A new T-compatibility condition and its application to the discretization of the damped time-harmonic Galbrun's equation

TL;DR

A new framework is reported for which the discrete operators T n only have to converge point-wise, but in addition a weak T-coercivity condition has to be satisfied on the discrete level to prove the convergence of certain H 1 -conforming element discretizations of the damped time-harmonic Galbrun’s equation.

Abstract

We consider the approximation of weakly T-coercive operators. The main property to ensure the convergence thereof is the regularity of the approximation (in the vocabulary of discrete approximation schemes). In a previous work the existence of discrete operators $T_n$ which converge to $T$ in a discrete norm was shown to be sufficient to obtain regularity. Although this framework proved useful for many applications for some instances the former assumption is too strong. Thus in the present article we report a weaker criterion for which the discrete operators $T_n$ only have to converge point-wise, but in addition a weak T-coercivity condition has to be satisfied on the discrete level. We apply the new framework to prove the convergence of certain $H^1$-conforming finite element discretizations of the damped time-harmonic Galbrun's equation, which is used to model the oscillations of stars. A main ingredient in the latter analysis is the uniformly stable invertibility of the divergence operator on certain spaces, which is related to the topic of divergence free elements for the Stokes equation.

Figures (6)

• Figure 1: The coarsest mesh of the sequence of unstructured simplicial meshes is shown on the left, with mesh size $h=0.5$. To construct the second sequence of meshes we apply barycentric mesh refinement, resulting in the mesh on the right.
• Figure 2: Convergence against a reference solution computed with polynomial degree $k=5$ and mesh size $h=1.5\cdot 2^{-6}$. We consider the setting described in \ref{['eq:nums']} and \ref{['eq:coeffs1']} with periodic boundary conditions and fixed $\alpha=0.2$ for the background flow $\mathbf{b}$ given in \ref{['eq:bflow1']}, and different polynomial degree $k=1,2,3,4$ and varying mesh size. From left to right we consider: unstructured meshes, barycentric refined meshes, and the Taylor-Hood variant, i.e. unstructured meshes with $Q_n\subset H^1$. The error is measured in the $\left\|\cdot\right\|_\mathbb{X}$-norm.
• Figure 3: On unstructured meshes with periodic boundary conditions we consider the error in the $\left\|\cdot\right\|_\mathbb{X}$-norm (left) and consistency-error (right) against a reference solution for different values of the coefficient in the background flow $\mathbf{b}$, given in \ref{['eq:bflow1']}, and fixed polynomial order $k=4$.
• Figure 4: The real part of the first entry of the reference solution computed with $k=5$ and $h=1.5 \cdot 2^{-6}$ for two different values of the coefficient of the flow field $\mathbf{b}$, $\alpha=0.2$ on the left and $\alpha=1.5$ on the right.
• Figure 5: Convergence towards an exact solution given in \ref{['eq:ex_sol']} for different polynomial orders $k=1,2,3,4$ for varying mesh sizes $h$. We consider homogeneous normal boundary contition with fixed $\alpha=0.1$ in the background flow $\mathbf{b}$, given in \ref{['eq:bflow2']}. From left to right we consider: unstructured meshes, barycentric refined meshes, and unstructured meshes with $Q_n\subset H^1$. The error is measured in the $\left\|\cdot\right\|_\mathbb{X}$-norm.

Theorems & Definitions (49)

• Definition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 2
• Theorem 3
• proof
• Proposition 4: Variation of Prop. 3.5 of Bredies:08
• Theorem 5