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Fully and semi-implicit robust space-time DG methods for the incompressible Navier-Stokes equations

L. Beirão da Veiga, F. Dassi, S. Gómez

TL;DR

This work develops robust space-time discontinuous Galerkin methods for the time-dependent incompressible Navier–Stokes equations by leveraging $H(\mathrm{div}, \Omega)$-conforming DG in space and DG time stepping. It proves pressure-robustness and Reynolds semi-robustness, establishes unconditional stability and quasi-optimal convergence for any time degree $\ell$, and introduces a high-order semi-implicit DG time discretization that avoids nonlinear solves after the first time slab. The analysis combines nonstandard test functions with Gauss–Radau time integration and a fixed-point argument to handle the nonlinear convective term, yielding existence, continuous data dependence, and a priori error estimates that are robust with respect to the pressure and the Reynolds number. Numerical experiments corroborate the theory, showing optimal convergence in diffusion- and convection-dominated regimes, Reynolds-robust velocity accuracy, and substantial efficiency gains for the semi-implicit scheme without compromising stability or accuracy.

Abstract

We carry out a stability and convergence analysis of a fully discrete scheme for the time-dependent Navier-Stokes equations resulting from combining an $H(\mathrm{div}, Ω)$-conforming discontinuous Galerkin spatial discretization, and a discontinuous Galerkin time stepping scheme. Such a scheme is proven to be pressure robust and Reynolds semi-robust. Standard techniques can be used to analyze only the case of lowest-order approximations in time. Therefore, we use some nonstandard test functions to prove existence of discrete solutions, unconditional stability, and quasi-optimal convergence rates for any degree of approximation in time. In particular, a continuous dependence of the discrete solution on the data of the problem, and quasi-optimal convergence rates for low and high Reynolds numbers are proven in an energy norm including the term $L^{\infty}(0, T; L^2(Ω)^d)$ for the velocity. In addition to the standard discontinuous Galerkin time stepping scheme, which is fully implicit, we propose and analyze a novel high-order semi-implicit version that avoids the need of solving nonlinear systems of equations after the first time slab, thus significantly improving the efficiency of the method. Some numerical experiments validating our theoretical results are presented.

Fully and semi-implicit robust space-time DG methods for the incompressible Navier-Stokes equations

TL;DR

This work develops robust space-time discontinuous Galerkin methods for the time-dependent incompressible Navier–Stokes equations by leveraging -conforming DG in space and DG time stepping. It proves pressure-robustness and Reynolds semi-robustness, establishes unconditional stability and quasi-optimal convergence for any time degree , and introduces a high-order semi-implicit DG time discretization that avoids nonlinear solves after the first time slab. The analysis combines nonstandard test functions with Gauss–Radau time integration and a fixed-point argument to handle the nonlinear convective term, yielding existence, continuous data dependence, and a priori error estimates that are robust with respect to the pressure and the Reynolds number. Numerical experiments corroborate the theory, showing optimal convergence in diffusion- and convection-dominated regimes, Reynolds-robust velocity accuracy, and substantial efficiency gains for the semi-implicit scheme without compromising stability or accuracy.

Abstract

We carry out a stability and convergence analysis of a fully discrete scheme for the time-dependent Navier-Stokes equations resulting from combining an -conforming discontinuous Galerkin spatial discretization, and a discontinuous Galerkin time stepping scheme. Such a scheme is proven to be pressure robust and Reynolds semi-robust. Standard techniques can be used to analyze only the case of lowest-order approximations in time. Therefore, we use some nonstandard test functions to prove existence of discrete solutions, unconditional stability, and quasi-optimal convergence rates for any degree of approximation in time. In particular, a continuous dependence of the discrete solution on the data of the problem, and quasi-optimal convergence rates for low and high Reynolds numbers are proven in an energy norm including the term for the velocity. In addition to the standard discontinuous Galerkin time stepping scheme, which is fully implicit, we propose and analyze a novel high-order semi-implicit version that avoids the need of solving nonlinear systems of equations after the first time slab, thus significantly improving the efficiency of the method. Some numerical experiments validating our theoretical results are presented.

Paper Structure

This paper contains 29 sections, 21 theorems, 175 equations, 5 figures.

Table of Contents

  1. Keywords.
  2. Mathematics Subject Classification.
  3. Introduction
  4. Previous works.
  5. Main contributions.
  6. Notation.
  7. Structure of the paper.
  8. Variational formulation and description of the method
  9. Mesh and DG notation
  10. Discrete spaces
  11. Fully discrete space--time formulation
  12. Existence of discrete solutions
  13. Some useful results for the stability analysis
  14. Inverse estimates.
  15. Properties of the discrete space--time forms.
  16. ...and 14 more sections

Key Result

Lemma 3.1

For any Banach space $(Z, \|\cdot\|_{Z})$ with $Z \subseteq L^1(\Omega)$, there exists a positive constant $C_{\mathrm{inv}}$ depending only on $\ell$ such that, for $n = 1, \ldots, N$, it holds

Figures (5)

  • Figure 1: Convergence rates of the fully implicit and the semi-implicit schemes for the error $\texttt{err}(\mathbf{u})$ varying $h$, considering different values of $\nu$ and approximations of degree $k=1$ and $k = 2$, for the problem with exact solution in \ref{['sol1']}, left. Resolution time in seconds for each simulation, right.
  • Figure 2: Behavior of the error $\texttt{err}(\mathbf{u})$ for the fully implicit and the semi-implicit schemes, considering different values of $\nu$ and approximations of degree $k=1$ and $k = 2$, for the problem with exact solution in \ref{['sol1']}.
  • Figure 3: Behavior of the error $\| \boldsymbol{e_u} \|_{L^\infty(0,T;L^2(\Omega)^d)}$ (left) and of the $L^2(\Omega)$ pressure error at the final time (right) for the fully implicit and the semi-implicit schemes, varying $h$, considering different values of $\nu$ and approximations of degree $k=1$ and $k = 2$, for the problem with exact solution in \ref{['sol1']}.
  • Figure 4: Convergence rates of the errors $\texttt{err}(\mathbf{u})$ (left) and $\| \boldsymbol{e_u} \|_{L^\infty(0,T;L^2(\Omega)^d)}$ (right) for the fully implicit and the semi-implicit schemes, varying $\tau$, considering different values of $\nu$ and approximations of degree $k=1$ and $k = 2$, for the problem with exact solution in \ref{['sol2']}.
  • Figure 5: Behavior of the error $\texttt{err}(\mathbf{u})$ (left) and of the $L^2(\Omega)$-error of the pressure at the final time (right) for the fully implicit and the semi-implicit schemes, varying $h$, considering different values of $\nu$ and approximations of degree $k=1$ and $k = 2$, for the problem with exact solution in \ref{['sol3']}.

Theorems & Definitions (48)

  • Remark 2.1: Use of the interpolant $\mathcal{I}_{\tau}^{\mathcal{R}}$
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3: Stability of $\mathcal{I}_{\tau}^{\mathcal{R}}$
  • proof
  • Lemma 3.4: Weak partial bound of the discrete solution
  • proof
  • Proposition 3.5: Continuous dependence on the data
  • ...and 38 more