementation of the HDG method.