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DG/splitting.html

@@ -475,7 +475,7 @@ <h1><span class="section-number">22. </span>Splitting Methods for the time-depen
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-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;ngsolve.comp.LinearForm at 0x1100b2270&gt;
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_sources/abstracttheory/Coercive.ipynb

@@ -15,10 +15,10 @@
     "A(u,u) \\geq \\alpha_1 \\| u \\|_V^2 \\qquad \\forall \\, u \\in V,\n",
     "\\end{equation}\n",
     "* and continuous\n",
+    "\n",
     "\\begin{equation}\n",
     "A(u,v) \\leq \\alpha_2 \\| u \\|_V  \\, \\| v \\|_V \\qquad \\forall \\, u , v \\in V,\n",
     "\\end{equation}\n",
-    "\\end{itemize}\n",
     "\n",
     "with bounds $\\alpha_1$ and $\\alpha_2$ in ${\\mathbb R}^+$. It is not necessarily symmetric.\n",
     "Let $f(.) : V \\rightarrow {\\mathbb R}$ be a continuous linear form on $V$, i.e., \n",
@@ -78,8 +78,9 @@
    ]
   },
   {
+   "attachments": {},
    "cell_type": "markdown",
-   "id": "e10c25d4-d37f-4d63-86be-c7ce12a01bf7",
+   "id": "9c0196f8-5315-4bf6-8474-5f2af7184258",
    "metadata": {},
    "source": [
     "Instead of the linear form $f(\\cdot)$, we will often write $f \\in V^\\ast$. The evaluation is written as the duality product \n",
@@ -88,7 +89,7 @@
     "\\left< f , v \\right>_{V^\\ast \\times V} = f(v).\n",
     "$$\n",
     "\n",
-    "**Lemma:** A continuous bilinear form $A(\\cdot,\\cdot) : V \\times V \\rightarrow \\setR$\n",
+    "**Lemma:** A continuous bilinear form $A(\\cdot,\\cdot) : V \\times V \\rightarrow {\\mathbb R}$\n",
     "induces a continuous linear operator $A : V \\rightarrow V^\\ast$ via\n",
     "\n",
     "$$\n",
@@ -106,95 +107,122 @@
     "$$\n",
     "Thus, we can define the operator $A : u \\in V \\rightarrow A(u,\\cdot) \\in V^\\ast$. \n",
     "It is linear, and its operator norm is bounded by\n",
-    "\\begin{eqnarray*}\n",
-    "\\| A \\|_{V \\rightarrow V^\\ast} & = &\n",
+    "\\begin{align*}\n",
+    "\\| A \\|_{V \\rightarrow V^\\ast} &= \n",
     "\\sup_{u\\in V} \\frac{ \\| A u \\|_{V^\\ast}}{\\| u \\|_V} =\n",
     "\\sup_{u\\in V} \\sup_{v \\in V} \\frac{ \\left< A u, v\\right>_{V^\\ast \\times V} } {\\| u \\|_V \\, \\| v \\|_V} \\\\\n",
-    "& = & \\sup_{u\\in V} \\sup_{v \\in V} \\frac{ A (u,v) } {\\| u \\|_V \\, \\| v \\|_V} \\leq\n",
+    "&= \\sup_{u\\in V} \\sup_{v \\in V} \\frac{ A (u,v) } {\\| u \\|_V \\, \\| v \\|_V} \\leq\n",
     "\\sup_{u\\in V} \\sup_{v\\in V} \\frac{ \\alpha_2 \\| u \\|_V \\| v \\|_V } {\\| u \\|_V \\, \\| v \\|_V} = \\alpha_2.\n",
-    "\\end{eqnarray*}\n",
-    "$\\Box$"
+    "\\end{align*}\n",
+    "$\\Box$\n",
+    "\n",
+    "Using this notation, we can write the variational problem as operator equation: find $u \\in V$ such that\n",
+    "$$\n",
+    "A u = f \\qquad (\\mbox{in } V^\\ast).\n",
+    "$$"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "7ee7c92a-a3b7-4548-8ab6-9b3bd734716f",
