Analysis of divergence-free 𝐻¹ conforming FEM with IMEX-SAV scheme for the Navier-Stokes equations at high Reynolds number
Yongbin Han, Yanren Hou, Min Zhang
Abstract
In this paper, we analyze the first-order implicit-explicit type scheme based on the scalar auxiliary variable (SAV) with divergence-free <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">H^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> conforming finite element method (FEM) in space for the evolutionary incompressible Navier-Stokes equations at high Reynolds number. The stability and a priori error estimates are given, in which the constants are independent of the Reynolds number. The velocity energy estimate is given without any condition on the time step, however, the a priori error estimates for the velocity are obtained with severe time step restrictions. In addition, a Reynolds-dependent error bound with convergence order of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k plus 1"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in space is also obtained for the velocity error in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm with no time step restrictions. Here, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the polynomial order of the velocity space. Some numerical experiments are carried out to verify the analytical results.