How can I use Block Forms, Multiphenicsx, and Lagrange Multipliers to Solve Poisson's Equation with Pure Neumann Boundary Conditions?

francesco-ballarin · December 4, 2023, 10:51am

The new tutorial in multiphenicsx about this topic is now available at

multiphenics/multiphenicsx/blob/main/tutorials/08_singular_poisson/tutorial_create_nullspace.ipynb

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 "cells": [
  {
   "cell_type": "markdown",
   "id": "675904d8",
   "metadata": {},
   "source": [
    "# Tutorial 08: creating nullspace vector for pure Neumann problem with restrictions\n",
    "\n",
    "In this tutorial we solve the problem\n",
    "\n",
    "$$\\begin{cases}\n",
    "-\\Delta u = f, & \\text{in } \\Omega,\\\\\n",
    "\\nabla u \\cdot\\mathbf{n} = g, & \\text{on } \\partial\\Omega,\n",
    "\\end{cases}$$\n",
    "\n",
    "where $\\Omega$ is a ball in 2D. The forcing term $f$ and the Neumann data $g$ are such that they satisfy the condition\n",
    "$$\\int_\\Omega f \\; dx = - \\int_{\\partial\\Omega} g \\; ds $$\n",
    "which is a necessary condition for the existence of the solution to this pure Neumann problem, and which can be easily obtained multiplying the first equation in the system by the constant 1 and integrating by parts. Note that the solution is determined up to a constant.\n",
    "\n",

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(joint work with @hherlyng )

Please keep in mind that the link above points to the notebook on the main branch. I do not backport new tutorials to branches related to older dolfinx versions. However, as of now (December 2023), I think that only a couple of minor changes are required for you to run that tutorial with dolfinx 0.6.0. Just download that tutorial and:

replace every dolfinx.fem.functionspace with dolfinx.fem.FunctionSpace,
drop the import of viskex, and every cell containing it. (I started working on viskex after the dolfinx 0.6.0 release).

(Future readers should be aware that items 1 and 2 may not necessarily apply in future, as the tutorial on the main branch may keep changing as dolfinx/multiphenicsx development continues)

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How can I use Block Forms, Multiphenicsx, and Lagrange Multipliers to Solve Poisson's Equation with Pure Neumann Boundary Conditions?

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