Convergence analysis of nonconforming $$H(\operatorname {div})$$-finite elements for the damped time-harmonic Galbrun’s equation

Abstract We consider the damped time-harmonic Galbrun’s equation, which is used to model stellar oscillations. We introduce a discontinuous Galerkin finite element method (DGFEM) with $$H(\operatorname {div})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>(</mml:mo> <mml:mo>div</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -elements, which is nonconforming with respect to the convection operator. We report a convergence analysis, which is based on the frameworks of discrete approximation schemes and T-compatibility. A novelty is that we show how to interpret a DGFEM as a discrete approximation scheme and this approach enables us to apply compact perturbation arguments in a DG-setting, and to circumvent any extra regularity assumptions on the solution. The advantage of the proposed $$H(\operatorname {div})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>(</mml:mo> <mml:mo>div</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -DGFEM compared to $$H^1$$ <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> -conforming methods is that we do not require a minimal polynomial order or any special assumptions on the mesh structure. Further, we extend the analysis of the symmetric interior penalty DGFEM to a DGFEM without a penalty term, which considerably improves the smallness assumption on the Mach number to a fairly explicit bound. In addition, the method is robust with respect to the drastic changes of magnitude of the density and sound speed, which occur in stars.