💻 computer science

Error estimates for finite element discretizations of the instationary Navier-Stokes equations

This paper establishes best approximation type error estimates for fully discrete finite element approximations of the two-dimensional instationary Navier-Stokes equations in the $L^\infty(I;L^2(\Omega))$ , $L^2(I;H^1(\Omega))$ , and $L^2(I;L^2(\Omega))$ norms by employing a discontinuous Galerkin time discretization, inf-sup stable spatial elements, an error splitting approach, a tailored duality argument, and a specialized discrete Gronwall lemma.

Original authors: Boris Vexler, Jakob Wagner

Published 2026-05-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Boris Vexler, Jakob Wagner

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the chaotic dance of wind swirling around a building or water flowing through a pipe. This is described by a set of complex mathematical rules called the Navier-Stokes equations. These equations are notoriously difficult to solve exactly, so scientists use computers to approximate the solution. They break time and space into tiny chunks (like a grid) and calculate what happens in each little box.

This paper is about how accurate those computer calculations are and, more importantly, how to prove they are accurate in the most critical way possible.

Here is the breakdown of their work using simple analogies:

1. The Problem: The "Perfect" vs. The "Pixelated"

Think of the real fluid flow as a smooth, continuous movie. The computer, however, sees this movie as a series of low-resolution pixels (the grid) and frames (the time steps).

The Goal: The authors want to know: "How close is our pixelated, frame-by-frame computer simulation to the real, smooth movie?"
The Challenge: In the past, when scientists tried to measure this "closeness" (error), they often had to mix two different types of measurements together. It was like trying to judge the quality of a photo by looking at both its sharpness and its color brightness at the same time. If the color was slightly off, it made the sharpness look worse, even if the sharpness was actually perfect. This made it hard to get a precise guarantee that the simulation was good.

2. The New Tool: The "Best Approximation" Mirror

The main achievement of this paper is a new way to measure the error that looks at the "sharpness" (specifically the $L^\infty(I; L^2)$ norm, which is a fancy way of saying "the worst-case difference at any single moment in time") all by itself.

They call this a "Best Approximation" type estimate.

The Analogy: Imagine you are trying to guess the height of a mountain. Previous methods said, "Your guess is off by X feet, but that includes the error in your ruler and the error in your eyesight."
This Paper's Method: They say, "Your guess is off by exactly how far your guess is from the best possible guess you could have made with your tools." They separate the error caused by the math from the error caused by the computer's limitations. This gives a much clearer, more honest picture of the simulation's quality.

3. The Secret Weapons

To pull this off, the authors had to invent and refine two specific tools:

A. The "Time-Traveling" Duality Argument

Usually, to check how wrong a prediction is, you look at the prediction itself. But for these fluid equations, that's too messy.

The Metaphor: Imagine you want to know how much a specific drop of water missed its target. Instead of chasing the drop forward, the authors run a "reverse movie" (a dual equation) starting from the target and working backward.
The Trick: By running this reverse movie, they can trace the error back to its source. The paper proves that this reverse movie is stable and doesn't blow up, allowing them to measure the error in the forward movie with high precision.

B. The "Growth-Stopper" (Discrete Gronwall Lemma)

When simulating fluids, small errors can sometimes grow exponentially, like a snowball rolling down a hill, until the whole simulation crashes.

The Metaphor: The authors developed a special "brake" (a new version of a mathematical lemma called Gronwall's Lemma).
How it works: This brake is smart enough to handle situations where the "fuel" for the error (the external forces pushing the fluid) is a bit messy or irregular. It proves that even with messy inputs, the error won't explode; it stays under control and grows only in a predictable, manageable way.

4. The Result: A Better Map

By combining these tools, the authors proved that their computer method (using Discontinuous Galerkin in time and Finite Elements in space) is highly accurate.

The Verdict: They showed that the error in their simulation is essentially proportional to the size of the grid and the time steps, multiplied by a tiny "logarithmic" factor (which is like a very small tax on the accuracy).
Why it matters: This is better than previous methods. For example, if you halve the size of your grid, you get a much more accurate result than you would have with older formulas. It confirms that their method is "optimal"—meaning you can't really do much better without using a completely different, more expensive approach.

Summary

In short, this paper is a rigorous proof that a specific way of simulating fluid flow on a computer is extremely reliable. The authors didn't just say "it works"; they built a new mathematical framework to prove exactly how close the computer's answer is to reality, isolating the error so it can be measured cleanly. They used a "reverse movie" technique to track errors and a special "brake" to ensure those errors don't run away, resulting in the most precise error estimates for this type of fluid simulation to date.

Problem Statement
This work addresses the numerical analysis of the two-dimensional instationary Navier-Stokes equations with homogeneous Dirichlet (no-slip) boundary conditions. The equations are discretized using a fully discrete scheme combining a discontinuous Galerkin (dG) method in time and inf-sup stable finite elements in space. The primary objective is to derive error estimates for the fully discrete solution, specifically focusing on the $L^\infty(I; L^2(\Omega))$ norm.

