A Dirac Source Is First a Failure of Duality

對混合有限元方法而言，Dirac 源首先是對偶失效，而不是正則性不足

Before approximation begins, the usual mixed formulation is already undefined.

在開始討論逼近之前，通常的混合格式本身就已經沒有定義。

The Usual Diagnosis Comes Too Late 通常的診斷來得太晚

A Dirac source is commonly regarded as a low-regularity approximation problem. The solution has a logarithmic singularity, so one studies weighted spaces, mesh grading, regularization, or specialized approximation near the pole.

Dirac 源通常被視為低正則性的逼近問題。在二維，解在源點具有對數奇異性，因此常見的處理方式包括加權空間、graded mesh、regularization，以及源點附近的特殊逼近。

For a mixed method, the first obstruction occurs before approximation: the conservation equation cannot be paired with the scalar test space.對混合方法而言，第一個障礙出現在逼近之前：守恆方程無法與純量測試空間配對。

The scalar test space in a standard mixed formulation consists of Lebesgue equivalence classes, with no point values. A Dirac measure cannot be paired with a generic scalar test function by point evaluation: the expression v(x₀) is not well defined on such equivalence classes.

標準混合格式中的純量測試函數屬於 Lebesgue 空間，只在幾乎處處相等的意義下確定，並沒有確定的點值。因而 Dirac 測度無法藉由點值與一般純量測試函數配對：v(x₀) 在這種等價類上沒有定義。

Why the Standard Mixed Formulation Fails 為什麼標準混合格式失效

Consider · 考慮

−∇·(A∇u) = f₁ + δ_x₀ in Ω,u = g on ∂Ω.

With the physical flux σ_phys := −A∇u, the conservation law is

取物理通量 σ_phys := −A∇u，守恆方程成為

∇·σ_phys = f₁ + δ_x₀.

A standard mixed method tests this equation against a scalar function in a Lebesgue space. The pairing with δ_x₀ would require a point value at x₀, which a general Lebesgue function does not possess.

標準混合方法用 Lebesgue 空間中的純量函數測試守恆方程。與 δ_x₀ 配對需要測試函數在 x₀ 的點值，但一般 Lebesgue 函數沒有這個點值。

What fails is duality, not approximation. The failure occurs at the continuous level, before any mesh, finite element space, or convergence estimate is introduced.真正失效的是對偶關係，而不是逼近。這個失效發生在連續層面，早於任何網格、有限元空間或收斂估計。

Move the Measure out of the Conservation Law 把測度移出守恆方程

Choose the explicit field · 選取顯式向量場

f₂(x) = (1/2π) (x−x₀)/|x−x₀|²,∇·f₂ = δ_x₀.

Then define the modified flux · 然後定義修正通量

σ := −A∇u − f₂.

The first-order system becomes · 一階系統便成為

A⁻¹σ + ∇u = −A⁻¹f₂,∇·σ = f₁.

The Dirac measure has disappeared from the conservation equation. The singular field remains only in the constitutive equation, where it is paired with a flux test function through ordinary L^p–L^p′ duality.

Dirac 測度已從守恆方程中消失。奇異向量場只留在本構方程中，並以普通的 L^p–L^p′ 對偶與通量測試函數配對。

A pairing that required continuity at one point has been replaced by a pairing that requires only integrability.原本需要測試函數在一個點連續的配對，被替換成只需要可積性的配對。

An Exact Reformulation, Not a Regularization 精確重寫，而不是正則化

Eliminating σ gives · 消去 σ，可得

−∇·(A∇u) = f₁ + ∇·f₂.

This is exactly the original equation because ∇·f₂ = δₓ₀. The continuous datum is not regularized, and no discrete delta is required. In computation, integrals of the singular field still have to be evaluated accurately.

因為 ∇·f₂ = δ_x₀，這正是原來的方程。原始數據沒有被正則化，也不需要構造離散 delta。實際計算時仍須準確積分這個奇異向量場。

Regularization routeReplace the Dirac source by a computable approximation and carry the data perturbation through the analysis.

把 Dirac 源替換成可計算的近似，並在分析中計入數據擾動誤差。

Divergence-splitting routeKeep the datum exact, rewrite it in divergence form, and absorb the difficulty into the flux norm.

