A P ₂ H-Div-Nonconforming-H-Curl Finite Element for the Stokes Equations on Triangular Meshes

The P2 BDM H-div finite element is enriched by five <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> P₅ divergence-free bubbles on each triangle. The resulting finite element remains H-div and is also H-curl nonconforming. Thus the new finite element combined with the discontinuous <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> P₁ element for pressure, is stabilizer-free and still divergence-free in solving the Stokes equations on triangular meshes. The stabilization can also be done by three, four, or six <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> P₅ divergence-free bubble-enrichment each triangle, shown both in theory and in computation. Numerical tests show the advantage of the divergence-free element over the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> P₂ Taylor–Hood finite element.