-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathindex.xml
443 lines (432 loc) · 33.2 KB
/
index.xml
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
<?xml version="1.0" encoding="utf-8" standalone="yes" ?>
<rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
<channel>
<title>Pierre Matalon</title>
<link>https://pmatalon.github.io/</link>
<atom:link href="https://pmatalon.github.io/index.xml" rel="self" type="application/rss+xml" />
<description>Pierre Matalon</description>
<generator>Wowchemy (https://wowchemy.com)</generator><language>en-us</language><copyright>©2023 Pierre Matalon</copyright><lastBuildDate>Wed, 10 Jul 2024 00:00:00 +0000</lastBuildDate>
<image>
<url>https://pmatalon.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
<title>Pierre Matalon</title>
<link>https://pmatalon.github.io/</link>
</image>
<item>
<title>Iterative solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order method</title>
<link>https://pmatalon.github.io/publication/2022-biharmonic-problem/</link>
<pubDate>Wed, 10 Jul 2024 00:00:00 +0000</pubDate>
<guid>https://pmatalon.github.io/publication/2022-biharmonic-problem/</guid>
<description><h2 id="reproduction-of-the-numerical-experiments">Reproduction of the numerical experiments</h2>
<p>The experiments can be reproduced with the open-source code <a href="https://pmatalon.github.io/software/fhhos4" target="_blank">fhhos4</a>, release 1.2.</p>
<h3 id="convergence-of-the-discrete-normal-derivative">Convergence of the discrete normal derivative</h3>
<p>Fig. 2</p>
<pre><code class="language-bash">-pb diff -geo square -source sine -s ch -mesh {cart|tri} -normalder -k {0|1|2|3} -n {16|32|64|128|256|512}
</code></pre>
<!--
### Well-posedness
```bash
# square cart
-pb bihar -geo square -source sine -s ch -nbh-depth 8 -mesh cart -cs r -kc {0|1} -k {0|1|2|3|4} -n {8|16}
# polygonal(2)
-pb bihar -geo squarecircularhole -tc default -s ch -bihar-prec p -nbh-depth 8 -mesh poly -cs r -polymesh-bfc c -polymesh-n-pass 1 -n 16 -kc {0|1} -k {0|1|2|3} -ut
# polygonal(4) -> retry until you get a max of 4 boundary faces per element
-pb bihar -geo squarecircularhole -tc default -s ch -bihar-prec p -nbh-depth 8 -mesh poly -cs r -polymesh-bfc c -polymesh-n-pass 2 -n 32 -kc {0|1} -k {0|1|2|3} -ut
-pb bihar -geo magnetism2 -tc default -s ch -bihar-prec p -nbh-depth 8 -mesh poly -cs r -polymesh-bfc c -polymesh-n-pass 1 -kc 1 -k 5 -n 8 -export mesh -ut -fc-coplanar-tol 1e-5 -threads 0 -f-basis monomials -f-ogb 0
```
-->
<h3 id="convergence-of-the-biharmonic-scheme">Convergence of the biharmonic scheme</h3>
<p>Fig. 3, 4, 5</p>
<pre><code class="language-bash"># Cartesian mesh
-pb bihar -geo square -source {exp|poly} -s ch -bihar-prec s -nbh-depth 8 -mesh cart -cs r -k {0|1|2|3} -n {16|32|64|128|256} -tol 1e-10
# Polygonal mesh
-pb bihar -geo square -source exp -s ch -bihar-prec s -nbh-depth 8 -bihar-prec-solver bicgstab -mesh poly -polymesh-init cart -polymesh-n-pass 1 -polymesh-fcs c -k {0|1|2|3} -n {16|32|64|128|256} -tol 1e-10
</code></pre>
<h3 id="preconditioner-convergence">Preconditioner convergence</h3>
<p>Fig. 7</p>
<pre><code class="language-bash">-pb bihar -geo square -source poly -s ch -mesh cart -cs r -k 1 -n 256 -tol 1e-14 -export iter -bihar-prec no #no preconditioner
-pb bihar -geo square -source poly -s ch -mesh cart -cs r -k 1 -n 256 -tol 1e-14 -export iter -nbh-depth {2|4|6|8|10} #with preconditioner
