Inf-Sup Stability of Parabolic TraceFEM

Abstract We develop a parabolic inf-sup theory for a modified TraceFEM semi-discretization in space of the heat equation posed on a stationary hypersurface embedded in $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> . We consider the normal derivative volume stabilization and add an $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -type stabilization to the time derivative. We assume that the representation of and the integration over the surface are exact, however, all our results are independent of how the surface cuts the bulk mesh. For any mesh for which the method is well-defined, we establish necessary and sufficient conditions for inf-sup stability of the proposed TraceFEM in terms of $$H^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -stability of a stabilized $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -projection and of an inverse inequality constant that accounts for the lack of conformity of TraceFEM. Furthermore, we prove that the latter two quantities are bounded uniformly for a sequence of shape-regular and quasi-uniform bulk meshes. We derive several consequences of uniform discrete inf-sup stability, namely uniform well-posedness, discrete maximal parabolic regularity, parabolic quasi-best approximation, convergence to minimal regularity solutions, and optimal order-regularity energy and $$L^2 L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> error estimates. We show that the additional stabilization of the time derivative restores optimal conditioning of time-discrete TraceFEM typical of fitted discretizations.