Update documentation

_sources/multigrid/analysisMG.ipynb

@@ -87,7 +87,7 @@
     "\\| (I - P_l A_{l-1}^{-1} P_l^T A_l) u_l \\|_{L_2} \\preceq h \\, \\| u_l \\|_{H^1}\n",
     "$$\n",
     "\n",
-    "We can prove this similar to the Nitsche-lemma using full elliptic regularity.\n",
+    "We can prove this similar to the Aubin-Nitsche lemma using full elliptic regularity.\n",
     "\n",
     "We introduce the coarse grid projector\n",
     "\n",
@@ -96,20 +96,20 @@
     "$$\n",
     "\n",
     "\n",
-    "**Theorem [Approximation property]** There holds with a constant $C > 0$ independent of the refinement level:\n",
-    "\n",
-    "$$\n",
-    "\\forall \\, u_l \\in V_l : \\qquad \n",
-    "\\| u_l - \\Pi_{l-1} u_l \\|_{D_l}^2 \\leq C \\, \\| u_l - \\Pi_{l-1} u_l \\|_{A_l}^2\n",
-    "$$\n",
+    "> **Theorem [Approximation property]** There holds with a constant $C > 0$ independent of the refinement level:\n",
+    ">\n",
+    "> $$\n",
+    "  \\forall \\, u_l \\in V_l : \\qquad \n",
+    "  \\| u_l - \\Pi_{l-1} u_l \\|_{D_l}^2 \\leq C \\, \\| u_l - \\Pi_{l-1} u_l \\|_{A_l}^2\n",
+    "  $$\n",
     "\n",
     "*Proof:* We write $u_l$ for the coefficient vector as well as for the corresponding finite element function in $V_l$. For a given $u_l$, define the $A(.,.)$-orthogonal projection $u_{l-1} \\in V_{l-1}$ via\n",
     "\n",
     "$$\n",
     "A(u_{l-1}, v_{l-1}) = A(u_l, v_{l-1}) \\qquad \\forall \\, v_{l-1} \\in V_{l-1}\n",
     "$$\n",
     "\n",
-    "The error $e_l = u_l - u_{l-1}$ is $A(.,.)$-orthogonal to $V_{l-1}$. As in the Nitsche-Lemma we define an artificial problem with the error as right hand side: find $\\varphi \\in H^1$ such that\n",
+    "The error $e_l = u_l - u_{l-1}$ is $A(.,.)$-orthogonal to $V_{l-1}$. As in the Aubin-Nitsche Lemma we define an artificial problem with the error as right hand side: find $\\varphi \\in H^1$ such that\n",
     "\n",
     "$$\n",
     "A(\\varphi, \\psi) = (e_l, \\psi)_{L_2} \\qquad \\forall \\, \\psi \\in H^1.\n",
@@ -188,7 +188,6 @@
    "source": [
     "from ngsolve import *\n",
     "from ngsolve.webgui import Draw\n",
-    "from netgen.geom2d import unit_square\n",
     "mesh = Mesh(unit_square.GenerateMesh(maxh=0.02))\n",
     "\n",
     "fes = H1(mesh, order=1)\n",
@@ -222,12 +221,14 @@
    "id": "public-hebrew",
    "metadata": {},
    "source": [
-    "**Theorem [Smoothing property]**\n",
-    "$$\n",
-    "\\| (I - D^{-1} A)^m u \\|_{A D^{-1} A}^2 \\leq \\frac{1}{2m+1} \\| u \\|_A^2\n",
-    "$$\n",
+    "> **Theorem [Smoothing property]**\n",
+    ">\n",
+    ">  $$\n",
+    "  \\| (I - D^{-1} A)^m u \\|_{A D^{-1} A}^2 \\leq \\frac{1}{2m+1} \\| u \\|_A^2\n",
+    "  $$\n",
     "\n",
     "*Proof:* We use spectral theory. Expanding $u$ in the eigensystem of $A z = \\lambda D z$ we can reformulate the inequality as\n",
+    "\n",
     "$$\n",
     "\\sum \\lambda_i^2 (1 - \\lambda_i)^{2m} u_i^2 \\leq \\frac{1}{2m+1} \\sum \\lambda_i u_i^2\n",
     "$$\n",
@@ -261,13 +262,14 @@
    "id": "broad-transport",
    "metadata": {},
    "source": [
-    "**Lemma [Improved smoothing property]**\n",
-    "\n",
-    "$$\n",
-    "\\| (I - D^{-1} A)^m u \\|_{A D^{-1} A}^2 \\leq \\frac{1}{2m} \\left( \\| u \\|_A^2 - \\| (I-D^{-1} A)^m u \\|_A^2 \\right)\n",
-    "$$\n",
+    "> **Lemma [Improved smoothing property]**\n",
+    ">\n",
+    "> $$\n",
+    "  \\| (I - D^{-1} A)^m u \\|_{A D^{-1} A}^2 \\leq \\frac{1}{2m} \\left( \\| u \\|_A^2 - \\| (I-D^{-1} A)^m u \\|_A^2   \\right)\n",
+    "  $$\n",
     "\n",
     "*Proof:* Very similar, we have to verify\n",
+    "\n",
     "$$\n",
     "\\lambda^2 (1-\\lambda)^{2m} \\leq \\frac{1}{2m} \\left( \\lambda - \\lambda (1-\\lambda)^{2m} \\right)\n",
     "$$"
@@ -298,20 +300,20 @@
     "\n",
     "We can now combine these properties to analyze the multigrid iteration:\n",
     "\n",
-    "**Theorem:** Assume that the approximation property and the smoothing property are satisfied. Then there holds\n",
+    "> **Theorem:** Assume that the approximation property (with constant $C$) and the smoothing property are satisfied. Then there holds\n",
+    ">\n",
+    "> $$\n",
+    "  \\| M_l \\|_A \\leq \\delta \n",
+    "  $$\n",
+    ">\n",
+    "> with \n",
+    ">\n",
+    "> $$\n",
+    "  \\delta = \\frac{C}{C+2m} < 1.\n",
+    "  $$\n",
     "\n",
-    "$$\n",
-    "\\| M_l \\|_A \\leq \\delta \n",
-    "$$\n",
     "\n",
-    "with \n",
-    "\n",
-    "$$\n",
-    "\\delta = \\frac{C}{C+2m} < 1.\n",
-    "$$\n",
-    "\n",
-    "\n",
-    "*Proof*: We prove the claim per induction. For $l=$ there holds $M_0 = 0$.\n",
+    "*Proof*: We prove the claim per induction. For $l=0$ there holds $M_0 = 0$.\n",
     "\n",
     "From $M_{l-1} = I - C_{l-1}^{-1} A_{l-1}$ there follows $C_{l-1}^{-1} = (I - M_{l-1}) A_{l-1}^{-1}$. \n",
     "\n",
@@ -361,6 +363,14 @@
     "\n",
     "We used that by the choice of $\\delta$ the factor $\\delta - \\frac{(1-\\delta) C}{2m}$ is positive. The proof is complete."
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2057ef66-ba2f-44a6-8ac4-ca8e7670e864",
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {

