Update documentation

_sources/time_integration/mixed_methods_py.ipynb

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "3aa9653d",
+   "metadata": {},
+   "source": [
+    "# Mixed methods in NGSolve"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "44eefa62",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from ngsolve import *\n",
+    "from ngsolve.webgui import Draw"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "781071c3",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "7211179e9b3848d4ac27a4ae95d392de",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "WebGuiWidget(value={'gui_settings': {}, 'ngsolve_version': '6.2.2402-37-g45ed24bde', 'mesh_dim': 2, 'order2d':…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/plain": [
+       "BaseWebGuiScene"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "mesh = Mesh(unit_square.GenerateMesh(maxh=0.3))\n",
+    "Draw(mesh)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a7932da4",
+   "metadata": {},
+   "source": [
+    "## Product spaces\n",
+    "In NGSolve spaces maybe tensorized by using the `*` symbol"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "f4193eee",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "h1 dofs:  61\n",
+      "l2 dofs:  144\n",
+      "total dofs:  205  =  205\n"
+     ]
+    }
+   ],
+   "source": [
+    "order = 2\n",
+    "fesh1 = H1(mesh, order = order)\n",
+    "print(\"h1 dofs: \", fesh1.ndof)\n",
+    "fesl2 = VectorL2(mesh, order = order-1)\n",
+    "print(\"l2 dofs: \", fesl2.ndof)\n",
+    "\n",
+    "fes = fesh1*fesl2\n",
+    "print(\"total dofs: \", fes.ndof,\" = \", fesh1.ndof+fesl2.ndof)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "21f78dbb",
+   "metadata": {},
+   "source": [
+    "To assemble bilinear forms we need test and trial functions. From product spaces test and trial functions are returned as tuples:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "9ca83c3a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "(p,v), (q,w) = fes.TnT()\n",
+    "bfgrad = BilinearForm(grad(p)*w*dx).Assemble()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "40449cbf",
+   "metadata": {},
+   "source": [
+    "The matrix is still an operator on the large space!"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "461940c7",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<matplotlib.image.AxesImage at 0x7d8fdbca3af0>"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    },
+    {
+     "data": {
+      "image/png": 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/gKQnY+BFSTfZ3jpPp6bV5GtdiaruNoJFbQ+Y90pUddaPBNo21zYG27dL+qiklyRtiYjL6aG3JG1J97dJerPysoupDUBPzBwMtj8k6buSPh8Rv6o+FoM/k3P9qbV9yPaK7ZXV1dV5XgqgZTMFg+0PaBAK34mI76XmK8MSId1eTe2XJG2vvPy21PZHIuJIRCxHxPLS0tLcHR93NGFJZikRSjtiExiaZa+EJT0h6dWI+GbloeOSDqb7ByU9V2l/MO2d2CvpnUrJAaAHZjnt+q8kfVrSy7ZPp7YvSfqapGdsPyTpDUn3pcdOSNov6bykX0v6zCI7DKB9LmEqa3tV0v9K+mXXfVmwW8WY+uBaGdOfRcRMdXsRwSBJtlciYrnrfiwSY+oHxpTjkGgAGYIBQKakYDjSdQdawJj6gTGNKGYbA4BylDRjAFAIggFAhmAAkCEYAGQIBgCZ/wd6orNo8zYTPgAAAABJRU5ErkJggg==\n",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as pl\n",
+    "pl.spy(bfgrad.mat.ToDense())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3380c813",
+   "metadata": {},
+   "source": [
+    "Gridfunctions on product spaces consist of components"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "17f55a9d",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "length of whole vector:  205\n",
+      "length of first component:  61\n",
+      "length of second component:  144\n"
+     ]
+    }
+   ],
+   "source": [
+    "gfu = GridFunction(fes)\n",
+    "print(\"length of whole vector: \",len(gfu.vec))\n",
+    "gfup = gfu.components[0]\n",
+    "print(\"length of first component: \",len(gfup.vec))\n",
+    "gfuv = gfu.components[1]\n",
+    "print(\"length of second component: \",len(gfuv.vec))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e7e5ba85",
+   "metadata": {},
+   "source": [
+    "Restriction and embedding operators are available:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "c314b8ac",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "(205, 61)\n",
+      "(205, 144)\n",
+      "(61, 205)\n",
+      "(144, 205)\n"
+     ]
+    }
+   ],
+   "source": [
+    "emb_p = fes.Embedding(0)\n",
+    "emb_v = fes.Embedding(1)\n",
+    "\n",
+    "print(emb_p.shape)\n",
+    "print(emb_v.shape)\n",
+    "\n",
+    "res_p = fes.Restriction(0)\n",
+    "res_v = fes.Restriction(1)\n",
+    "\n",
+    "print(res_p.shape)\n",
+    "print(res_v.shape)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "127988c2",
+   "metadata": {},
+   "source": [
+    "## Mixed operators\n",
+    "\n",
+    "Alternatively one may define mixed operators. Note that in this case the test and trial functions must be obtained from the base spaces."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 27,
+   "id": "f829c585",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fesh1 = H1(mesh,order=order)\n",
+    "feshdiv = HDiv(mesh,order=order)\n",
+    "\n",
+    "p_,q_ = fesh1.TnT()\n",
+    "v_,w_ = feshdiv.TnT()\n",
+    "\n",
+    "bfgrad = BilinearForm(grad(p_)*w_*dx).Assemble()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 28,
+   "id": "82c547e2",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<matplotlib.image.AxesImage at 0x7d8fd31fd3f0>"
+      ]
+     },
+     "execution_count": 28,
+     "metadata": {},
+     "output_type": "execute_result"
+    },
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "pl.spy(bfgrad.mat.ToDense())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "07678ea5",
+   "metadata": {},
+   "source": [
+    "Matrices may also be transposed:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "id": "7e840338",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "6.409741197293665e-15\n"
+     ]
+    }
+   ],
+   "source": [
+    "n = specialcf.normal(2)\n",
+    "bfdiv = BilinearForm(-div(v_)*q_*dx+v_.Trace()*n*q_*ds).Assemble()\n",
+    "print((bfdiv.mat.T.ToDense()-bfgrad.mat.ToDense()).Norm())"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0e334862",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}