High-Order BDDC Paper

papers/copper-mountain-2021/lfa_pmultigrid_high_order_fem.tex

@@ -1022,7 +1022,7 @@ \section{Conclusions}\label{sec:conclusion}
 
 % -----------------------------------------------------------------------------
 \bibliographystyle{siamplain}
-\bibliography{references}
+\bibliography{../references}
 % -----------------------------------------------------------------------------
 
 \end{document}

papers/high-order-bddc-2021/.gitignore

@@ -0,0 +1,12 @@
+.*.swp
+*.aux
+*.bbl
+*.blg
+*.fdb_latexmk
+*.fls
+*.log
+*.out
+*.pdf
+*.synctex.gz
+*.thm
+siamart.cls

papers/high-order-bddc-2021/data/bddc-condition-numbers-32.csv

@@ -0,0 +1,33 @@
+p,m,kind,θ,λ_min,λ_max,κ
+1,4,lumped,1,0.999999999999919,4.44354670514879,4.44354670514915
+1,4,lumped,1,0.999999999999919,4.44354670514879,4.44354670514915
+1,4,dirichlet,0,0.999999999999946,2.34861818767369,2.34861818767381
+1,4,dirichlet,-0.0625,0.999999999999946,2.34861818767369,2.34861818767381
+1,6,lumped,1,0.99999999999986,8.16263332834066,8.16263332834181
+1,6,lumped,1,0.99999999999986,8.16263332834066,8.16263332834181
+1,6,dirichlet,0,0.999999999999777,2.82522182919664,2.82522182919727
+1,6,dirichlet,-0.0625,0.999999999999777,2.82522182919664,2.82522182919727
+1,8,lumped,1,0.99999999999969,12.2685887833641,12.2685887833679
+1,8,lumped,1,0.99999999999969,12.2685887833641,12.2685887833679
+1,8,dirichlet,-0.0625,0.999999999999503,3.19260786785099,3.19260786785258
+1,8,dirichlet,0,0.999999999999503,3.19260786785099,3.19260786785258
+1,10,lumped,1,0.999999999999688,16.6712358297221,16.6712358297273
+1,10,lumped,1,0.999999999999688,16.6712358297221,16.6712358297273
+1,10,dirichlet,-0.0625,0.999999999999389,3.49525132179121,3.49525132179335
+1,10,dirichlet,0,0.999999999999389,3.49525132179121,3.49525132179335
+1,12,lumped,1,0.999999999999248,21.314347684091,21.314347684107
+1,12,lumped,1,0.999999999999248,21.314347684091,21.314347684107
+1,12,dirichlet,-0.0625,0.999999999999161,3.75422422223343,3.75422422223658
+1,12,dirichlet,0,0.999999999999161,3.75422422223343,3.75422422223658
+1,14,lumped,1,0.99999999999943,26.1594484432303,26.1594484432452
+1,14,lumped,1,0.99999999999943,26.1594484432303,26.1594484432452
+1,14,dirichlet,0,0.999999999998964,3.98147178391333,3.98147178391745
+1,14,dirichlet,-0.0625,0.999999999998964,3.98147178391333,3.98147178391745
+1,16,lumped,1,0.999999999999456,31.1786033591328,31.1786033591497
+1,16,lumped,1,0.999999999999456,31.1786033591328,31.1786033591497
+1,16,dirichlet,0,0.999999999998699,4.18449860326451,4.18449860326995
+1,16,dirichlet,-0.0625,0.999999999998699,4.18449860326451,4.18449860326995
+1,32,lumped,1,0.999999999998981,75.761107494851,75.7611074949282
+1,32,lumped,1,0.999999999998981,75.761107494851,75.7611074949282
+1,32,dirichlet,0,0.999999999995734,5.33120628628446,5.3312062863072
+1,32,dirichlet,-0.0625,0.999999999995734,5.33120628628446,5.3312062863072

papers/high-order-bddc-2021/data/bddc-condition-numbers-64.csv

@@ -0,0 +1,17 @@
+p,m,kind,θ,λ_min,λ_max,κ
+1,4,lumped,1.0,0.9999999999995242,4.443546705148792,4.443546705150907
+1,4,lumped,1.0,0.9999999999995242,4.443546705148792,4.443546705150907
+1,4,dirichlet,0.0,0.9999999999996358,2.3508069384136467,2.350806938414503
+1,4,dirichlet,-0.031250000000000035,0.9999999999996358,2.3508069384136467,2.350806938414503
+1,8,lumped,1.0,0.999999999999014,12.268588783364079,12.268588783376176
+1,8,lumped,1.0,0.999999999999014,12.268588783364079,12.268588783376176
+1,8,dirichlet,-0.031250000000000035,0.999999999998345,3.1956201298283142,3.195620129833603
+1,8,dirichlet,0.0,0.999999999998345,3.1956201298283142,3.195620129833603
+1,16,lumped,1.0,0.9999999999973945,31.17860335913278,31.178603359214016
+1,16,lumped,1.0,0.9999999999973945,31.17860335913278,31.178603359214016
+1,16,dirichlet,0.0,0.9999999999945208,4.188166252021851,4.188166252044799
+1,16,dirichlet,-0.031250000000000035,0.9999999999945208,4.188166252021851,4.188166252044799
+1,32,lumped,1.0,0.9999999999961624,75.76110749485096,75.7611074951417
+1,32,lumped,1.0,0.9999999999961624,75.76110749485096,75.7611074951417
+1,32,dirichlet,0.0,0.9999999999831699,5.3354780255459175,5.335478025635714
+1,32,dirichlet,-0.031250000000000035,0.9999999999831699,5.3354780255459175,5.335478025635714