</code></pre>
<h3 id="asymptotic-behaviour">Asymptotic behaviour</h3>
<p>Tables 1, 2, 3</p>
<pre><code class="language-bash">-pb bihar -geo square -source exp -mesh cart -cs r -s ch -nbh-depth 8 -tol 1e-8 -bihar-prec {s|no} -k {0,1,2,3} -n {32,64,128,256,512}
-pb bihar -geo cube -source exp -mesh tetra -not-compute-errors -s fcguamg -hp-cs p_h -nbh-depth 2 -bihar-prec-solver bicgstab -tol 1e-8 -k 0 -bihar-prec {s|no} -n {8,16,32,64}
</code></pre>
<!-- -pb bihar -geo {disk|magnetism2} -tc default -mesh poly -polymesh-bfc c -polymesh-n-pass 2 -s ch -nbh-depth 8 -tol 1e-8 -k 0 -n {64,128,256,512,1024} -->
<!--
### Heuristics
```bash
-pb bihar -geo square -source poly -s fcguamg -mesh tri -cs r -k 3 -n 32 -tol 1e-8 -bihar-prec p -nbh-depth 2 -export iter -opt2 {0|2}
``` -->
</description>
</item>
<item>
<title>Homogeneous multigrid for hybrid discretizations: application to HHO methods</title>
<link>https://pmatalon.github.io/publication/2023-hho-hdg-demo-framework/</link>
<pubDate>Tue, 02 Apr 2024 00:00:00 +0000</pubDate>
<guid>https://pmatalon.github.io/publication/2023-hho-hdg-demo-framework/</guid>
<description><h2 id="reproduction-of-the-numerical-experiments">Reproduction of the numerical experiments</h2>
<h3 id="square">Square</h3>
<p>The experiments can be reproduced with the open-source code <a href="https://pmatalon.github.io/software/fhhos4" target="_blank">fhhos4</a>, release 1.2.1.</p>
<h4 id="injection-operator-i_ell1">Injection operator $I_\ell^1$</h4>
<p>This injection operator is enabled by the parameters <code>-prolong 2 -disable-hor</code>.</p>
<pre><code class="language-bash">-pb diff -geo square -mesh stri -mesher inhouse -s mg -cs s -tol 1e-6 -smoothers ags -cycle {V,1,1|V,2,2} -prolong 2 -disable-hor -k {1|2|3} -n {32|64|128|256|512}
</code></pre>
<h4 id="injection-operator-i_ell2">Injection operator $I_\ell^2$</h4>
<p>This injection operator is enabled by the parameters <code>-prolong 1 -disable-hor</code>.</p>
<pre><code class="language-bash">-pb diff -geo square -mesh stri -mesher inhouse -s mg -cs s -tol 1e-6 -smoothers ags -cycle {V,1,1|V,2,2} -prolong 1 -disable-hor -k {1|2|3} -n {32|64|128|256|512}
</code></pre>
<h4 id="injection-operator-i_ell3">Injection operator $I_\ell^3$</h4>
<p>This injection operator is enabled by the parameter <code>-prolong 1</code>.</p>
<pre><code class="language-bash">-pb diff -geo square -mesh stri -mesher inhouse -s mg -cs s -tol 1e-6 -smoothers ags -cycle {V,1,1|V,2,2} -prolong 1 -k {1|2|3} -n {32|64|128|256|512}
</code></pre>
<h3 id="l-shape">L-shape</h3>
<h4 id="injection-operator-i_ell1-1">Injection operator $I_\ell^1$</h4>
<pre><code class="language-bash">-pb diff -geo L_shape -mesh tri -mesher gmsh -s mg -cs r -tol 1e-6 -smoothers ags -cycle {V,1,1|V,2,2} -prolong 2 -disable-hor -k {1|2|3} -n {16|32|64|128|256}
</code></pre>
<h4 id="injection-operator-i_ell2-1">Injection operator $I_\ell^2$</h4>
<pre><code class="language-bash">-pb diff -geo L_shape -mesh tri -mesher gmsh -s mg -cs r -tol 1e-6 -smoothers ags -cycle {V,1,1|V,2,2} -prolong 1 -disable-hor -k {1|2|3} -n {16|32|64|128|256}
</code></pre>
<h4 id="injection-operator-i_ell3-1">Injection operator $I_\ell^3$</h4>
