15 Jun 2021

15 Jun 2021

Review status: a revised version of this preprint was accepted for the journal SE.

On the choice of finite element for applications in geodynamics

Cedric Thieulot1 and Wolfgang Bangerth2 Cedric Thieulot and Wolfgang Bangerth
  • 1Department of Earth Sciences, Utrecht University, Utrecht, The Netherlands
  • 2Department of Mathematics, Department of Geosciences, Colorado State University, Fort Collins, CO, USA

Abstract. 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.

Cedric Thieulot and Wolfgang Bangerth

Status: final response (author comments only)

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • RC1: 'Comment on se-2021-78', Anonymous Referee #1, 30 Jul 2021
    • AC1: 'Reply on RC1', Cedric THIEULOT, 05 Oct 2021
  • RC2: 'Comment on se-2021-78', Dave May, 30 Aug 2021
    • AC2: 'Reply on RC2', Cedric THIEULOT, 05 Oct 2021

Cedric Thieulot and Wolfgang Bangerth

Cedric Thieulot and Wolfgang Bangerth


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Short summary
One of the main numerical methods to solve the mass, momentum and energy conservation equations in geodynamics is the Finite Element Method. Four main types of elements have been used in the past decades in hundreds of publications. We here for the first time compare results obtained with these four elements on a series of geodynamical benchmarks and applications and draw conclusions as to which are the best ones, and which are to be preferably avoided.