_sources/multigrid/multilevel-extension.ipynb

@@ -46,11 +46,11 @@
     "\n",
     "which is a decomposition $u = u_0 + \\sum w_l$.\n",
     "\n",
-    "**Theorem** There holds\n",
-    "\n",
-    "$$\n",
-    "\\| u_0 \\|_{H^1}^2 + \\sum_{l=1}^L h_l^{-2} \\| w_l \\|_{L_2}^2 \\preceq L^2 \\, \\| u \\|_{H^1}^2\n",
-    "$$\n",
+    "> **Theorem** There holds\n",
+    ">\n",
+    "> $$ \n",
+    "  \\| u_0 \\|_{H^1}^2 + \\sum_{l=1}^L h_l^{-2} \\| w_l \\|_{L_2}^2 \\preceq L^2 \\, \\| u \\|_{H^1}^2\n",
+    "  $$\n",
     "\n",
     "*Proof:* We use that there exists a stable decomposition $u = \\tilde u_0 + \\sum_{k=1}^L \\tilde w_k$. The computable choice is\n",
     "\n",
@@ -93,7 +93,7 @@
    "source": [
     "Algorithm\n",
     "---\n",
-    "The quasi-interpolant of $u in V_L$ to $u_l \\in V_l$,\n",
+    "The quasi-interpolant of $u \\in V_L$ to $u_l \\in V_l$,\n",
     "\n",
     "$$\n",
     "\\Pi_l u := \\sum_{i=1}^{N_l} \\frac{ (u, \\varphi_{l,i})_{L_2} } { (1, \\varphi_{l,i} )_{L_2} } \\varphi_{l,i}\n",
@@ -130,7 +130,6 @@
    "outputs": [],
    "source": [
     "from ngsolve import *\n",
-    "from netgen.geom2d import unit_square\n",
     "from ngsolve.webgui import Draw"
    ]
   },
@@ -160,9 +159,9 @@
    "outputs": [],
    "source": [
     "bnd = mesh.Boundaries(\"left|bottom\")\n",
-    "gfu.Set (1-x-y+0.3*sin(30*x), definedon=bnd)\n",
+    "gfu.Set (1-x-y+0.3*sin(10*pi*x), definedon=bnd)\n",
     "\n",
-    "Draw (gfu)\n",
+    "Draw (gfu, deformation=True)\n",
     "print (\"Norm(u) = \", InnerProduct((a.mat*gfu.vec).Evaluate(), gfu.vec))"
    ]
   },
@@ -229,16 +228,14 @@
    "cell_type": "code",
    "execution_count": null,
    "id": "advance-wages",
-   "metadata": {
-    "scrolled": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
-    "bndmass = BilinearForm(u*v*ds(bnd)).Assemble().mat\n",
+    "bndmass = BilinearForm(u*v*ds(bnd), check_unused=False).Assemble().mat\n",
     "ext = MLExtension(fes, fes.mesh.levels-1, bndmass, fes.GetDofs(bnd))\n",
     "ext.Extend(gfu.vec)\n",
     "\n",
-    "Draw (gfu)\n",
+    "Draw (gfu, deformation=True)\n",
     "print (\"Norm(uext) = \", InnerProduct(a.mat*gfu.vec, gfu.vec))"
    ]
   },
@@ -261,7 +258,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3",
+   "display_name": "Python 3 (ipykernel)",
    "language": "python",
    "name": "python3"
   },
@@ -275,7 +272,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.4"
+   "version": "3.12.2"
   }
  },
  "nbformat": 4,