<pre><code class="language-bash">-pb diff -geo L_shape -mesh tri -mesher gmsh -s mg -cs r -tol 1e-6 -smoothers ags -cycle {V,1,1|V,2,2} -prolong 1 -k {1|2|3} -n {16|32|64|128|256}
</code></pre>
<h3 id="cube">Cube</h3>
<pre><code class="language-bash">-pb diff -geo cube -mesh stetra -mesher inhouse -s mg -cs s -tol 1e-6 -smoothers ags -cycle {V,1,1|V,2,2} -prolong 1 -k {1|2|3} -n {8|16|32}
</code></pre>
</description>
</item>
<item>
<title>Algebraic multigrid preconditioner for statically condensed systems arising from lowest-order hybrid discretizations</title>
<link>https://pmatalon.github.io/publication/2021-amg-for-hybrid-methods/</link>
<pubDate>Wed, 01 Mar 2023 00:00:00 +0000</pubDate>
<guid>https://pmatalon.github.io/publication/2021-amg-for-hybrid-methods/</guid>
<description><h2 id="reproduction-of-the-numerical-experiments">Reproduction of the numerical experiments</h2>
<p>The experiments can be reproduced with the open-source code <a href="https://pmatalon.github.io/software/fhhos4" target="_blank">fhhos4</a>, release 1.0.</p>
<h3 id="figure-42-and-table-42">Figure 4.2 and Table 4.2</h3>
<p>Execute the following command line</p>
<pre><code class="language-bash">&gt; fhhos4 [[test_case]] [[solver]]
</code></pre>
<p>where <code>[[test_case]]</code> corresponds to</p>
<pre><code class="language-bash"># Cube-cart
-geo cube -mesh cart -mesher inhouse -k 0 -n 128
# Cube-tet
-geo cube -mesh tetra -k 0 -n 64
# Complex-tet
-geo platewith4holes -k 0 -n 16
# Heterog1e8
-geo square4quadrants -k 0 -n 2048 -heterog 1e8
# Cube-cart-aniso100
-geo cube -mesh cart -mesher inhouse -k 0 -n 128 -aniso 100
# Cube-tet-aniso100
-geo cube -mesh tetra -k 0 -n 64 -aniso 20
</code></pre>
<p>and <code>[[solver]]</code> corresponds to</p>
<pre><code class="language-bash"># U-AMG
-s fcguamg
# C-AMG
-s fcgaggregamg
# AGMG
-s agmg
</code></pre>
<h3 id="figure-44">Figure 4.4</h3>
<pre><code class="language-bash">&gt; fhhos4 -geo cube -mesh tetra -k 0 -n {16|32|64|128} [[solver]]
</code></pre>
<h3 id="section-433-table-43-of-the-submitted-version">Section 4.3.3 (Table 4.3 of the submitted version)</h3>
<p>Details of the adaptive multiple coarsening strategy of U-AMG for the test
case Cube-tet.</p>
<pre><code class="language-bash">&gt; fhhos4 -geo cube -mesh tetra -k 0 -n 64 -s fcguamg
</code></pre>
<h3 id="section-433-table-44-of-the-submitted-version">Section 4.3.3 (Table 4.4 of the submitted version)</h3>
<p>Details of the fixed double coarsening strategy of U-AMG for the test case
Cube-tet.
Important parameter: <code>-cs dpa</code>.</p>
<pre><code class="language-bash">&gt; fhhos4 -geo cube -mesh tetra -k 0 -n 64 -s fcguamg -cs dpa
</code></pre>
<h3 id="table-43">Table 4.3</h3>
<pre><code class="language-bash"># U-AMG (multiple coarsening)
&gt; fhhos4 -geo cube -mesh tetra -k 0 -n 64 -s fcguamg -cs mpa
# U-AMG (double coarsening)
&gt; fhhos4 -geo cube -mesh tetra -k 0 -n 64 -s fcguamg -cs dpa
# C-AMG
&gt; fhhos4 -geo cube -mesh tetra -k 0 -n 64 -s fcgaggregamg
# AGMG
&gt; fhhos4 -geo cube -mesh tetra -k 0 -n 64 -s agmg
</code></pre>
<h3 id="table-44">Table 4.4</h3>
<p>Important parameter: <code>-coarsening-prolong {3|4|5|6}</code>.</p>
<pre><code class="language-bash"># Cube-tet
&gt; fhhos4 -geo cube -mesh tetra -k 0 -n 64 -s fcguamg -coarsening-prolong 6
&gt; fhhos4 -geo cube -mesh tetra -k 0 -n 64 -s fcguamg -coarsening-prolong 4
&gt; fhhos4 -geo cube -mesh tetra -k 0 -n 64 -s fcguamg -coarsening-prolong 5
