Divergence-conforming HDG methods for Stokes flows

<p>In this paper, we show that by sending the normal stabilization function to infinity in the hybridizable discontinuous Galerkin methods previously proposed in [Comput. Methods Appl. Mech. Engrg. 199 (2010), 582–597], for Stokes flows, a new class of divergence-conforming methods is obtained which maintains the convergence properties of the original methods. Thus, all the components of the approximate solution, which use polynomial spaces of degree <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>, converge with the optimal 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 <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> for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k \ge 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, the postprocessed velocity approximation is also divergence-conforming, exactly divergence-free and converges with order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k plus 2"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k+2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than-or-equal-to 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\ge 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and with order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals 0"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The novelty of the analysis is that it proceeds by taking the limit when the normal stabilization goes to infinity in the error estimates recently obtained in [Math. Comp., 80 (2011) 723–760].</p>