Copper Mountain 2021 SISC Special Issue (#22)

Initial submission to Copper Mountain 2021 SISC Special Issue

Co-authored-by: Yunhui He <[email protected]>
Co-authored-by: Jed Brown <[email protected]>

Author: 3 people

examples/jupyter/LFAToolkit_Introduction.ipynb

@@ -145,8 +145,8 @@
     "\n",
     "$$\n",
     "\\color{maroon}{\\mathbf{A}} u = \\sum_e \\mathcal{E}^T_e \\left[\n",
-    "  \\left( \\color{royalblue}{\\mathbf{B}}_0^e \\right)^T \\color{royalblue}{\\mathbf{W}}^e \\Lambda \\left( \\color{yellowgreen}{\\mathbf{f_0}} \\left( u^e, \\nabla u^e \\right) \\right)\n",
-    "  + \\sum_{i = 1}^d \\left( \\color{royalblue}{\\mathbf{B}}_i^e \\right)^T \\color{royalblue}{\\mathbf{W}}^e \\Lambda \\left( \\color{yellowgreen}{\\mathbf{f_1}} \\left( u^e, \\nabla u^e \\right) \\right)\n",
+    "  \\left( \\color{royalblue}{\\mathbf{B}}_0^e \\right)^T \\color{yellowgreen}{\\mathbf{W}}^e \\Lambda \\left( \\color{yellowgreen}{\\mathbf{f_0}} \\left( u^e, \\nabla u^e \\right) \\right)\n",
+    "  + \\sum_{i = 1}^d \\left( \\color{royalblue}{\\mathbf{B}}_i^e \\right)^T \\color{yellowgreen}{\\mathbf{W}}^e \\Lambda \\left( \\color{yellowgreen}{\\mathbf{f_1}} \\left( u^e, \\nabla u^e \\right) \\right)\n",
     "\\right]\n",
     "$$\n",
     "\n",
@@ -274,20 +274,18 @@
     "## <center>Mathematical Background</center>\n",
     "### <center>Local Fourier Analysis</center>\n",
     "\n",
-    "For a system of equations, such as the $p \\times p$ linear system of equations for a scalar problem on a single $p$ order finite element\n",
+    "For a $q \\times q$ system of equations\n",
     "\n",
     "<br>\n",
     "\n",
     "$$\n",
     "\\tilde{L}_h =\n",
     "\\begin{bmatrix}\n",
-    "    \\tilde{L}_h^{1, 1}  &&  \\cdots  &&  \\tilde{L}_h^{1, p}  \\\\\n",
+    "    \\tilde{L}_h^{1, 1}  &&  \\cdots  &&  \\tilde{L}_h^{1, q}  \\\\\n",
     "    \\vdots              &&  \\vdots  &&  \\vdots              \\\\\n",
-    "    \\tilde{L}_h^{p, 1}  &&  \\cdots  &&  \\tilde{L}_h^{p, p}  \\\\\n",
+    "    \\tilde{L}_h^{q, 1}  &&  \\cdots  &&  \\tilde{L}_h^{q, q}  \\\\\n",
     "\\end{bmatrix}\n",
-    "$$\n",
-    "\n",
-    "Multi-component equations are represented in a similar fashion"
+    "$$"
    ]
   },
   {
@@ -826,7 +824,7 @@
     "<br>\n",
     "\n",
     "$$\n",
-    "\\tilde{\\mathbf{S}} \\left( \\omega, \\mathbf{\\theta} \\right) = \\mathbf{I} - \\omega \\space \\left( \\mathbf{Q}^T \\text{diag} \\left( \\color{maroon}{\\mathbf{A}}_e \\right) \\mathbf{Q} \\right) \\tilde{\\color{maroon}{\\mathbf{A}}} \\left( \\mathbf{\\theta} \\right)\n",
+    "\\tilde{\\mathbf{S}} \\left( \\omega, \\mathbf{\\theta} \\right) = \\mathbf{I} - \\omega \\space \\left( \\mathbf{Q}^T \\text{diag} \\left( \\color{maroon}{\\mathbf{A}}_e \\right) \\mathbf{Q} \\right)^{-1} \\tilde{\\color{maroon}{\\mathbf{A}}} \\left( \\mathbf{\\theta} \\right)\n",
     "$$"
    ]
   },
@@ -1439,7 +1437,7 @@
     "<br>\n",
     "\n",
     "$$\n",
-    "\\mathbf{P}_{ctof} = \\mathbf{P}_f^T \\color{yellowgreen}{\\mathbf{D}}_{scale} \\color{royalblue}{\\mathbf{B}}_{ctof} \\mathbf{P}_c, \\ \\ \\mathbf{R}_{ftoc} = \\mathbf{P}_{ctof}^T\n",
+    "\\mathbf{P}_{ctof} = \\mathbf{P}_f^T \\color{royalblue}{\\mathbf{I}} \\color{yellowgreen}{\\mathbf{D}}_{scale} \\color{royalblue}{\\mathbf{B}}_{ctof} \\mathbf{P}_c, \\ \\ \\mathbf{R}_{ftoc} = \\mathbf{P}_{ctof}^T\n",
     "$$"
    ]
   },

examples/jupyter/demo223_dirichlet_bddc.ipynb

@@ -64,15 +64,15 @@
     "\n",
     "# compute and plot smoothing factor\n",
     "# setup\n",
-    "ω = [0.2]\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",
+    "        eigenvalues = [abs(val) for val in eigvals(A)]\n",
     "        maxeigenvalues[i, j] = max(eigenvalues...)\n",
     "        maxeigenvalue = max(maxeigenvalue, maxeigenvalues[i, j])\n",
     "    end\n",

papers/copper-mountain-2021/.gitignore

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

papers/copper-mountain-2021/data/jacobi-smoothing-factor-1d.csv

@@ -0,0 +1,5 @@
+p,ω,θ,ρ
+2,1.0,0.5000000000000001,0.7559289460184544
+4,0.91,0.5000000000000001,0.954993451479252
+8,0.82,0.5000000000000001,0.9918585345934762
+16,0.75,0.5000000000000001,0.9984846471940714

papers/copper-mountain-2021/data/jacobi-smoothing-factor-2d.csv

@@ -0,0 +1,7 @@
+p,ω,θ,ρ
+2,1.05,7.067899292141149e-17,0.8392165313210713
+2,1.05,0.5000000000000001,0.8392165313210713
+4,1.0,7.067899292141149e-17,0.9718657882287578
+4,1.0,0.5000000000000001,0.9718657882287578
+8,0.87,0.5000000000000001,0.9952895206977193
+8,0.87,7.067899292141149e-17,0.9952895206977193

papers/copper-mountain-2021/data/jacobi-smoothing-factor-estimate-1d.csv

@@ -0,0 +1,5 @@
+p,ω_classical,ω_highorder
+2,1.0000000000000002,0.8381346147406376
+4,0.9114731737026649,0.8545152818638532
+8,0.8235654204608711,0.8066714492123008
+16,0.7515343568509513,0.7485946457236492

papers/copper-mountain-2021/data/jacobi-smoothing-factor-estimate-2d.csv

@@ -0,0 +1,5 @@
+p,ω_classical,ω_highorder
+2,1.2181176114168595,1.1728762637162735
+4,1.008526953705753,1.008526953705753
+8,0.8795806340156368,0.8795806340156368
+16,0.8046736861365273,0.8046736861365273