&gt; fhhos4 -geo cube -mesh tetra -k 0 -n 64 -s fcguamg -coarsening-prolong 3
# Cube-cart-aniso100
&gt; fhhos4 -geo cube -mesh cart -mesher inhouse -k 0 -n 128 -aniso 100 -s fcguamg -coarsening-prolong 6
&gt; fhhos4 -geo cube -mesh cart -mesher inhouse -k 0 -n 128 -aniso 100 -s fcguamg -coarsening-prolong 4
&gt; fhhos4 -geo cube -mesh cart -mesher inhouse -k 0 -n 128 -aniso 100 -s fcguamg -coarsening-prolong 5
&gt; fhhos4 -geo cube -mesh cart -mesher inhouse -k 0 -n 128 -aniso 100 -s fcguamg -coarsening-prolong 3
</code></pre>
<h3 id="figure-44-1">Figure 4.4</h3>
<p>Same commands as for Table 4.4, also with the solver <code>-s fcgaggregamg</code>.</p>
</description>
</item>
<item>
<title>High-order multigrid strategies for HHO discretizations of elliptic equations</title>
<link>https://pmatalon.github.io/publication/2021-high-order-strategies/</link>
<pubDate>Sun, 01 Jan 2023 00:00:00 +0000</pubDate>
<guid>https://pmatalon.github.io/publication/2021-high-order-strategies/</guid>
<description><h2 id="reproduction-of-the-numerical-experiments">Reproduction of the numerical experiments</h2>
<p>The experiments can be reproduced with the open-source code <a href="https://pmatalon.github.io/software/fhhos4" target="_blank">fhhos4</a>, release 1.0.</p>
<p>In the following command lines, the strategies are configured by the parameter <code>-hp-config {1|2|3|4}</code>:</p>
<pre><code class="language-bash">-hp-config 1 # h-only
-hp-config 2 # p-h
-hp-config 3 # p-h*
-hp-config 4 # hp-h
</code></pre>
<h3 id="section-341-effect-of-basis-normalization">Section 3.4.1 (effect of basis normalization)</h3>
<p>Execute the following command to show that the multigrid method diverges when orthonormal bases are used with local refinement, even for low degree and small problem size:</p>
<pre><code class="language-bash">&gt; fhhos4 -geo square4quadrants_tri_localref -no-cache -tc square -cs r -k 1 -n 32 -e-ogb 3
</code></pre>
<p>The parameter <code>-e-ogb 3</code> orthonormalizes the element bases. Replace it with <code>-e-ogb 1</code> (orthogonalization without normalization) to make the multigrid method converge properly.</p>
<p>Note that divergence occurs if the refinement ratio is &gt;= 1e5. To see it, edit the file <code>data/meshes/2D/square4quadrants_tri_localref.geo</code> to play with the refinement ratio:</p>
<pre><code class="language-bash">h_center = h*1e-5; # divergence
h_center = h*1e-4; # convergence
</code></pre>
<p>It is interesting to see that normalization does not seem to affect the conditioning of the matrix, and that using a direct solver seems to work. Indeed, execute the following command and check the L2-error obtained after the system is solved by a Cholesky factorization.</p>
<pre><code class="language-bash">&gt; fhhos4 -geo square4quadrants_tri_localref -no-cache -tc square -cs r -k 1 -n 256 -e-ogb 3 -s ch
</code></pre>
<p>Replace the last parameters with <code>-e-ogb 1 -s mg</code> to check that you obtain the same L2-error using the multigrid and without normalization.</p>
<h3 id="figure-3-advantages-of-high-order-smooth-solution">Figure 3 (advantages of high-order, smooth solution)</h3>
<pre><code class="language-bash">&gt; fhhos4 -geo square -mesh cart -cs r -tol 1e-12 -s fcgmg -hp-config 2 -k 2 -n 1024