保持原始數據精確不變，改寫成散度形式，並在通量範數中處理奇異性。

f₂ depends only on the source point x₀, not on the coefficient A.f₂ 只依賴源點 x₀，而不依賴係數 A。
No coefficient-dependent singular solution is required.不需要任何依賴係數的奇異解析解。
For the RT0–P0 discretization considered here, the matrix is the standard mixed matrix; only the right-hand side changes. Integrals involving the singular lifting must still be evaluated with appropriate care.對這裡的 RT0–P0 離散，矩陣仍是標準混合矩陣，只有右端項改變；涉及奇異 lifting 的積分仍須適當計算。
The source point may lie anywhere relative to the mesh, including at a vertex.源點可以位於網格的任何位置，包括網格頂點。

The Splitting Is Not Specific to Mixed Methods 這個分解並不只適用於混合方法

The divergence-form splitting is a reformulation of the differential equation itself, not a device tied to RT0–P0 or to any particular mixed discretization. Once

散度型分解是對微分方程本身的重寫，並不依附於 RT0–P0 或任何特定的混合離散。將

δ_x₀ = ∇·f₂

is inserted, the original measure-data problem is rewritten exactly as an elliptic problem with prescribed divergence-form data generated by an L^p field:

代入後，原來的測度數據問題便被精確寫成以 L^p 向量場的散度為數據的橢圓問題：

−∇·(A∇u) = f₁ + ∇·f₂.

This rewritten equation is available before any discretization is selected. Each numerical method can therefore use its own formulation and analysis for divergence-form data.

這個重寫在選擇任何離散方法之前就已成立。不同的數值方法可以針對散度型數據採用各自的格式與分析。

Mixed FEMAbsorb f₂ into the flux unknown. The conservation equation then has an ordinary Lebesgue-space right-hand side.

把 f₂ 吸收到通量未知量中，使守恆方程的右端回到普通的 Lebesgue 空間。

Conforming FEMUse the discrete action −(f₂,∇v_h) against suitable finite-element test functions. Since f₂ is not in L², this does not automatically define the usual weak functional on all of H¹; separate stability and consistency arguments are needed.

對適當的有限元測試函數使用 −(f₂,∇v_h)。由於 f₂ 不屬於 L²，這並不直接成為定義在整個 H¹ 上的通常弱泛函；仍須有相應的穩定性及一致性分析。

Discontinuous Galerkin / HDGInsert the divergence-form load into the method's volume and numerical-flux structure, while leaving the datum exact.

把散度型載荷納入體積項與 numerical flux 的結構，同時保留原始 Dirac 數據的精確表示。

CR and other nonconforming methodsUse the broken gradient pairing with f₂, together with the method's standard consistency treatment.

使用 f₂ 與 broken gradient 的配對，並配合方法本身的一致性處理。

Finite volume methodsInterpret the divergence lifting through flux balance over control volumes rather than through a discrete point source.

通過控制體的通量平衡處理散度 lifting，而不是構造離散點源。

Method-independent messageThe Dirac measure is replaced nowhere: it is represented exactly as the divergence of an explicit singular field.

Dirac 測度並未被近似替換，而是精確表示為一個顯式奇異向量場的散度。

Mixed FEM motivates the modified-flux variable, but the divergence-form splitting belongs to the PDE and is available to conforming, DG, HDG, CR/nonconforming, and finite-volume discretizations.混合有限元方法啟發了修正通量變量，但 divergence-form splitting 屬於 PDE 本身，同樣可供 conforming、DG、HDG、CR／非協調以及有限體積離散使用。

Exact divergence splitting does not automatically validate every standard variational formulation. For singular f₂, the test-space integrability, numerical fluxes, and evaluation of the singular load must be handled in the framework of the particular method.

精確的散度分解本身並不自動確保每一種標準變分格式都適用。對奇異的 f₂，測試空間的可積性、numerical flux 及奇異載荷的數值積分，須按照所採用的方法處理。

The price paid after the splitting depends on the method. For the mixed method analyzed here, the modified flux may leave L², so the flux error must be studied in L^p, 1 < p < 2. Other methods must likewise use, or develop, the appropriate theory for W^−1,p data in divergence form. What is universal is the exact reformulation; what is method-dependent is the subsequent stability and error analysis.

分解之後的分析取決於具體方法。對這裡的混合方法，修正通量可能不屬於 L²，因此通量誤差須在 1 < p < 2 的 L^p 尺度中研究。其他方法也需要相應的散度型 W^−1,p 數據理論。共同之處是精確的重寫；各方法不同的是後續的穩定性與誤差分析。

Why the Analysis Leaves the Hilbert Scale 為什麼分析必須離開 Hilbert 尺度

In two dimensions, δ_x₀ is not in H⁻¹(Ω). Equivalently, no square-integrable vector field can have divergence δ_x₀. The explicit lifting satisfies

在二維，δ_x₀ 不屬於 H⁻¹(Ω)。等價地，不存在散度等於 δ_x₀ 的平方可積向量場。這裡的顯式 lifting 滿足

f₂ ∈ L^p(Ω)² for every 1 < p < 2,f₂ ∉ L²(Ω)².

