Where Does LSFEM Coercivity Come From?

PDE Stability Is the Starting Point 從 PDE 的穩定性出發

Consider the general second-order elliptic equation

考慮一般二階橢圓方程

The standard weak form may be indefinite because of the lower-order terms. A real Helmholtz problem away from its spectrum is a familiar example. What matters is the stability of the PDE, not coercivity of its usual bilinear form. Under the standard elliptic assumptions, uniqueness of the homogeneous problem and the Fredholm alternative provide this stability.

由於低階項，標準弱形式可以是不定的。避開特徵值的實 Helmholtz 問題就是典型例子。真正重要的是原始 PDE 的 stability，而不是其雙線性形式本身 coercive。在通常的橢圓性假設下，由齊次問題的唯一性及 Fredholm 理論便能得到這個穩定性。

For the linear weak operator defined by a, the PDE stability estimate can be written in the equivalent inf-sup form:

對由雙線性形式 a 定義的線性弱算子，PDE stability estimate 可以等價地寫成 inf-sup 形式：

The First-Order Residuals 一階系統的兩個殘差

For the diffusion-flux formulation, set the unknown pair in $H_N(\operatorname{div};\Omega)\times H_D^1(\Omega)$ and define

對 diffusion-flux formulation，取未知量於 $H_N(\operatorname{div};\Omega)\times H_D^1(\Omega)$，定義

Here the subscripted spaces encode homogeneous Neumann and Dirichlet conditions on the corresponding boundary parts. A standard least-squares functional is

這裡的下標表示相應邊界部分上的齊次 Neumann 與 Dirichlet 條件。標準 least-squares functional 為

Coercivity means that the two residuals control the natural $H(\operatorname{div})\times H^1$ norm. The proof begins with the original PDE stability estimate.

所謂 coercivity，就是這兩個殘差能控制自然的 $H(\operatorname{div})\times H^1$ 範數。這個證明直接從原 PDE 的 stability estimate 出發。

A Two-Line Coercivity Proof 兩三行完成的 Coercivity 證明

Let $a(\cdot,\cdot)$ be the standard variational bilinear form. Green’s formula and the homogeneous boundary conditions give the exact identity

記 $a(\cdot,\cdot)$ 為標準變分雙線性形式。由 Green 公式及齊次邊界條件得到

Applying the PDE stability estimate to this residual identity immediately gives

因此，把原 PDE 的 stability estimate 用於上述殘差恆等式，立即得到

The definitions of $r_1$ and $r_2$ then bound $\boldsymbol\tau$ and $\nabla\!\cdot\boldsymbol\tau$. Hence

再利用 $r_1,r_2$ 的定義控制 $\boldsymbol\tau$ 和 $\nabla\!\cdot\boldsymbol\tau$，便得到

This is the main idea of the 2023 coercivity paper. The paper also treats the total-flux formulation and both $L^2$- and $H^{-1}$-based residuals.

這就是 2023 年 coercivity 論文的核心。文章同時處理 total-flux formulation，以及 $L^2$ 和 $H^{-1}$ 兩種殘差範數。

Why Conforming LSFEM Inherits Stability 為什麼 Conforming LSFEM 自動穩定

Once the continuous coercivity estimate holds, it holds on every conforming finite element subspace of $H_N(\operatorname{div})\times H_D^1$. No sufficiently-small-mesh assumption is needed, even when the original elliptic variational problem is indefinite.

連續 coercivity 一旦成立，對 $H_N(\operatorname{div})\times H_D^1$ 的任何 conforming finite element subspace 都成立。即使原橢圓問題是不定的，也不要求網格尺寸充分小。

The consequences are familiar: a symmetric positive definite algebraic system, quasi-best approximation, and a least-squares functional that provides the natural built-in error measure.

因此可得到對稱正定代數系統、準最佳逼近，以及 least-squares functional 本身所提供的內建誤差度量。

What Changes with Crouzeix–Raviart? 換成 Crouzeix–Raviart 後改變了什麼？

The nonconforming case is different. In the two-field potential–flux div-LSFEM, the flux uses Raviart–Thomas elements and the potential uses Crouzeix–Raviart elements:

Nonconforming 情形就不同。Two-field potential–flux div-LSFEM 以 Raviart–Thomas 元逼近通量、Crouzeix–Raviart 元逼近勢函數：

Since $CR_1\not\subset H^1(\Omega)$, continuous norm equivalence cannot simply be restricted to this discrete space. The two-field least-squares functional retains the ordinary elementwise residuals, but its discrete solvability requires a new argument.

由於 $CR_1\not\subset H^1(\Omega)$，不能把連續 norm equivalence 直接限制到這個離散空間。Two-field functional 仍然採用通常的逐單元殘差，但必須重新證明離散問題的可解性。

The Liang–Zhang paper gives three proofs of discrete solvability for sufficiently fine meshes: one based on coercivity of the original bilinear form and two under the minimal uniqueness assumption for the elliptic PDE.

Liang–Zhang 論文對充分細的網格給出三種離散可解性證明：一種基於原雙線性形式的 coercivity，另外兩種只使用橢圓方程的 minimal uniqueness assumption。

There is a second, more surprising distinction. A counterexample shows that the two-field div least-squares functional has no norm equivalence on the sum space $H^1+CR_1$. Therefore, unlike conforming FOSLS, this functional cannot serve directly as a built-in a posteriori error estimator for the natural error. The paper develops separate reliable and efficient estimators.

還有一個更重要的區別：論文給出反例，證明 two-field div least-squares functional 在 $H^1+CR_1$ 的和空間上沒有 norm equivalence。因此它不能像 conforming FOSLS 那樣直接充當自然誤差的內建 a posteriori estimator；文章另外構造了可靠且有效率的誤差估計量。

The Three-Field Div–Curl Formulation Three-Field Div–Curl 格式

The same JSC paper introduces a potential–flux–intensity three-field div–curl least-squares formulation. Its stability mechanism is stronger: under the minimal uniqueness assumption, the least-squares functional is norm equivalent in the appropriate abstract nonconforming piecewise $H^1$ space.

同一篇 JSC 論文還引入 potential–flux–intensity 的 three-field div–curl least-squares formulation。它的穩定性結果更強：在 minimal uniqueness assumption 下，least-squares functional 在相應的 abstract nonconforming piecewise $H^1$ 空間中具有 norm equivalence。

As a result, this three-field formulation has no restriction requiring a sufficiently small mesh, and its least-squares functional is a built-in a posteriori error estimator.

因此，這個 three-field formulation 不需要網格尺寸充分小的限制，而且其 least-squares functional 可以直接作為內建 a posteriori error estimator。

The Mathematical Message 數學上的主線

The underlying PDE supplies stability; for linear problems, the inf-sup form of that estimate gives a short coercivity proof for conforming FOSLS. Nonconforming approximation changes the underlying space, so the stability question must be answered again. The answer depends on the first-order formulation, not merely on minimizing a sum of squared residuals.

原始 PDE 的 stability 是 conforming FOSLS coercivity 的根本來源；對線性問題，用其 inf-sup 形式便可直接證明。當逼近空間變成 nonconforming，穩定性問題就必須重新處理；答案取決於一階系統如何構造，而不只是把殘差平方相加。

Papers 相關論文