+   "id": "491f09d8-8cec-4f28-b601-3bb266624fc7",
    "metadata": {},
    "source": [
-    "Using this notation, we can write the variational problem as operator equation: find $u \\in V$ such that\n",
-    "$$\n",
-    "A u = f \\qquad (\\mbox{in } V^\\ast).\n",
-    "$$\n",
-    "\n",
-    "\\begin{theorem}[Banach's contraction mapping theorem] Given a Banach space\n",
+    "**Theorem: (Banach's contraction mapping theorem)** Given a Banach space\n",
     "$V$ and a mapping $T : V \\rightarrow V$, satisfying the Lipschitz condition\n",
+    "\n",
     "$$\n",
     "\\| T(v_1) - T(v_2) \\| \\leq L \\, \\| v_1 - v_2 \\| \\qquad \\forall  \\, v_1, v_2 \\in V\n",
     "$$\n",
+    "\n",
     "for a fixed $L \\in [0,1)$. Then there exists a unique $u \\in V$ such that\n",
+    "\n",
     "$$\n",
     "u = T(u),\n",
     "$$\n",
+    "\n",
     "i.e. the mapping $T$ has a unique fixed point $u$. The iteration $u^1 \\in V$ given, compute\n",
+    "\n",
     "$$\n",
     "u^{k+1} := T(u^k)\n",
     "$$\n",
+    "\n",
     "converges to $u$ with convergence rate $L$:\n",
+    "\n",
     "$$\n",
     "\\| u - u^{k+1} \\| \\leq L \\| u - u^k \\|\n",
-    "$$\n",
-    "\\end{theorem}\n",
-    "\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d6a58687-792b-4be3-8a32-7aaff43e68d8",
+   "metadata": {},
+   "source": [
+    "**Theorem:(Lax Milgram)** Given a Hilbert space $V$, a coercive and continuous bilinear form $A(\\cdot,\\cdot)$, and a continuous linear form $f(.)$. Then there exists a unique $u \\in V$ solving\n",
     "\n",
-    "\\begin{theorem}[Lax Milgram] Given a Hilbert space $V$, a coercive and continuous bilinear form $A(\\cdot,\\cdot)$, and a continuous linear form $f(.)$. Then there exists a unique $u \\in V$ solving\n",
     "$$\n",
     "A(u,v) = f(v) \\qquad \\forall \\, v \\in V.\n",
     "$$\n",
     "There holds \n",
-    "\\begin{equation} \\label{equ_lax_milgram_bound}\n",
+    "\n",
+    "$$\n",
     "\\| u \\|_V \\leq \\alpha_1^{-1} \\| f \\|_{V^\\ast}\n",
-    "\\end{equation}\n",
-    "\\end{theorem}\n",
-    "{\\em Proof:} Start from the operator equation $A u = f$. Let $J_V : V^\\ast \\rightarrow V$ be the Riesz isomorphism defined by\n",
+    "$$\n",
+    "\n",
+    "*Proof:* Start from the operator equation $A u = f$. Let $J_V : V^\\ast \\rightarrow V$ be the Riesz isomorphism defined by\n",
+    "\n",
     "$$\n",
     "(J_V g, v)_V = g(v) \\qquad \\forall \\, v \\in V, \\;  \\forall \\, g \\in V^\\ast.\n",
     "$$\n",
+    "\n",
     "Then the operator equation is equivalent to\n",
+    "\n",
     "$$\n",
     "J_V A u = J_V f \\qquad (\\mbox{in } V),\n",
     "$$\n",
-    "and to the fixed point equation (with some $0 \\neq \\tau \\in \\setR$ chosen below)\n",
-    "\\begin{equation} \\label{equ_fixedpointequ}\n",
+    "\n",
+    "and to the fixed point equation (with some $0 \\neq \\tau \\in {\\mathbb R}$ chosen below)\n",
+    "\n",
+    "$$\n",
