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Abstract. Many geodynamical models are formulated in terms of the Stokes equations that are then coupled to other equations. For the numerical solution of the Stokes equations, geodynamics codes over the past decades have used essentially every finite element that has ever been proposed for the solution of this equation, on both triangular/tetrahedral (“simplex”) and quadrilaterals/hexahedral (“hypercube”) meshes. However, in many and perhaps most cases, the specific choice of element does not seem to have been the result of careful benchmarking efforts but based on implementation efficiency or the implementers' background. In a first part of this paper (Thieulot and Bangerth, 2022), we have provided a comprehensive comparison of the accuracy and efficiency of the most widely used hypercube elements for the Stokes equations. We have done so using a number of benchmarks that illustrate “typical” geodynamic situations, specifically taking into account spatially variable viscosities. Our findings there showed that only Taylor–Hood-type elements with either continuous (Q2×Q1) or discontinuous (Q2×P-1) pressure are able to adequately and efficiently approximate the solution of the Stokes equations. In this, the second part of this work, we extend the comparison to simplex meshes. In particular, we compare triangular Taylor–Hood elements against the MINI element and one often referred to as the “Crouzeix–Raviart” element. We compare these choices against the accuracy obtained on hypercube Taylor–Hood elements with approximately the same computational cost. Our results show that, like on hypercubes, the Taylor–Hood element is substantially more accurate and efficient than the other choices. Our results also indicate that hypercube meshes yield slightly more accurate results than simplex meshes but that the difference is relatively small and likely unimportant given that hypercube meshes often lead to slightly denser (and consequently more expensive) matrices.
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On the choice of finite element for applications in geodynamics
Abstract. Geodynamical simulations over the past decades have widely beenbuilt on quadrilateral and hexahedral finite elements. For thediscretization of the key Stokes equation describing slow, viscousflow, most codes use either the unstable Q1×P0 element, astabilized version of the equal-order Q1×Q1 element, ormore 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 isactually the best at accurately simulating “typical” geodynamicsituations. Herein, we provide a systematic comparison of all of theseelements for the first time. We use a series of benchmarks that illuminate differentaspects of the features we consider typical of mantle convectionand geodynamical simulations. We will show in particular that the stabilizedQ1×Q1 element has great difficulty producing accuratesolutions for buoyancy-driven flows – the dominant forcing formantle convection flow – and that the Q1×P0 element istoo unstable and inaccurate in practice. As a consequence, webelieve that the Q2×Q1 and Q2×P-1 elementsprovide the most robust and reliable choice for geodynamical simulations,despite the greater complexity in their implementation and thesubstantially higher computational cost when solving linearsystems.
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- PAR ID:
- 10359238
- Date Published:
- Journal Name:
- Solid Earth
- Volume:
- 13
- Issue:
- 1
- ISSN:
- 1869-9529
- Page Range / eLocation ID:
- 229 to 249
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.5194/se-13-229-2022
