pbm formulation

Author: campospinto

.gitignore

@@ -19,3 +19,5 @@ usr
 .env
 .iga-python
 *_autosummary*
+
+.DS_Store

exercises/harmonic-feec.ipynb

@@ -264,11 +264,7 @@
    ]
   }
  ],
- "metadata": {
-  "language_info": {
-   "name": "python"
-  }
- },
+ "metadata": {},
  "nbformat": 4,
  "nbformat_minor": 5
 }

exercises/harmonic-fem.ipynb

@@ -7,12 +7,13 @@
    "source": [
     "# Harmonic fields with H1 fem spaces\n",
     "\n",
-    "In this example we .... consider the vector Poisson equation with homogeneous Dirichlet boundary conditions:\n",
+    "In this exercise we consider the following vector problem with homogeneous boundary conditions in H(curl):\n",
     "\n",
     "$$\n",
     "\\begin{align}\n",
-    "  - \\nabla^2 \\mathbf{u} = \\mathbf{f} \\quad \\mbox{in} ~ \\Omega, \\quad \\quad \n",
-    "  \\mathbf{u} = 0            \\quad \\mbox{on} ~ \\partial \\Omega.\n",
+    "    \\nabla \\times \\mathbf{u} = 0 \\quad \\mbox{in} ~ \\Omega \\\\\n",
+    "    \\nabla \\cdot \\mathbf{u} = 0 \\quad \\mbox{in} ~ \\Omega \\\\\n",
+    "    \\mathbf{n} \\times \\mathbf{u} = 0            \\quad \\mbox{on} ~ \\partial \\Omega.\n",
     "\\end{align}\n",
     "$$\n",
     "\n",
@@ -23,15 +24,24 @@
     "$$\n",
     "\\begin{align}\n",
     "  \\text{find $\\mathbf{u} \\in V$ such that} \\quad \n",
-    "  a(\\mathbf{u},\\mathbf{v}) = l(\\mathbf{v}) \\quad \\forall \\mathbf{v} \\in V,\n",
+    "  a(\\mathbf{u},\\mathbf{v}) = 0 \\quad \\forall \\mathbf{v} \\in V,\n",
     "\\end{align}\n",
     "$$\n",
     "\n",
     "where \n",
     "\n",
-    "- $V \\subset \\mathbf{H}_0^1(\\Omega)$, \n",
-    "- $a(\\mathbf{u},\\mathbf{v}) := \\int_{\\Omega} \\nabla \\mathbf{u} : \\nabla \\mathbf{v} ~ d\\Omega$,\n",
-    "- $l(\\mathbf{v}) := \\int_{\\Omega} \\mathbf{f} \\cdot \\mathbf{v} ~ d\\Omega$."
+    "- $V \\subset (\\mathbf{H}^1(\\Omega))^2$, \n",
+    "- $a := a_{\\rm curl} + a_{\\rm div} + a_{\\rm bc}$\n",
+    "- $a_{\\rm curl}(\\mathbf{u},\\mathbf{v}) = \\int_{\\Omega} (\\nabla \\times \\mathbf{u}) \\cdot (\\nabla \\times \\mathbf{v})$,\n",
+    "- $a_{\\rm div}(\\mathbf{u},\\mathbf{v}) = \\int_{\\Omega} (\\nabla \\cdot \\mathbf{u}) (\\nabla \\cdot \\mathbf{v})$,\n",
+    "- $a_{\\rm bc}(\\mathbf{u},\\mathbf{v}) = \\kappa \\int_{\\partial\\Omega} (\\mathbf{n} \\times \\mathbf{u}) \\cdot (\\mathbf{n} \\times \\mathbf{v})$.\n",
+    "\n",
+    "with $\\kappa$ a large penalization constant.\n",
+    "\n",
+    "The exercise is to assemble the matrix of $a$ and compute the first eigenmode using scipy.\n",
+    "\n",
+    "The same problem will be addressed in another notebook with structure preserving finite elements, where the solution will only be in H(curl)\n",
+    "and the divergence-free equation will only be imposed in a weak sense.\n"
    ]
   },
   {
@@ -264,11 +274,7 @@
    ]
   }
  ],
- "metadata": {
-  "language_info": {
-   "name": "python"
-  }
- },
+ "metadata": {},
  "nbformat": 4,
  "nbformat_minor": 5
 }