     "u = u - \\tau J_V (A u - f).\n",
-    "\\end{equation}\n",
+    "$$\n",
+    "\n",
     "We will verify that\n",
+    "\n",
     "$$\n",
     "T (v) := v - \\tau J_V (A v - f)\n",
     "$$\n",
+    "\n",
     "is a contraction mapping, i.e., $\\| T (v_1) - T (v_2) \\|_V \\leq L \\| v_1 - v_2 \\|_V$ \n",
     "with some Lipschitz constant $L \\in [0,1)$.\n",
     "Let $v_1, v_2 \\in V$, and set $v = v_1 - v_2$. Then\n",
-    "\\begin{eqnarray*}\n",
+    "\n",
+    "\\begin{align*}\n",
     "\\| T (v_1) - T (v_2) \\|_V^2 \n",
-    "& = & \\| \\{ v_1 - \\tau J_V (A v_1 - f) \\} - \\{ v_2 - \\tau J_V (A v_2 -f) \\} \\|_V^2 \\\\\n",
-    "& = & \\| v - \\tau J_V A v \\|_V^2 \\\\\n",
-    "& = & \\| v \\|_V^2 - 2 \\tau (J_V A v, v)_V + \\tau^2 \\| J_V A v \\|_V^2 \\\\\n",
-    "& = & \\| v \\|_V^2 - 2 \\tau \\left< A v , v \\right> + \\tau^2 \\| A v \\|_{V^\\ast}^2 \\\\\n",
-    "& = & \\| v \\|_V^2 - 2 \\tau A(v,v) + \\tau^2 \\| A v \\|_{V^\\ast}^2 \\\\\n",
-    "& \\leq & \\| v \\|_V^2 - 2 \\tau \\alpha_1 \\| v \\|_V^2 + \\tau^2 \\alpha_2^2 \\| v \\|_V^2 \\\\\n",
-    "& = & (1 - 2 \\tau \\alpha_1 + \\tau^2 \\alpha_2^2) \\| v_1 - v_2 \\|_V^2\n",
-    "\\end{eqnarray*}\n",
+    "&=  \\| \\{ v_1 - \\tau J_V (A v_1 - f) \\} - \\{ v_2 - \\tau J_V (A v_2 -f) \\} \\|_V^2 \\\\\n",
+    "&=  \\| v - \\tau J_V A v \\|_V^2 \\\\\n",
+    "&=  \\| v \\|_V^2 - 2 \\tau (J_V A v, v)_V + \\tau^2 \\| J_V A v \\|_V^2 \\\\\n",
+    "&=  \\| v \\|_V^2 - 2 \\tau \\left< A v , v \\right> + \\tau^2 \\| A v \\|_{V^\\ast}^2 \\\\\n",
+    "&=  \\| v \\|_V^2 - 2 \\tau A(v,v) + \\tau^2 \\| A v \\|_{V^\\ast}^2 \\\\\n",
+    "& \\leq  \\| v \\|_V^2 - 2 \\tau \\alpha_1 \\| v \\|_V^2 + \\tau^2 \\alpha_2^2 \\| v \\|_V^2 \\\\\n",
+    "&=  (1 - 2 \\tau \\alpha_1 + \\tau^2 \\alpha_2^2) \\| v_1 - v_2 \\|_V^2\n",
+    "\\end{align*}\n",
+    "\n",
     "Now, we choose $\\tau = \\alpha_1 / \\alpha_2^2$, and obtain a Lipschitz constant\n",
+    "\n",
     "$$\n",
     "L^2 = 1 - \\alpha_1^2 / \\alpha_2^2 \\in [0,1).\n",
     "$$\n",
+    "\n",
     "Banach's contraction mapping theorem state that (\\ref{equ_fixedpointequ}) \n",
     "has a unique fixed point. Finally, we obtain the bound (\\ref{equ_lax_milgram_bound}) from\n",
+    "\n",
     "$$\n",
     "\\| u \\|_V^2 \\leq \\alpha_1^{-1} A(u,u) = \\alpha_1^{-1} \\, f(u) \\leq \\alpha_1^{-1} \\| f\\|_{V^\\ast} \\| u \\|_V,\n",
     "$$\n",
+    "\n",
     "and dividing by one factor $\\|u\\|$.\n",
-    "\\hfill $\\Box$"
+    "$\\Box$"
    ]
   },
   {
@@ -203,11 +231,14 @@
    "metadata": {},
    "source": [
     "## Approximation of coercive variational problems\n",
+    "\n",
     "Now, let $V_h$ be a closed subspace of $V$. We compute the\n",
     "approximation $u_h \\in V_h$ by the Galerkin method\n",
-    "\\begin{equation}\n",
+    "\n",
+    "$$\n",
     "A(u_h, v_h) = f(v_h) \\qquad \\forall \\, v_h \\in V_h.\n",
-    "\\end{equation}\n",
+    "$$\n",
+    "\n",
     "This variational problem is uniquely solvable by Lax-Milgram, \n",