papers/high-order-bddc-2021/data/spectral-bddc-condition-numbers-1.csv

@@ -0,0 +1,13 @@
+p,m,kind,θ,λ_min,λ_max,κ
+2,1,Dirichlet,1.0,0.9999999999999472,2.0416666666666665,2.0416666666667744
+2,1,Dirichlet,0.0,0.9999999999999472,2.0416666666666665,2.0416666666667744
+3,1,Dirichlet,-0.0625,0.999999999999844,2.614399208665368,2.6143992086657755
+3,1,Dirichlet,0.0,0.999999999999844,2.614399208665368,2.6143992086657755
+4,1,Dirichlet,-0.0625,0.9999999999996337,3.2414831312996704,3.2414831313008574
+4,1,Dirichlet,0.0,0.9999999999996337,3.2414831312996704,3.2414831313008574
+8,1,Dirichlet,-0.0625,0.999999999998532,5.194529913448625,5.194529913456251
+8,1,Dirichlet,0.0,0.999999999998532,5.194529913448625,5.194529913456251
+16,1,Dirichlet,0.0,0.999999999993277,7.756317490204899,7.756317490257044
+16,1,Dirichlet,-0.0625,0.999999999993277,7.756317490204899,7.756317490257044
+32,1,Dirichlet,-0.0625,0.9999999999587119,10.96078129475398,10.96078129520653
+32,1,Dirichlet,0.0,0.9999999999587119,10.96078129475398,10.96078129520653

papers/high-order-bddc-2021/data/spectral-bddc-condition-numbers-2.csv

@@ -0,0 +1,13 @@
+p,m,kind,θ,λ_min,λ_max,κ
+2,2,Dirichlet,0.0,0.9999999999998268,2.7000272676727954,2.700027267673263
+2,2,Dirichlet,-0.0625,0.9999999999998268,2.7000272676727954,2.700027267673263
+3,2,Dirichlet,-0.0625,0.9999999999992926,3.567928437112254,3.5679284371147784
+3,2,Dirichlet,0.0,0.9999999999992926,3.567928437112254,3.5679284371147784
+4,2,Dirichlet,-0.0625,0.9999999999988743,4.305060011561867,4.305060011566713
+4,2,Dirichlet,0.0,0.9999999999988743,4.305060011561867,4.305060011566713
+8,2,Dirichlet,-0.0625,0.9999999999952132,6.510880021845141,6.510880021876307
+8,2,Dirichlet,0.0,0.9999999999952132,6.510880021845141,6.510880021876307
+16,2,Dirichlet,-0.0625,0.9999999999808564,9.353764059470207,9.35376405964927
+16,2,Dirichlet,0.0,0.9999999999808564,9.353764059470207,9.35376405964927
+32,2,Dirichlet,0.0,0.9999999998855446,12.85606582669229,12.856065828163738
+32,2,Dirichlet,-0.0625,0.9999999998855446,12.85606582669229,12.856065828163738

papers/high-order-bddc-2021/jupyter/.gitignore

@@ -0,0 +1,5 @@
+# Jupyter notebook checkpoint files
+.ipynb_checkpoints/
+
+# File generated by Pkg, the package manager, based on a corresponding Project.toml
+Manifest.toml