When the singular part of f₂ is not cancelled by the operator-dependent physical flux, the modified flux has the same status:

當 f₂ 的奇異部分沒有被依賴係數的物理通量抵消時，修正通量也具有相同的可積性：

σ ∈ L^p(Ω)² for 1 < p < 2,σ ∉ L²(Ω)².

The discrete method remains the classical RT0–P0 method. The exponent p enters the error analysis, not the algebraic system.離散方法仍然是經典的 RT0–P0 方法。指數 p 只進入誤差分析，不進入代數系統。

A Sharp Local Barrier 源點單元上的尖銳障礙

For a nonzero homogeneous |x−x₀|⁻¹ leading term, no RT0 field can approximate the singular flux on the pole element better than

若通量的主導項是非零的 |x−x₀|⁻¹ 齊次奇異項，那麼在包含源點的單元上，任何 RT0 函數的最佳逼近誤差都不可能優於

inf_{τ_h∈RT0} ‖σ−τ_h‖_L^p ≃ h^2/p−1.

The matching lower bound already comes from the single element carrying the pole. It is a property of the singularity and the local approximation space, not an artifact of the proof or the numerical method.

相應的下界僅從包含源點的那個單元就能得到。這是奇異性和局部逼近空間共同決定的，而不是證明技巧或數值方法造成的。

On a quasi-uniform mesh, the flux rate is decided by one element.在準一致網格上，通量收斂率由一個單元決定。

Grading Removes the Barrier 網格分級突破局部障礙

The barrier is local, so the remedy is local. Refining toward the pole and balancing the local contributions restores the first-order complexity of the lowest-order space:

障礙是局部的，因此加密也應針對局部。向源點作 graded refinement，並平衡各單元的誤差貢獻，就能恢復最低階空間的一階複雜度：

‖σ−σ_h‖_L^p(Ω) = O(N^−1/2).

one pole element fixes the quasi-uniform rate準一致網格上的通量誤差受源點單元支配↓shrink the pole scale and grade outward縮小源點尺度，向外逐級加密↓recover first-order complexity恢復一階複雜度

The scalar variable behaves differently. Under dual regularity, the mixed discretization component is first order, while the total scalar error is asymptotically the best piecewise-constant approximation of the logarithmic potential:

純量變量的情形不同。在對偶正則性假設下，混合離散誤差部分具有一階收斂；而純量總誤差的漸近階由對數奇異解的最佳分片常數逼近決定：

‖u−u_h‖_L²(Ω) ≃ h|log h|^1/2.

A Computable L^p Estimator 可計算的 L^p 誤差估計

A residual norm equivalence in the Lebesgue scale yields a computable estimator for the augmented flux–potential pair. For the pure Dirac problem, the conservation residual is zero and the estimator reduces to the constitutive residual.

Lebesgue 尺度下的殘差範數等價性給出 augmented 通量–勢變量對的可計算誤差估計量。對純 Dirac 問題，守恆殘差為零，估計量只剩本構殘差。

Reliable for the mixed flux.對混合通量具有 reliability。
Locally efficient for the augmented flux–potential pair.對 augmented 通量–勢變量對具有局部效率。
No bubble functions.不需要 bubble functions。
No jump terms.不需要 jump terms。
No flux reconstruction.不需要 flux reconstruction。

The Conceptual Chain 概念鏈條

Dirac source in the conservation law守恆方程含有 Dirac 源↓failure of duality in the scalar Lebesgue test space純量 Lebesgue 測試空間中的對偶配對失效↓exact divergence-form splitting精確的散度型分解↓standard mixed system with a modified flux採用修正通量的標準混合系統↓L^p analysis below the Hilbert exponent在 Hilbert 指數以下進行 L^p 分析

The Dirac source is represented exactly as the divergence of an L^p vector field. Absorbing that field into the mixed flux removes the Dirac measure from the conservation equation, without changing the original PDE.Dirac 源可以精確表示為一個 L^p 向量場的散度。將這個向量場吸收到混合通量中，守恆方程便不再直接含有 Dirac 測度，而原來的 PDE 完全不變。

Related Structure 相關的數據分解

For H⁻¹ data with an L² divergence lifting, the same modified-flux idea restores the standard L²-residual least-squares system. See the H⁻¹ LSFEM note.

對於能以 L² 向量場作散度分解的 H⁻¹ 數據，同樣的修正通量思想可以恢復標準的 L² 殘差最小二乘一階系統。參見 H⁻¹ LSFEM 專頁。

Core Paper 核心論文

Mixed Finite Element Methods for a Dirac Source: Divergence-Form Splitting and L^p Error Analysis

Dirac 源的混合有限元方法：散度型分解與 Lᵖ 誤差分析

Yueyao Wu and Shun Zhang, 2026.

arXiv · PDF · DOI