     "since, $(V_h,\\|.\\|_V)$ is \n",
     "a Hilbert space, and continuity and coercivity on $V_h$ are inherited\n",
@@ -218,42 +249,50 @@
     "up to a constant factor, as good as the best possible approximation in the\n",
     "finite dimensional space.\n",
     "\n",
-    "\\begin{theorem}[Cea] The approximation error of the Galerkin method\n",
+    "**Theorem:(Cea)** The approximation error of the Galerkin method\n",
     "is quasi optimal\n",
+    "\n",
     "$$\n",
     "\\| u - u_h \\|_V \\leq \\alpha_2 / \\alpha_1 \\inf_{v \\in V_h} \\| u - v_h \\|_V\n",
     "$$\n",
-    "\\end{theorem}\n",
-    "{\\em Proof:} A fundamental property is the Galerkin orthogonality\n",
+    "\n",
+    "*Proof:* A fundamental property is the Galerkin orthogonality\n",
+    "\n",
     "$$\n",
     "A(u-u_h, w_h) = A(u,w_h) - A(u_h, w_h) = f(w_h) - f(w_h) = 0\n",
     "\\qquad \\forall \\, w_h \\in V_h.\n",
     "$$\n",
+    "\n",
     "Now, pick an arbitrary $v_h \\in V_h$, and bound\n",
-    "\\begin{eqnarray*}\n",
-    "\\| u - u_h \\|_V^2 & \\leq & \\alpha_1^{-1} A(u-u_h, u-u_h) \\\\\n",
-    "& = & \\alpha_1^{-1} A(u-u_h, u-v_h) + \\alpha_1^{-1} A(u-u_h, \\underbrace{v_h-u_h}_{\\in V_h}) \\\\\n",
-    "& \\leq &  \\alpha_2 / \\alpha_1 \\, \\| u - u_h \\|_V \\| u - v_h \\|_V.\n",
-    "\\end{eqnarray*}\n",
+    "\\begin{align*}\n",
+    "\\| u - u_h \\|_V^2 & \\leq  \\alpha_1^{-1} A(u-u_h, u-u_h) \\\\\n",
+    "& =  \\alpha_1^{-1} A(u-u_h, u-v_h) + \\alpha_1^{-1} A(u-u_h, \\underbrace{v_h-u_h}_{\\in V_h}) \\\\\n",
+    "& \\leq   \\alpha_2 / \\alpha_1 \\, \\| u - u_h \\|_V \\| u - v_h \\|_V.\n",
+    "\\end{align*}\n",
+    "\n",
     "Divide one factor $\\|u - u_h\\|$. Since $v_h \\in V_h$ was arbitrary, the estimation\n",
     "holds true also for the infimum in $V_h$.\n",
-    "\\hfill $\\Box$\n",
+    " $\\Box$\n",
+    "\n",
     "\n",
-    "\\bigskip\n",
     "If $A(\\cdot,\\cdot)$ is additionally symmetric, then it is an inner product. In this\n",
     "case, the coercivity and continuity properties are equivalent to\n",
+    "\n",
     "$$\n",
     "\\alpha_1 \\| u \\|_V^2 \\leq A(u,u) \\leq \\alpha_2 \\, \\| u \\|_V^2\n",
     "\\qquad \\forall \\, u \\in V.\n",
     "$$\n",
+    "\n",
     "The generated norm $\\|.\\|_A$ is an equivalent norm to $\\|.\\|_V$. In the \n",
     "symmetric case, we can use the orthogonal projection with respect to \n",
     "$(.,.)_A$ to improve the bounds to\n",
+    "\n",
     "$$\n",
     "\\| u - u_h \\|_V^2 \\leq \\alpha_1^{-1} \\| u - u_h \\|_A^2 \\leq\n",
     "        \\alpha_1^{-1} \\inf_{v_h \\in V_h} \\| u - v_h \\|_A^2 \\leq\n",
     "        \\alpha_2 / \\alpha_1 \\| u - v_h \\|_V^2.\n",
     "$$\n",
+    "\n",
     "The factor in the quasi-optimality estimate is now the square root of the\n",
     "general, non-symmetric case."
    ]

_sources/abstracttheory/Untitled.ipynb

@@ -0,0 +1,6 @@
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