The stabilized lower-order and equal-order finite element methods for the hydrostatic Stokes problems

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.