A subspace of linear nonconforming finite element for nearly incompressible elasticity and Stokes flow

The linear nonconforming finite element, combined with constant finite element for pressure, is stable for the Stokes problem. But it does not satisfy the discrete Korn inequality. The linear conforming finite element satisfies the discrete Korn inequality, but is not stable for the Stokes problem and fails for the nearly incompressible elasticity problems. We enrich the linear conforming finite element by some nonconforming P1 bubbles, i.e., select a subspace of the linear nonconforming finite element space, so that the resulting linear nonconforming element is both stable and conforming enough to satisfy the Korn inequality, on HTC-type triangular and tetrahedral grids. Numerical tests in 2D and 3D are presented, confirming the analysis.

Automated summary

The paper proposes a stable and effective finite element method that satisfies both the discrete Korn inequality and the requirements of the Stokes problem. The linear conforming finite element is enriched by introducing some nonconforming bubbles.