While best approximation type error estimates for the instationary Stokes problem in $L^\infty(I; L^2(\Omega))$ , $L^2(I; H^1(\Omega))$ , and $L^2(I; L^2(\Omega))$ norms have been recently established, extending these results to the nonlinear Navier-Stokes equations presents significant challenges. Existing literature often estimates errors in $L^\infty(I; L^2(\Omega))$ and $L^2(I; H^1(\Omega))$ in a combined fashion, which leads to an order reduction for the spatial $L^2$ error estimate due to the necessity of accounting for the $H^1$ spatial error. Furthermore, previous results for Navier-Stokes often require stronger regularity assumptions on the source term $f$ (e.g., $f \in W^{1,\infty}$ or $W^{2,\infty}$ ) or yield suboptimal convergence orders (e.g., $O(\sqrt{k} + h)$ ).

Methodology
The authors employ a combination of error splitting, duality arguments, and stability analysis tailored to the dG time discretization.

Regularity Analysis: The paper first establishes necessary regularity results for the continuous Navier-Stokes solution. It proves that under convex domain assumptions and appropriate data regularity ( $f \in L^2(I; L^2(\Omega)^2)$ , $u_0 \in V$ ), the solution possesses $H^2$ spatial regularity and the nonlinear term $(u \cdot \nabla)u$ inherits sufficient regularity to be treated as a source term in the Stokes operator framework.
Discretization: The spatial discretization uses inf-sup stable pairs of finite element spaces $(U_h, M_h)$ satisfying the discrete LBB condition. The temporal discretization utilizes the dG( $q$ ) method with $q=0$ or $q=1$ , allowing for discontinuities at time nodes.
Stability and Existence: A key technical component is the development of a specially tailored discrete Gronwall lemma (Lemma 4.11). This lemma is adapted to handle right-hand sides that are only $L^1$ in time and allows for the derivation of stability bounds that are linear in the data, avoiding exponential growth with respect to the final time $T$ . This is used to prove the existence and uniqueness of discrete solutions and their stability in strong norms ( $L^\infty(I; H^1(\Omega))$ and $L^2(I; H^2(\Omega))$ ).
Error Splitting and Duality: The error $e = u - u_{kh}$ $e = u - u_{k h}$ is split into two parts: $\xi = u - \tilde{u}_{kh}$ $ξ = u - \tilde{u}_{k h}$ (the error of an instationary Stokes projection) and $\eta_{kh} = \tilde{u}_{kh} - u_{kh}$ $η_{k h} = \tilde{u}_{k h} - u_{k h}$ (the remainder).
- The Stokes part $\xi$ is estimated using existing best approximation results for the Stokes problem.
- The remainder $\eta_{kh}$ is estimated using a duality argument. A discrete dual problem is formulated, linearized around the average of the continuous and discrete solutions ( $u_{ukh} = \frac{1}{2}(u + u_{kh})$ ). The stability of this dual problem is established using the discrete Gronwall lemma and the previously proven regularity of the primal solutions.

Key Contributions and Results
The paper derives "best approximation type" error estimates for the Navier-Stokes equations, isolating the error in the $L^\infty(I; L^2(\Omega))$ norm without coupling it to the $H^1$ error in a way that reduces convergence order.

The main theoretical results are:

$L^2(I; H^1(\Omega))$ Estimate: For $f \in L^2(I; L^2(\Omega)^2)$ and $u_0 \in V$ , the error satisfies $\|u - u_{kh}\|_{L^2(I; H^1(\Omega))} \leq C(\sqrt{k} + h)$ .
$L^\infty(I; L^2(\Omega))$ Estimate: For $f \in L^\infty(I; L^2(\Omega)^2)$ and $u_0 \in V \cap H^2(\Omega)^2$ , the error satisfies:
$\|u - u_{kh}\|_{L^\infty(I; L^2(\Omega))} \leq C \ln(T/k)^2 (k + h^2)$
This result is presented as a best approximation type estimate involving the infimum over discrete spaces and the stationary Stokes projection error.
$L^2(I; L^2(\Omega))$ Estimate: For $f \in L^2(I; L^2(\Omega)^2)$ and $u_0 \in V$ , the error satisfies $\|u - u_{kh}\|_{L^2(I; L^2(\Omega))} \leq C(k + h^2)$ .

These estimates are optimal in terms of convergence order (up to logarithmic factors) and required regularity, improving upon previous results that either combined norms (leading to order reduction) or required stronger regularity assumptions on the data.

Significance
The authors state that the primary motivation for this work is the analysis of optimal control problems subject to state constraints that are pointwise in time. In such problems, the corresponding Lagrange multiplier is a measure in time, necessitating error estimates in the $L^\infty(I; L^2(\Omega))$ norm. The derived estimates provide the necessary theoretical foundation for analyzing these control problems using dG time discretizations.

The work also highlights that the techniques developed, particularly the tailored discrete Gronwall lemma and the duality argument for the nonlinear term, allow for the extension of Stokes error estimates to the nonlinear Navier-Stokes equations while maintaining optimal convergence rates and minimal regularity requirements. The paper emphasizes that these results are achieved without assuming the high regularity of the source term $f$ often required in previous literature.