add notebook example mixed

doc/notebooks/embTrefftz.ipynb

@@ -172,6 +172,14 @@
     "    f.Assemble()"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "4f1de24c",
+   "metadata": {},
+   "source": [
+    "### Trefftz embedding"
+   ]
+  },
   {
    "cell_type": "markdown",
    "id": "19d5d04a",
@@ -235,6 +243,61 @@
     "error = sqrt(Integrate((tgfu-exact)**2, mesh))\n",
     "print(\"embedded Trefftz error \",error)"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8f34efc5",
+   "metadata": {},
+   "source": [
+    "### Trefftz embedding"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c02fb15b",
+   "metadata": {},
+   "source": [
+    "We can also construct the embedding by finding the kernel of\n",
+    "$$\n",
+    "\\begin{align} \\label{def:matrices}\n",
+    "    (\\bW)_{ij}&=\\inner{\\calL\\phi_j,\\tilde\\phi_i}_{0,h},\\qquad \\forall \\tilde\\phi_i\\in V^{p-2}(\\Th)\n",
+    "\\end{align}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "558432aa",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "eps=10**-9\n",
+    "Lap = lambda u : sum(Trace(u.Operator('hesse')))\n",
+    "u = fes.TrialFunction()\n",
+    "fes2 = L2(mesh, order=order-2,  dgjumps=True)\n",
+    "v = fes2.TestFunction()\n",
+    "op = Lap(u)*v*dx\n",
+    "with TaskManager():\n",
+    "    PP = TrefftzEmbedding(op,fes,test_fes=fes2)\n",
+    "PPT = PP.CreateTranspose()\n",
+    "with TaskManager():\n",
+    "    TA = PPT@a.mat@PP\n",
+    "    TU = TA.Inverse(inverse='sparsecholesky')*(PPT*f.vec)\n",
+    "    tgfu = GridFunction(fes)\n",
+    "    tgfu.vec.data = PP*TU"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c58dd2a5",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "error = sqrt(Integrate((tgfu-exact)**2, mesh))\n",
+    "print(\"embedded Trefftz error \",error)"
+   ]
   }
  ],
  "metadata": {