[Paper Review] Green's functions for Neumann boundary conditions

Jerrold Franklin|arXiv (Cornell University)|Jan 29, 2012

Advanced Mathematical Modeling in Engineering1 references3 citations

TL;DR

This paper derives a rigorous Neumann Green's function for partial differential equations with Neumann boundary conditions, correcting widespread textbook omissions. It introduces a modified Green's function satisfying $\mathcal{L}_{\mathbf{r}'}G_N(\mathbf{r},\mathbf{r}') = \delta(\mathbf{r}-\mathbf{r}') - 1/V$, ensuring compatibility with the solvability constraint $\int \mathbf{dS} \cdot p(\mathbf{r}) \nabla\phi(\mathbf{r}) = \int \rho \, d\tau$, and provides a solution formula involving an arbitrary constant average value $\langle \phi \rangle$. The method ensures consistency for finite-volume problems and resolves inconsistencies in prior approaches.

ABSTRACT

Green's functions for Neumann boundary conditions have been considered in Math Physics and Electromagnetism textbooks, but special constraints and other properties required for Neumann boundary conditions have generally not been noticed or treated correctly. In this paper, we derive an appropriate Neumann Green's function with these constraints and properties incorporated.

Motivation & Objective

• To correct widespread textbook inaccuracies in the treatment of Neumann Green's functions, particularly the omission of critical solvability constraints.
• To derive a Neumann Green's function that satisfies both the inhomogeneous differential equation $\mathcal{L}_{\mathbf{r}'}G_N = \delta(\mathbf{r}-\mathbf{r}') - 1/V$ and homogeneous Neumann boundary conditions.
• To provide a consistent solution framework for Neumann problems in finite volumes, where the solution includes an arbitrary constant $\langle \phi \rangle$ equal to the average of $\phi$.
• To demonstrate that the standard approach of imposing a constant normal derivative on $G_N$ fails for non-symmetric geometries, unlike the proposed eigenfunction-based method.

Proposed method

• Derives the Neumann Green's function using a complete set of Neumann eigenfunctions, ensuring orthogonality and completeness in the expansion.
• Incorporates the $-1/V$ term in the differential equation to satisfy the solvability constraint $\int \mathbf{dS} \cdot p(\mathbf{r}) \nabla\phi(\mathbf{r}) = \int \rho \, d\tau$, which is essential for existence of solutions.
• Applies Green's theorem to the modified Green's function to derive the solution formula: $\phi(\mathbf{r}) = \langle \phi \rangle + \int G_N \rho \, d\tau' - \int \mathbf{dS}' \cdot G_N p \nabla' \phi$.
• Ensures the Green's function satisfies homogeneous Neumann boundary conditions $\mathbf{\hat{n}}' \cdot \nabla' G_N = 0$ on all bounding surfaces.
• Demonstrates that the Neumann Green's function is not necessarily symmetric, but can be made symmetric by adding a function of $\mathbf{r}$, which does not affect its action.
• Contrasts the proposed method with the flawed alternative approach that imposes $\mathbf{\hat{n}}' \cdot \nabla' G = 1/S$, showing it fails for general geometries like rectangular boxes.

Experimental results

Research questions

• RQ1 Why do standard textbook treatments of Neumann Green's functions fail to account for the solvability constraint $\int \mathbf{dS} \cdot p \nabla\phi = \int \rho \, d\tau$?
• RQ2 Can a Neumann Green's function be consistently defined for finite domains that satisfies both the inhomogeneous equation $\mathcal{L}_{\mathbf{r}'}G_N = \delta(\mathbf{r}-\mathbf{r}') - 1/V$ and homogeneous boundary conditions?
• RQ3 Why does the method of imposing a constant normal derivative $\mathbf{\hat{n}} \cdot \nabla G = 1/S$ fail for non-symmetric geometries such as rectangular parallelepipeds?
• RQ4 How does the presence of the $-1/V$ term in the Green's function equation ensure compatibility with the constraint for Neumann problems?
• RQ5 What is the role of the arbitrary constant $\langle \phi \rangle$ in the solution, and how is it related to the average value of $\phi$ in the domain?

Key findings

• The Neumann Green's function must satisfy $\mathcal{L}_{\mathbf{r}'}G_N(\mathbf{r},\mathbf{r}') = \delta(\mathbf{r}-\mathbf{r}') - 1/V$ to ensure compatibility with the solvability condition for Neumann problems.
• The solution to the Neumann problem includes an arbitrary constant $\langle \phi \rangle$, which equals the spatial average of $\phi$ over the volume, and can be set to zero for simplification.
• The proposed Green's function satisfies homogeneous Neumann boundary conditions $\mathbf{\hat{n}}' \cdot \nabla' G_N = 0$ on all bounding surfaces, ensuring consistency with the physical requirement of zero net flux for the Green's function.
• The standard method of imposing a constant normal derivative $\mathbf{\hat{n}} \cdot \nabla G = 1/S$ fails for general geometries, as it leads to incompatible requirements on derivatives and cannot be satisfied by a smooth function.
• The Neumann Green's function is not inherently symmetric, but can be made symmetric by adding a function of $\mathbf{r}$, which does not alter its action in the solution formula.
• The $-1/V$ term arises naturally from the expansion in Neumann eigenfunctions and is essential for the interior problem (finite volume), vanishing only in the exterior problem (infinite domain).