Update documentation

_sources/FEM/finiteelements.ipynb

@@ -0,0 +1,206 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "519ffe5b-0ec6-4603-9d8c-82dbfa888149",
+   "metadata": {},
+   "source": [
+    "# Finite Element Method\n",
+    "\n",
+    "Ciarlet's definition of a finite element is:\n",
+    "\n",
+    "> **Definition: Finite element** A finite element is a triple $(T, V_{T}, \\Psi_{T})$, where\n",
+    "> * $T$ is a bounded set\n",
+    "> * $V_{T}$ is function space on $T$ of finite dimension $N_T$ \n",
+    "> * $\\Psi_{T} = \\{ \\psi^1_T, \\ldots , \\psi^{N_T}_T \\}$ is a set of linearly independent functionals on $V_{T}$.\n",
+    "\n",
+    "The nodal basis $\\{\\varphi^1_T\\, \\ldots \\varphi^{N_T}_T\\}$ for $V_T$ is the basis \n",
+    "dual to $\\Psi_T$, i.e., \n",
+    "\n",
+    "$$\n",
+    "\\psi^i_T (\\varphi^j_T) = \\delta_{ij}\n",
+    "$$\n",
+    "Barycentric coordinates are useful to express the nodal basis functions.\n",
+    "\n",
+    "\n",
+    "Finite elements with point evaluation functionals are called Lagrange  \n",
+    "finite elements, elements using also derivatives are called Hermite finite elements."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a76719fd-f7de-4d43-8e0b-ae670b6628b5",
+   "metadata": {},
+   "source": [
+    "Usual function spaces on $T \\subset {\\mathbb R}^2$ are\n",
+    "\n",
+    "\\begin{align*}\n",
+    "P^p & := & \\mbox{span} \\{ x^i y^j : 0 \\leq i, \\, 0 \\leq j, \\, i+j \\leq p \\} \\\\\n",
+    "Q^p & := & \\mbox{span} \\{ x^i y^j : 0 \\leq i \\leq p, \\, 0 \\leq j \\leq p  \\}\n",
+    "\\end{align*}\n",
+    "\n",
+    "Examples for finite elements are\n",
+    "* A linear line segment \n",
+    "* A quadratic line segment\n",
+    "* A Hermite line segment\n",
+    "* A constant triangle\n",
+    "* A linear triangle\n",
+    "* A non-conforming triangle\n",
+    "* A Morley triangle\n",
+    "* A Raviart-Thomas triangle"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "80a7e67d-8f75-41fa-b162-a9d3cd355588",
+   "metadata": {},
+   "source": [
+    "The local nodal interpolation operator defined for functions $v \\in C^m(\\overline T)$ is\n",
+    "\n",
+    "$$\n",
+    "I_T v := \\sum_{\\alpha = 1}^{N_T} \\psi^\\alpha_T(v) \\varphi^\\alpha_T\n",
+    "$$\n",
+    "It is a projection.\n",
+    "\n",
+    "\n",
+    "Two finite elements $(T,V_T, \\Psi_T)$ and \n",
+    "$(\\widehat T, V_{\\widehat T}, \\Psi_{\\widehat T})$ are called \n",
+    "**equivalent** if there exists an invertible function $F$ such that\n",
+    "* $T = F (\\widehat T)$\n",
+    "* $V_T = \\{ \\hat v \\circ F^{-1} : \\hat v \\in V_{\\widehat T} \\}$\n",
+    "% \\item $\\Psi_T = \\{ \\psi_{\\widehat T} (. \\circ F) \\}$\n",
+    "* $\\Psi_T = \\{ \\psi^T_i : V_T \\rightarrow {\\mathbb R} : v \\rightarrow \\psi^{\\hat T}_i (v \\circ F) \\}$\n",
+    "\n",
+    "Two elements are called affine equivalent, if $F$ is an affine-linear function.\n",
+    "\n",
+    "Lagrangian finite elements defined above are equivalent. The Hermite elements are not equivalent.\n",
+    "\n",
+    "Two finite elements are called interpolation equivalent if there holds\n",
+    "$$\n",
+    "I_T (v) \\circ F = I_{\\widehat T} (v \\circ F)\n",
+    "$$\n",
+    "\n",
+    "> **Lemma:** Equivalent elements are interpolation equivalent\n",
+    "\n",
+    "The Hermite elements define above are also interpolation equivalent."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b72e87ce-09cc-4681-a7b5-fe2ac0b922ce",
+   "metadata": {},
+   "source": [
+    "A regular triangulation ${\\cal T} = \\{ T_1, \\ldots, T_M \\}$ of a domain \n",
