NGSolve24

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    "# The Basics"
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   "source": [
    "We learn the basic concepts to program in NGSolve, we start with an easy example and we explore the single steps to solve a PDE problem. Then we apply the same steps and concepts to a more complex problem."
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    "\n",
    "**basic topics**\n",
    "- [easy problem]()\n",
    "- [coefficient functions]()\n",
    "- [modelling domains with occ]()\n",
    "- [apply what we have learned]() Modelling a capacitor in 2D. \n",
    "\n"
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    "\n",
    "\n",
    "### Finite Element Method in a Nutshell\n",
    "\n",
    "\n",
    "**Problem:** (Strong)\\\n",
    "Let $\\Omega$ be a domain in $\\mathbb{R}^n$ .\n",
    "\n",
    "Find $u$ such that\n",
    "\\begin{align*} -\\Delta u &= f \\quad \\text{in } \\Omega, \\\\ u &= u_D \\quad \\text{on } \\Gamma_D,\\\\ n\\cdot \\nabla u &= g \\quad \\text{on } \\Gamma_N, \\end{align*}\n",
    "With $\\Gamma_D \\sqcup \\Gamma_N = \\partial \\Omega$ \n",
    "\n",
    "\n",
    "Multiply the equation by a test function $v$ and integrate by parts we get the weak formulation:\n",
    "\n",
    "\n",
    "**Problem:** (Weak) \\\n",
    "Find $u \\in H^1(\\Omega)$ such that $u = u_D$ on $\\Gamma_D$ and\n",
    "\\begin{align*} \\int_{\\Omega} \\nabla u \\cdot \\nabla v \\, dx &= \\int_{\\Omega} f v \\, dx - \\int_{\\Gamma_N} g v \\, ds \\quad \\forall v \\in H^1_{\\Gamma_D}(\\Omega), \\end{align*}\n",
    "\n",
    "Let $\\mathcal{T}$ ba a triangulation of the space $\\Omega$ the discrete weak formulation is given by:\n",
    "\n",
    "\n",
    "**Problem:** (Discrete-Weak)\\\n",
    "Find $u_h \\in V_{h}$ such that $u_h = u_D$ on $\\Gamma_D$ and\n",
    "\\begin{align*} \\int_{\\Omega} \\nabla u_h \\cdot \\nabla v_h \\, dx &= \\int_{\\Omega} f v_h \\, dx  - \\int_{\\Gamma_N} g v_h \\, ds\\quad \\forall v_h \\in V_{h,\\Gamma_D}(\\mathcal{T}), \\end{align*}\n",
    "\n",
    "Applying the Galerkin method:\n",
    "Suppose that on the triangulation $\\mathcal{T}$ we the finite  dimensional function space $V_{h}(\\mathcal{T})$ has a basis $\\{\\phi^i\\}_{i=1}^{N}$, and\n",
    "let \n",
    "\\begin{align*}\n",
    "v_h = \\sum_{i=1}^{N} v^i \\phi_i\\quad \\text{and} \\quad\n",
    "u_h = \\sum_{j=1}^{N} u^j \\phi_j\n",
    "\\end{align*}\n",
    "Then, by substituting $v_h$ and $u_h$ in the discrete weak formulation we obtain the following linear system:\n",
    "\n",
    "\\begin{align*} \n",
    "\\sum_{i,j} v^i u^j \\overbrace{\\int_{\\Omega} \\nabla \\phi_j \\cdot \\nabla \\phi_i \\, dx }^{A_{ij}}&= \\sum_{i} v^i \\overbrace{ \\left(\\int_{\\Omega} f \\phi_i \\, dx -  \\int_{\\Gamma_N} g \\phi_i \\, ds \\right)}^{f_i}\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "Then the problem reduces to solve the linear system $A u = f$.\n",
    "\n"
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