Abstract In this paper, we propose a family of stabilized lower-order and equal-order finite elements(FE) schemes for the hydrostatic Stokes problems or primitive equations of the ocean. It is known that two “inf-sup” conditions appear associated to the two constraints of this problem: namely incompressibility and hydrostatic pressure. The focus of this paper is to develop the stabilized lower-order and equal-order(velocity-velocity)-pressure pairs for the hydrostatic Stokes problems. Then, the new schemes offer a number of attractive properties: avoiding extra “inf-sup” condition, achieving optimal accuracy with respect to the solution regularity and unconditional stability, implementing simply and straightforward. Finally, ample numerical experiments are presented supporting the excellent stability and accuracy of the newly proposed methods.

中文翻译：

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静压Stokes问题的稳定低阶和等阶有限元方法

摘要 在本文中，我们针对海洋的静水 Stokes 问题或原始方程提出了一系列稳定的低阶和等阶有限元 (FE) 方案。众所周知，两个“inf-sup”条件似乎与该问题的两个约束条件相关：即不可压缩性和静水压力。本文的重点是为静水 Stokes 问题开发稳定的低阶和等阶（速度-速度）-压力对。然后，新方案提供了许多有吸引力的特性：避免额外的“inf-sup”条件，在解决方案规律性和无条件稳定性方面实现最佳精度，实现简单直接。最后，提供了大量的数值实验来支持新提出的方法的出色稳定性和准确性。