+    "$\\Omega$ is the subdivision of a domain $\\Omega$ into closed triangles $T_i$\n",
+    "such that $\\overline \\Omega = \\cup T_i$ and $T_i \\cap T_j$ is \n",
+    "* either empty\n",
+    "* or a common vertex of $T_i$ and $T_j$\n",
+    "* or a common edge of $T_i$ and $T_j$\n",
+    "* or $T_i = T_j$ in the case $i = j$.\n",
+    "\\end{itemize}\n",
+    "%\n",
+    "In a wider sense, a triangulation may consist of different element shapes such\n",
+    "as segments, triangles,  quadrilaterals, tetrahedra, hexhedra, prisms, pyramids."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "42533d56-095e-4898-b859-bf07d05ee811",
+   "metadata": {},
+   "source": [
+    "A finite element complex $\\{ (T, V_T, \\Psi_T) \\}$ is a set of finite elements\n",
+    "defined on the geometric elements of the triangulation ${\\cal T}$.\n",
+    "\n",
+    "It is convenient to construct finite element complexes such that all\n",
+    "its finite elements are affine equivalent to one {\\em reference finite element} $(\\widehat T, \\hat V_T, \\hat \\Psi_T)$. The transformation $F_T$ is such that\n",
+    "$T = F_T (\\widehat T)$.\n",
+    "\n",
+    "\n",
+    "*Examples:* linear reference line segment on $(0,1)$.\n",
+    "\n",
+    "\n",
+    "The finite element complex allows the definition of the global \n",
+    "interpolation operator for $C^m$-smooth functions by\n",
+    "\n",
+    "$$\n",
+    "I_{\\cal T} v_{|T} = I_T v_T \\qquad \\forall \\, T \\in {\\cal T}\n",
+    "$$\n",
+    "%\n",
+    "The finite element space is \n",
+    "\n",
+    "$$\n",
+    "V_{\\cal T} := \\{ v = I_{\\cal T} w : w \\in C^m(\\overline \\Omega) \\}\n",
+    "$$\n",
+    "\n",
+    "We say that $V_{\\cal T}$ has regularity $r$ if $V_{\\cal T} \\subset C^r$.\n",
+    "If $V_{\\cal T} \\neq C^0$, the regularity is defined as~$-1$.\n",
+    "\n",
+    "Examples:\n",
+    "* The $P^1$ - triangle with vertex nodes leads to regularity $0$.\n",
+    "* The $P^1$ - triangle with edge midpoint nodes leads to regularity $-1$.\n",
+    "* The $P^0$ - triangle leads to regularity $-1$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "25da5cd8-b54f-4904-9526-9468f0733392",
+   "metadata": {},
+   "source": [
+    "For smooth functions, functionals $\\psi_{T,\\alpha}$ and $\\psi_{\\widetilde T, \\tilde \\alpha}$ sitting in the same location are equivalent. The set of global \n",
+    "functionals $\\Psi  = \\{ \\psi_1, \\ldots, \\psi_N\\}$ is the linearly independent\n",
+    "set of functionals containing all (equivalence classes of) local functionals.\n",
+    "\n",
+    "The connectivity matrix $C_T \\in {\\mathbb R}^{N \\times N_T}$ is defined such\n",
+    "that the local functionals are derived from the global ones by\n",
+    "\n",
+    "$$\n",
+    "\\Psi_T (u) = C_T^t \\Psi (u)\n",
+    "$$\n",
+    "Examples in 1D and 2D\n",
+    "\n",
+    "\n",
+    "The nodal basis for the global finite element space is the\n",
+    "basis in $V_{\\cal T}$ dual to the global functionals $\\psi_j$, i.e., \n",
+    "$$\n",
+    "\\psi_j(\\varphi_i) = \\delta_{ij}\n",
+    "$$\n",
+    "There holds\n",
+    "\\begin{align*}\n",
+    "\\varphi_i|_T & = \n",
+    "I_T \\varphi_i = \\sum_{\\alpha = 1}^{N_T} \\psi_T^\\alpha (\\varphi_i) \\varphi^\\alpha_T \\\\\n",
+    "& =  \\sum_{\\alpha = 1}^{N_T} (C_T^t \\psi(\\varphi_i))_\\alpha \\varphi_T^\\alpha \\\\\n",
+    "& =  \\sum_{\\alpha = 1}^{N_T} (C_T^t e_i)_\\alpha \\varphi_T^\\alpha = \\sum_{\\alpha = 1}^{N_T} C_{T,i\\alpha} \\varphi_T^\\alpha \n",
+    "\\end{align*}"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8f8359d8-cea0-4423-91b1-7b0feb4db8bb",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}