Update documentation

_sources/saddlepoint/fewconstraints.ipynb

@@ -369,9 +369,10 @@
     "* instead of applying the preconditioner $\\hat A$ twice, one could store the matrix $\\hat A^{-1} B$\n",
     "\n",
     "* the Lagrange parameter $p$ can be computed by multiplying the equation $A u + B^T p = f$ from left with $B \\hat A^{-1}$, and then solving with the inexact Schur complement\n",
-    "$$\n",
-    "p = (B \\hat A^{-1} B^T)^{-1} (B \\hat A^{-1} (f - A u))\n",
-    "$$"
+    " \n",
+    "  $$\n",
+    "  p = (B \\hat A^{-1} B^T)^{-1} (B \\hat A^{-1} (f - A u))\n",
+    "  $$"
    ]
   },
   {
@@ -385,7 +386,7 @@
  ],
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-   "display_name": "Python 3",
+   "display_name": "Python 3 (ipykernel)",
    "language": "python",
    "name": "python3"
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@@ -399,7 +400,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.4"
+   "version": "3.12.2"
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_sources/saddlepoint/parameterdependent.ipynb

@@ -322,6 +322,7 @@
     "$$\n",
     "\n",
     "The operator-norm of the prolongation is\n",
+    "\n",
     "$$\n",
     "\\| P \\|_{(V_H, A_H^\\varepsilon) \\rightarrow (V_h, A_h^\\varepsilon)}\n",
     " = \\sup_{v_H} \\frac{ \\| P v_H \\|_{A_h^\\varepsilon}}{\\| v_H \\|_{A_H^\\varepsilon}}\n",
@@ -340,7 +341,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "induced-salon",
+   "id": "63f52087-cb49-4f3d-82b9-8bb4597e4f4a",
    "metadata": {},
    "source": [
     "### Two-level analysis for Maxwell equations\n",
@@ -378,9 +379,14 @@
     "\n",
     "On the other hand, every function from $[H^1]^3$, and every gradient from $H^1$ is in $H(\\operatorname{curl})$.\n",
     "\n",
-    "*Proof:* Pasciak, J. E.; Zhao, J.: Overlapping Schwarz methods in H(curl) on polyhedral domains. J. Numer. Math. 10 (2002)\n",
-    "\n",
-    "\n",
+    "*Proof:* Pasciak, J. E.; Zhao, J.: Overlapping Schwarz methods in H(curl) on polyhedral domains. J. Numer. Math. 10 (2002)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b6f3c533-0b8f-4249-bb5c-148c86faee01",
+   "metadata": {},
+   "source": [
     "**Theorem: (Commuting quasi-interpolation operators)** There exist projections $\\pi_{V_h} : H(\\operatorname{curl}) \\rightarrow V_h $ and $\\pi_{W_h} : H^1 \\rightarrow W_h$ which commute with the gradient:\n",
     "\n",
     "$$\n",
@@ -389,15 +395,12 @@
     "\n",
     "They are stable in norms and semi-norms, and provide optimal approximation error estimates.\n",
     "\n",
-    "*Proof:* Schöberl, A multilevel decomposition result in H(curl). in Multigrid, Multilevel and Multiscale Methods, EMG 2005 \n",
-    "\n",
-    "\n",
-    "\n"
+    "*Proof:* Schöberl, A multilevel decomposition result in H(curl). in Multigrid, Multilevel and Multiscale Methods, EMG 2005 "
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "grand-stopping",
+   "id": "b6d4fa0f-b365-4803-8b8e-2819c816f843",
    "metadata": {},
    "source": [
     "We are ready to prove that the condition-number of the two-level preconditioner is robust in the mesh-size $h$, as well as the regularization parameter $\\varepsilon$. We assume that $H = ch$. We transfer results from the $H^1$-case to the $H(\\operatorname{curl})$ case.\n",
@@ -417,10 +420,12 @@
     "$$\n",
     "\n",
     "where\n",
+    "\n",
     "$$\n",
     "\\| r \\|_{L_2}^2 \\preceq h^2 \\| \\operatorname{curl} u \\|^2\n",
     "$$\n",
     "and\n",
+    "\n",
     "$$\n",
     "\\varepsilon \\| \\psi \\|_{L_2}^2 \\preceq h^2 \\| u \\|_{A^\\varepsilon}^2\n",
     "$$\n",
@@ -437,6 +442,7 @@
     "\\| z \\|_{H^1}^2 \\preceq \\| \\operatorname{curl} u \\|_{L_2}^2\n",
     "$$\n",
     "and \n",
+    "\n",
     "$$\n",
     "\\varepsilon \\| \\Phi \\|_{H^1}^2 \\preceq \\| u \\|_{A^\\varepsilon}^2\n",
     "$$\n",
@@ -454,11 +460,14 @@
     "r = z - \\pi_{V_H} z \\qquad \\text{and} \\qquad \\psi = \\Phi - \\pi_{W_H} \\Phi\n",
     "$$\n",
     "\n",
-    "and using the approximation properties of the interpolation operators we obtain the splitting.\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
+    "and using the approximation properties of the interpolation operators we obtain the splitting."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d35bf357-a695-4567-a2eb-554e86be6619",
+   "metadata": {},
+   "source": [
     "**Lemma:** For the coarse-grid interpolation error there exists a discrete decomposition\n",
     "\n",
     "$$\n",