The Aubin–Nitsche Trick in Finite Element Analysis

1. Introduction

In finite element analysis, we often begin by establishing error estimates in an energy norm (typically the $H^1$-norm). However, many applications require sharper $L^2$ error estimates. The Aubin–Nitsche trick (or duality argument) is a powerful tool that leverages the extra regularity of a dual (or adjoint) problem to upgrade an $H^1$ error estimate of order $h^p$ to an $L^2$ error estimate of order $h^{p+1}$.

2. Background and Motivation

2.1 The Elliptic Problem

Consider the Poisson problem:

Let $u$ be the exact solution and $u_h$ its finite element approximation. We define the error as

2.2 Standard Error Estimate

A common energy norm (or $H^1$-seminorm) error estimate is

where $h$ is the mesh size and $p$ denotes the polynomial degree. Our goal is to improve the convergence order in the $L^2$ norm.

3. The Aubin–Nitsche Trick

The key idea is to introduce a dual problem whose solution exhibits higher regularity than the error $e$. This extra smoothness can be exploited to "transfer" the $H^1$ error estimate into a sharper $L^2$ estimate.

3.1 Defining the Dual (Adjoint) Problem

Introduce $\psi$ as the solution to the dual problem:

Under suitable regularity conditions (e.g., if $\Omega$ is convex), elliptic regularity gives:

3.2 Testing the Dual Problem with the Error

Multiply the dual equation by the error $e$ and integrate over $\Omega$:

Using integration by parts (and noting that $\psi = 0$ on $\partial\Omega$), we obtain:

3.3 Using Boundedness and Approximation Properties

In a finite element context, one typically employs:

• Galerkin Orthogonality:
  $B_h(e, v) = 0 \quad \forall v \in V_h,$
  where $B_h(\cdot,\cdot)$ is the bilinear form and $V_h$ is the finite element space.
• Interpolation Estimate:
  Let $\psi_I \in V_h$ be an interpolation of $\psi$ so that
  $\|\psi - \psi_I\|_{H^1(\Omega)} \leq C\, h\, \|\psi\|_{H^2(\Omega)}.$

Using the boundedness of the bilinear form, one can relate the error in the dual problem to the energy error:

Thus,

By the elliptic regularity,

so we deduce

Dividing by $\|e\|_{L^2(\Omega)}$ (assuming it is nonzero) gives:

3.4 Combining with the Energy Norm Error

Substitute the known energy norm error estimate:

to obtain:

This result demonstrates that the $L^2$ error converges one order faster than the $H^1$ error.

4. Summary and Conclusion

• Objective:
  Improve the $L^2$ error estimate in finite element methods from $O(h^p)$ (energy norm) to $O(h^{p+1})$.

• Key Steps:

  1. Dual Problem Formulation:
     Define $\psi$ such that

     $-\Delta \psi = e \quad \text{with} \quad \psi = 0 \quad \text{on } \partial\Omega.$

  2. Testing and Integration:
     Test the dual problem with the error $e$ and use integration by parts to relate $\|e\|_{L^2(\Omega)}$ to $\nabla e$ and $\nabla \psi$.

  3. Approximation and Regularity:
     Utilize the approximation properties of the finite element space and the elliptic regularity result

     $\|\psi\|_{H^2(\Omega)} \leq C\, \|e\|_{L^2(\Omega)}.$

  4. Derivation of the Improved Estimate:
     Combine these results with the energy norm error estimate to derive:

     $\|u - u_h\|_{L^2(\Omega)} \leq C\, h^{p+1}\, |u|_{H^{p+1}(\Omega)}.$

• Conclusion:
  The Aubin–Nitsche trick is an essential technique in finite element analysis, allowing one to harness the extra regularity of the dual problem to achieve optimal $L^2$ convergence rates. This is particularly useful in applications where $L^2$ accuracy is critical, such as in post-processing or in the analysis of discontinuous Galerkin methods.