Do We Really Need an $H^{-1}$ -Norm LSFEM?

我們真的需要 $H^{-1}$ -範數 LSFEM 嗎？

Not for divergence-form $H^{-1}$ data. A modified flux restores a standard $L^2$ -residual first-order system, without replacing the least-squares norm.

對具有散度表示的 $H^{-1}$ 數據，不需要。 把數據中的散度部分吸收到通量變量後，仍可使用標準的 $L^2$ -殘差一階最小二乘方法。

The Starting Point: Mixed-Boundary Elliptic Problems 從混合邊界問題出發

Let $\Omega\subset\mathbb R^d$ be a bounded Lipschitz domain whose boundary is split into relatively open Dirichlet and Neumann parts $\Gamma_D$ and $\Gamma_N$ , with $|\Gamma_D|>0$ . Consider the sign convention used in our goal-oriented LSFEM paper:

$\begin{aligned} -\nabla\!\cdot(A\nabla u)+\mathbf b\!\cdot\nabla u+cu &=f_1+\nabla\!\cdot\mathbf f_2 &&\text{in }\Omega,\\ u&=\phi &&\text{on }\Gamma_D,\\ -(A\nabla u+\mathbf f_2)\cdot\mathbf n &=\psi &&\text{on }\Gamma_N. \end{aligned}$

Here $f_1\in L^2(\Omega)$ , $\mathbf f_2\in L^2(\Omega)^d$ , $\phi\in H^{1/2}(\Gamma_D)$ , and $\psi\in L^2(\Gamma_N)$ . We take $A$ symmetric and uniformly positive definite, $A,\mathbf b,c$ bounded, and assume the mixed-boundary weak operator is an isomorphism on the homogeneous Dirichlet space. This includes well-posed problems with lower-order terms, not only coercive diffusion equations.

這裡 $f_1$ 是體積數據， $\mathbf f_2$ 是梯度配對中的向量數據； $\phi$ 和 $\psi$ 分別是 Dirichlet 與 Neumann 邊界數據。假設相應的混合邊界弱問題適定。

Four Contributions to an H¹ Functional H¹ 線性泛函的四類數據

Volume and boundary data act on H¹(Ω) through scalar, gradient, Dirichlet-boundary, and Neumann-boundary terms:

$\mathscr F(w)=(f_1,w)_\Omega-(\mathbf f_2,\nabla w)_\Omega +\langle h_D,w\rangle_{\Gamma_D} +\langle h_N,w\rangle_{\Gamma_N},\qquad w\in H^1(\Omega).$

一個 H¹(Ω) 線性泛函可以包含四類數據：體積項 f₁、梯度項 f₂，以及 Dirichlet、Neumann 邊界上的跡配對。

For the mixed-boundary problem above, let

$H_D^1(\Omega):=\{w\in H^1(\Omega):w|_{\Gamma_D}=0\},\quad X:=\{w\in H^1(\Omega):w|_{\Gamma_D}=\phi\}.$

The prescribed Dirichlet value fixes the trial space; the Neumann datum enters the weak right-hand side:

$\boxed{\ell(w)=(f_1,w)_\Omega-(\mathbf f_2,\nabla w)_\Omega -\langle\psi,w\rangle_{\Gamma_N},\qquad w\in H^1(\Omega).}$

Find u ∈ X_p such that

$a(u,w_0)=\ell(w_0)\quad(\forall w_0\in H_D^1(\Omega)),\qquad a(u,w)=(A\nabla u,\nabla w)_\Omega+(\mathbf b\!\cdot\nabla u+cu,w)_\Omega.$

Why the Unshifted Flux May Fail 為什麼原來的梯度通量可能失效

Because $\mathbf f_2$ need only be in $L^2$ , its distributional divergence need not lie in $L^2$ . The conventional physical gradient flux $-A\nabla u$ is square integrable, but may not belong to $H(\mathrm{div};\Omega)$ ; its divergence still contains the non- $L^2$ term $\nabla\!\cdot\mathbf f_2$ .

The difficulty is not that the least-squares residual norm is too weak. It is that the unshifted flux is generally not in the function space required by the standard $H(\mathrm{div})\times H^1$ formulation.

真正的問題不是 $L^2$ 殘差範數太弱，而是未修正的梯度通量不一定屬於標準一階最小二乘格式所要求的 $H(\mathrm{div})$ 空間。

Absorb the Divergence Data into the Flux 把散度數據吸收到通量中

Define the shifted flux

$\boxed{\boldsymbol\sigma:=-A\nabla u-\mathbf f_2.}$

Using the plus sign in the original PDE is essential: since

$-\nabla\!\cdot(A\nabla u+\mathbf f_2)+\mathbf b\!\cdot\nabla u+cu=f_1,$

the equations are exactly equivalent to

$\boxed{\begin{aligned} \boldsymbol\sigma+A\nabla u+\mathbf f_2&=0,\\ \nabla\!\cdot\boldsymbol\sigma+\mathbf b\!\cdot\nabla u+cu&=f_1,\\ u|_{\Gamma_D}&=\phi,\\ \boldsymbol\sigma\cdot\mathbf n|_{\Gamma_N}&=\psi. \end{aligned}}$

Now $\boldsymbol\sigma\in L^2(\Omega)^d$ , and

$\nabla\!\cdot\boldsymbol\sigma=f_1-\mathbf b\!\cdot\nabla u-cu\in L^2(\Omega).$

Therefore $\boldsymbol\sigma\in H(\mathrm{div};\Omega)$ . This is an exact change of variable, not a regularization or approximation of the right-hand side.

