SUMMARYIn this paper we study the stability and approximability of the P 1 -P 0 element (continuous piecewise linear for the velocity and piecewise constant for the pressure on triangles) for Stokes equations. Although this element is unstable for all meshes, it provides optimal approximations for the velocity and the pressure in many cases. We establish a relation between the stabilities of the Q 1 -P 0 element (bilinear/constant on quadrilaterals) and the P 1 -P 0 element. We apply many stability results on the Q 1 -P 0 element to the analysis of the P 1 -P 0 element. We prove that the element has the optimal order of approximations for the velocity and the pressure on a variety of mesh families.As a byproduct, we also obtain a basis of divergence-free piecewise linear functions on a mesh family on squares. Numerical tests are provided to support the theory and to show the efficiency of the newly discovered, truly divergence-free, P 1 finite element spaces in computation.
SUMMARYIn this paper we study the stability and performance of the quadrilateral finite element Q 1 -P 0 (bilinear/constant) for the Stokes equations. We set up a framework to show the stability of the element for a wide range of meshes with macroelement patches. We apply the new theory to show the stability of Q 1 -P 0 elements on some previously studied meshes and on some newly suggested meshes. Nevertheless such earlier and newly suggested meshes are not effective in practice, compared to the traditional unstable meshes for the Q 1 -P 0 element. The new theory leads naturally to a general idea in treating instability of square Q 1 -P 0 elements by the local stabilization on macroelement patches of larger, but fixed sizes. The good performance of the traditional Q 1 -P 0 square elements with filtering can be kept in some cases after the local stabilization. Some numerical tests are provided to support the theory and to show the performance of stabilized Q 1 -P 0 elements.
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