papers/high-order-bddc-2021/jupyter/Dirichlet-BDDC-High-Order-Histogram.ipynb

@@ -0,0 +1,115 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Dirichlet BDDC Example"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# dependencies\n",
+    "using LFAToolkit\n",
+    "using LinearAlgebra\n",
+    "using Pkg\n",
+    "Pkg.activate(\"./\")\n",
+    "Pkg.instantiate()\n",
+    "using Plots"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "code_folding": []
+   },
+   "outputs": [],
+   "source": [
+    "# setup\n",
+    "p = 4\n",
+    "mesh = Mesh2D(1.0, 1.0)\n",
+    "\n",
+    "# diffusion operator\n",
+    "diffusion = GalleryOperator(\"diffusion\", p + 1, p + 2, mesh)\n",
+    "\n",
+    "# Dirichlet BDDC smoother\n",
+    "bddc = DirichletBDDC(diffusion)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "code_folding": [
+     14,
+     16,
+     27
+    ],
+    "scrolled": true
+   },
+   "outputs": [],
+   "source": [
+    "# full operator symbols\n",
+    "numbersteps = 4\n",
+    "maxeigenvalue = 0\n",
+    "θ_min = -π/2\n",
+    "θ_max = 3π/2\n",
+    "θ_step = 2π/(numbersteps-1)\n",
+    "θ_range = θ_min:θ_step:θ_max\n",
+    "\n",
+    "# compute and plot smoothing factor\n",
+    "# setup\n",
+    "ω = [1.0]\n",
+    "eigenvalues = []\n",
+    "\n",
+    "# compute\n",
+    "for i in 1:numbersteps, j in 1:numbersteps\n",
+    "    θ = [θ_range[i], θ_range[j]]\n",
+    "    if sqrt(abs(θ[1])^2 + abs(θ[2])^2) > π/128\n",
+    "        A = computesymbols(bddc, ω, θ)\n",
+    "        currenteigenvalues = [abs(val) for val in eigvals(I - A)]\n",
+    "        push!(eigenvalues, currenteigenvalues)\n",
+    "    end\n",
+    "end\n",
+    "\n",
+    "# plot\n",
+    "histogram(\n",
+    "    eigenvalues,\n",
+    "    title=\"Dirichlet BDDC Eigenvalues\",\n",
+    "    legend=false,\n",
+    "    aspect_ratio=:equal,\n",
+    "    color=:cividis\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "savefig(\"DirichletBDDCHighOrderEigenvalues\")"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Julia 1.6.1",
+   "language": "julia",
+   "name": "julia-1.6"
+  },
+  "language_info": {
+   "file_extension": ".jl",
+   "mimetype": "application/julia",
+   "name": "julia",
+   "version": "1.6.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}

papers/high-order-bddc-2021/jupyter/Dirichlet-BDDC-High-Order.ipynb

@@ -0,0 +1,126 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Dirichlet BDDC Example"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# dependencies\n",
+    "using LFAToolkit\n",
+    "using LinearAlgebra\n",
+    "using Pkg\n",
+    "Pkg.activate(\"./\")\n",
+    "Pkg.instantiate()\n",
+    "using Plots"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "code_folding": []
+   },
+   "outputs": [],
+   "source": [
+    "# setup\n",
+    "p = 4\n",
+    "mesh = Mesh2D(1.0, 1.0)\n",
+    "\n",
+    "# diffusion operator\n",
+    "diffusion = GalleryOperator(\"diffusion\", p + 1, p + 2, mesh)\n",
+    "\n",
+    "# Dirichlet BDDC smoother\n",
+    "bddc = DirichletBDDC(diffusion)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "code_folding": [
+     14,
+     16,
+     27
+    ],
+    "scrolled": true
+   },
+   "outputs": [],
+   "source": [
+    "# full operator symbols\n",
+    "numbersteps = 250\n",
+    "maxeigenvalue = 0\n",
+    "θ_min = -π/2\n",
+    "θ_max = 3π/2\n",
+    "θ_step = 2π/(numbersteps-1)\n",
+    "θ_range = θ_min:θ_step:θ_max\n",
+    "\n",
+    "# compute and plot smoothing factor\n",
+    "# setup\n",
+    "ω = [1.0]\n",
+    "maxeigenvalues = zeros(numbersteps, numbersteps)\n",
+    "\n",
+    "# compute\n",
+    "for i in 1:numbersteps, j in 1:numbersteps\n",
+    "    θ = [θ_range[i], θ_range[j]]\n",
+    "    if sqrt(abs(θ[1])^2 + abs(θ[2])^2) > π/128\n",
+    "        A = computesymbols(bddc, ω, θ)\n",
+    "        eigenvalues = [abs(val) for val in eigvals(I - A)]\n",
+    "        maxeigenvalues[i, j] = max(eigenvalues...)\n",
+    "        maxeigenvalue = max(maxeigenvalue, maxeigenvalues[i, j])\n",
+    "    end\n",
+    "end\n",
+    "\n",
+    "# plot\n",
+    "println(\"max eigenvalue: \", maxeigenvalue)\n",
+    "xrange = θ_range/π\n",
+    "heatmap(\n",
+    "    xrange,\n",
+    "    xlabel=\"θ/π\",\n",
+    "    xtickfont=font(12, \"Courier\"),\n",
+    "    xrange,\n",
+    "    ylabel=\"θ/π\",\n",
+    "    ytickfont=font(12, \"Courier\"),\n",
+    "    maxeigenvalues,\n",
+    "    title=\"Dirichlet BDDC Symbol\",\n",
+    "    transpose=true,\n",
+    "    aspect_ratio=:equal,\n",
+    "    color=:cividis\n",
+    ")\n",
+    "xlims!(θ_min/π, θ_max/π)\n",
+    "ylims!(θ_min/π, θ_max/π)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "savefig(\"DirichletBDDCHighOrder\")"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Julia 1.6.1",
+   "language": "julia",
+   "name": "julia-1.6"
+  },
+  "language_info": {
+   "file_extension": ".jl",
+   "mimetype": "application/julia",
+   "name": "julia",
+   "version": "1.6.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}