A low-degree strictly conservative finite element method for incompressible flows

Abstract

In this paper, a new P 2 − P 1 finite element pair is proposed for incompressible fluid. For this pair, the discrete inf-sup condition and the discrete Korn's inequality hold on general triangulations. It yields exactly divergence-free velocity approximations when applied to models of incompressible flows. The robust capacity of the pair for incompressible flows are verified theoretically and numerically.

A lowest-degree strictly conservative finite element scheme for incompressible Stokes problem on general triangulations

“…We refer to [7,17,33] for pairs on rectangular grids and [21] for ones on cubic grids where full advantage of the geometric symmetry of the cells are taken. The approaches given in [30] and the present paper can be generalized to higher dimensions and non-simplicial grids. This will be discussed in future.…”

“…The auxiliary pair V sBDFM h0 − P 1 h0 is constructed by reducing H(div) finite element spaces which was firstly adopted in [30]. It is interesting to notice that, the sBDFM element has the same nodal parameters as ones given in [20,28] (the lowest-degree) and [10] (the lowest-degree), but it uses the lowest-degree polynomials among these four, and only the sBDFM element space can accompany the piecewise linear polynomial space to form a stable pair, while the other three can only accompany the piecewise constant space.…”

“…As revealed by [30], the space can be viewed as a reduced Brezzi-Douglas-Marini element space with enhanced smoothness.…”

“…The pair is stable and immediately strictly conservative on general triangulations, and is of the lowest degree ever known. Meanwhile, as is pointed out in [30], this P 2 − P 1 pair can be viewed as a smoothened reduction from the famous Brezzi-Douglas-Marini pair, and this idea can be carried on for other H(div) pairs so that the degree of finite element pairs may be reduced further.…”

“…We mainly utilize a two-step argument. For the auxiliary pair V sBDFM h0 − P 1 h0 , we mainly utilize Stenberg's macroelement argument by following the procedures of [30]; then the stability of the pair V el h0 − P 0 h0 , which is a sub-pair of V sBDFM h0 − P 1 h0 , is proved just by inheriting the stability of the V sBDFM h0 − P 1 h0 . This "reduce and inherit" procedure can be found in, e.g., [36,37] where some low degree optimal schemes are designed for other problems.…”