On the choice of finite element for applications in geodynamics

Geodynamical simulations over the past decades have widely been built on quadrilateral and hexahedral finite elements. For the discretisation of the key Stokes equation describing slow, viscous flow, most codes use either the unstable Q1 × P0 element, a stabilised version of the equal-order Q1 × Q1 element, or more recently the stable Taylor-Hood element with continuous (Q2 × Q1) or discontinuous (Q2 × P−1) pressure. However, it is not clear which of these choices is actually the best at accurately simulating typical geodynamic situations. Herein, we are providing for the first time a systematic comparison of all of these elements. We use a series of benchmarks that illuminate different aspects of the features we consider typical of mantle convection and geodynamical simulations. We will show in particular that the stabilised Q1 × Q1 element has great difficulty producing accurate solutions for buoyancy-driven flows – the dominant forcing for mantle convection flow – and that the Q1 × P0 element is too unstable and inaccurate in practice. As a consequence, we believe that the Q2 × Q1 and Q2 × P−1 elements provide the most robust and reliable choice for geodynamical simulations, despite the greater complexity in their implementation and the substantially higher computational cost when solving linear systems.

Slow viscous flow of two porous spherical particles translating along the axis of a cylinder

We describe the motion of two freely moving porous spherical particles located along the axis of a cylindrical tube with background Poiseuille flow at low Reynolds number. The stream function and a framework based on cylindrical harmonics are adopted to solve the flow field around the particles and the flow within the tube, respectively. The two solutions are employed in an iterated framework using the method of reflections. We first consider the case of two identical particles, followed by two particles with different dimensions. In both cases, the drag force coefficients of the particles are solved as functions of the separation distance between the particles and the permeability of the particles. The detailed flow field in the vicinity of the two particles is investigated by plotting the streamlines and velocity contours. We find that the particle–particle interaction is dependent on the separation distance, particle sizes and permeability of the particles. Our analysis reveals that when the permeability of the particles is large, the streamlines are more parallel and the particle–particle interaction has less effect on the particle motion. We further show that a smaller permeability and bigger particle size generally tend to squeeze the streamlines and velocity contour towards the wall.

Computational Software: Simple FMM Libraries for Electrostatics, Slow Viscous Flow, and Frequency-Domain Wave Propagation

We have developed easy to use fast multipole method (FMM) libraries for the Laplace, low-frequency Helmholtz, and Stokes equations in two and three dimensions. The codes are based on a new method for applying translation operators and provide reasonable performance on either single core processors, or small multi-core systems using OpenMP.