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.

中文翻译：

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有限元在地球动力学应用中的选择

摘要。过去几十年的地球动力学模拟广泛建立在四边形和六面体有限元上。对于描述缓慢粘性流的关键斯托克斯方程的离散化，大多数代码使用不稳定的Q ₁  ×  P ₀元素、等阶Q ₁  ×  Q ₁元素的稳定版本，或者最近使用稳定的 Taylor-Hood具有连续 ( Q ₂  ×  Q ₁ ) 或不连续 ( Q ₂  ×  P _-1 ) 的元素） 压力。然而，尚不清楚这些选择中的哪一个实际上最适合准确模拟典型的地球动力学情况。在此，我们首次对所有这些要素进行系统比较。我们使用一系列基准来阐明我们认为典型的地幔对流和地球动力学模拟特征的不同方面。我们将特别展示稳定的Q ₁  ×  Q ₁元素很难为浮力驱动的流动（地幔对流的主要强迫）产生准确的解，并且Q ₁  ×  P ₀元素在实践中太不稳定和不准确。因此，我们认为Q ₂  ×  Q ₁和Q ₂  ×  P _-1元素为地球动力学模拟提供了最稳健和可靠的选择，尽管它们的实现更加复杂，并且在求解线性系统时计算成本显着提高。