因為 $\mathbf f_2$ 已被吸收到 $\boldsymbol\sigma$ ，守恆方程的右端現在屬於 $L^2$ ，因此修正通量自然屬於 $H(\mathrm{div};\Omega)$ 。這是完全等價的重寫，並沒有修改原始數據。

The Neumann Trace Must Be the Shifted-Flux Trace Neumann 條件必須施加在修正通量上

The boundary condition is $\boldsymbol\sigma\cdot\mathbf n=\psi$ on $\Gamma_N$ — not a separate condition on $-A\nabla u$ . In the normal-trace sense, this means

$(\boldsymbol\sigma,\nabla w_0)_\Omega+(\nabla\!\cdot\boldsymbol\sigma,w_0)_\Omega =\langle\psi,w_0\rangle_{\Gamma_N},\quad w_0\in H_D^1(\Omega).$

For a flux lifting with this trace, define

$\Sigma:=\{\boldsymbol\tau\in H(\mathrm{div};\Omega): \boldsymbol\tau\cdot\mathbf n|_{\Gamma_N}=\psi\}.$

Applying Green’s formula to the first-order system gives precisely

$a(u,w_0)=(f_1,w_0)_\Omega-(\mathbf f_2,\nabla w_0)_\Omega -\langle\psi,w_0\rangle_{\Gamma_N}=\ell(w_0).$

Conversely, the weak equation identifies the normal trace of the shifted flux and recovers the first-order system. Thus $u\in X$ solves the mixed-boundary weak problem if and only if $(\boldsymbol\sigma,u)\in\Sigma\times X$ solves the shifted first-order system.

修正通量的定義、弱形式中的 Neumann 配對、以及 $H(\mathrm{div})$ 正常跡，是同一個結構。

A Standard $L^2$ Least-Squares Functional 保留標準的 $L^2$ 最小二乘泛函

There is now no reason to replace the $L^2$ divergence residual by a negative norm. Minimize, over $\Sigma\times X$ ,

$\boxed{\begin{aligned} \mathcal L(\boldsymbol\tau,v) ={}&\big\|A^{-1/2}(\boldsymbol\tau+A\nabla v+\mathbf f_2)\big\|_{L^2(\Omega)}^2\\ &+\big\|\nabla\!\cdot\boldsymbol\tau+\mathbf b\!\cdot\nabla v+cv-f_1\big\|_{L^2(\Omega)}^2. \end{aligned}}$

With boundary liftings, the error pair belongs to $H_N(\mathrm{div};\Omega)\times H_D^1(\Omega)$ . Under the assumed isomorphism / inf–sup stability of the mixed-boundary elliptic operator, the homogeneous least-squares residuals satisfy the norm equivalence

$\begin{aligned} C_1\big(\|\boldsymbol\tau\|_{H(\mathrm{div})}^2+\|v\|_{H^1}^2\big) \ &\le \|A^{-1/2}(\boldsymbol\tau+A\nabla v)\|_{L^2}^2\\ &\quad+\|\nabla\!\cdot\boldsymbol\tau+\mathbf b\!\cdot\nabla v+cv\|_{L^2}^2\\ &\le C_2\big(\|\boldsymbol\tau\|_{H(\mathrm{div})}^2+\|v\|_{H^1}^2\big), \end{aligned}$

for $\boldsymbol\tau\in H_N(\mathrm{div};\Omega)$ and $v\in H_D^1(\Omega)$ . Thus conforming least-squares discretizations retain symmetric positive-definite systems, quasi-optimality, and a native residual-based error estimator, subject to the stated well-posedness and conforming-space assumptions.

The negative norm belongs to the data space, not necessarily to the residual. A divergence splitting keeps the familiar $L^2$ FOSLS structure.

負範數出現在數據空間，不一定要放進殘差泛函。 正確的散度分解與通量重定義，已足以恢復標準的 $L^2$ -based FOSLS 架構。

Where This Stops: A Dirac Source Hilbert 尺度的邊界：Dirac 源

In two dimensions, a point Dirac measure is not in $H^{-1}(\Omega)$ . Its explicit divergence lifting behaves like $|x-x_0|^{-1}$ and belongs to $L^p$ for every $1<p<2$ , but not to $L^2$ . The same structural splitting still makes sense, but the analysis must leave the Hilbert scale: see A Dirac Source Is First a Failure of Duality.

二維 Dirac 測度不屬於 $H^{-1}$ ，所以不能直接套用上面的 $L^2$ 分解。相同的結構性思路仍成立，但必須轉到 $L^p$ 尺度。這正是另一篇 Dirac mixed 專題的起點。

The discussion here concerns volume data and the external mixed boundary. Sources supported on interior interfaces or embedded surfaces require their own jump and trace accounting and are left for a later treatment.

Papers 相關論文

A simple proof of coerciveness of first-order system least-squares methods for general second-order elliptic PDEs

Shun Zhang. Computers & Mathematics with Applications 130 (2023), 98–104.

Journal DOI · PDF

Goal-Oriented Error Estimation for Least-Squares Finite Element Methods via Physically Meaningful Adjoint PDEs

Yueyao Wu and Shun Zhang. Preprint, 2026. See Sections 2.1–2.2 for the mixed-boundary weak problem and shifted-flux formulation.

arXiv · PDF

Do We Really Need an H−1H^{-1}-Norm LSFEM?