&gt; fhhos4 -geo square -mesh cart -cs r -tol 1e-12 -s fcgmg -hp-config 2 -k 3 -n 512
&gt; fhhos4 -geo square -mesh cart -cs r -tol 1e-12 -s fcgmg -hp-config 2 -k 4 -n 256
&gt; fhhos4 -geo square -mesh cart -cs r -tol 1e-12 -s fcgmg -hp-config 2 -k 5 -n 128
</code></pre>
<h3 id="figure-5-equivalent-test-non-smooth-solution">Figure 5 (equivalent test, non-smooth solution)</h3>
<pre><code class="language-bash">&gt; fhhos4 -geo square4quadrants_tri_localref -tc kellogg -no-cache -cs r -s fcgmg -hp-config 1 -e-ogb 0 -f-ogb 0 -e-basis monomials -f-basis monomials -tol 1e-3 -k 1 -n 256
&gt; fhhos4 -geo square4quadrants_tri_localref -tc kellogg -no-cache -cs r -s fcgmg -hp-config 1 -e-ogb 0 -f-ogb 0 -e-basis monomials -f-basis monomials -tol 1e-3 -k 2 -n 128
&gt; fhhos4 -geo square4quadrants_tri_localref -tc kellogg -no-cache -cs r -s fcgmg -hp-config 1 -e-ogb 0 -f-ogb 0 -e-basis monomials -f-basis monomials -tol 1e-3 -k 3 -n 64
&gt; fhhos4 -geo square4quadrants_tri_localref -tc kellogg -no-cache -cs r -s fcgmg -hp-config 1 -e-ogb 0 -f-ogb 0 -e-basis monomials -f-basis monomials -tol 1e-3 -k 4 -n 64
&gt; fhhos4 -geo square4quadrants_tri_localref -tc kellogg -no-cache -cs r -s fcgmg -hp-config 1 -e-ogb 0 -f-ogb 0 -e-basis monomials -f-basis monomials -tol 1e-3 -k 5 -n 32
</code></pre>
<h3 id="figures-7-and-8-square-cart-mesh">Figures 7 and 8 (square, Cart. mesh)</h3>
<p>Set <code>-s mg</code> for Fig. 6 and <code>-s fcgmg</code> for Fig. 7.</p>
<pre><code class="language-bash">&gt; fhhos4 -geo square -mesh cart -cs r -k 5 -n 128 -tol 1e-12 -s {mg|fcgmg} -hp-config {1|2|3|4}
</code></pre>
<h3 id="figure-9-and-10-square-tri-mesh">Figure 9 and 10 (square, tri. mesh)</h3>
<p>Set <code>-s mg</code> for Fig. 8 and <code>-s fcgmg</code> for Fig. 9.</p>
<pre><code class="language-bash">&gt; fhhos4 -geo square -cs r -k 5 -n 64 -tol 1e-12 -s {mg|fcgmg} -hp-config {1|2|3|4}
</code></pre>
<h3 id="figure-11-and-12-kellogg">Figure 11 and 12 (Kellogg)</h3>
<p>Set <code>-s mg</code> for Fig. 10 and <code>-s fcgmg</code> for Fig. 11.</p>
<pre><code class="language-bash">&gt; fhhos4 -geo square4quadrants_tri_localref -tc kellogg -no-cache -cs r -k 5 -n 256 -tol 1e-12 -s {mg|fcgmg} -hp-config {1|2|3|4}
</code></pre>
<h3 id="figure-13-and-14-cube-cart-mesh">Figure 13 and 14 (cube, Cart. mesh)</h3>
<p>Set <code>-s mg</code> for Fig. 12 and <code>-s fcgmg</code> for Fig. 13.</p>
<pre><code class="language-bash">&gt; fhhos4 -geo cube -k 3 -n 64 -mesh cart -mesher inhouse -cs s -tol 1e-12 -s {mg|fcgmg} -hp-config {1|2|3|4}
</code></pre>
<h3 id="figure-15-and-16-cube-tet-mesh">Figure 15 and 16 (cube, tet. mesh)</h3>
<p>Set <code>-s mg</code> for Fig. 14 and <code>-s fcgmg</code> for Fig. 15.</p>
<pre><code class="language-bash">&gt; fhhos4 -geo cube -k 3 -n 32 -mesh stetra -mesher inhouse -cs s -tol 1e-12 -s {mg|fcgmg} -hp-config {1|2|3|4}
</code></pre>
<h3 id="figure-17-and-18-geneva-wheel">Figure 17 and 18 (Geneva wheel)</h3>
<p>Set <code>-s mg</code> for Fig. 16 and <code>-s fcgmg</code> for Fig. 17.</p>
<pre><code class="language-bash">&gt; fhhos4 -geo genevawheel.geo -tc default -k 2 -n 32 -cs m -tol 1e-12 -s {mg|fcgmg} -hp-config {1|2|3|4}
</code></pre>
</description>
</item>
<item>
<title>Fast solvers for robust discretizations in computational fluid dynamics</title>
<link>https://pmatalon.github.io/publication/2021-phd-thesis/</link>
<pubDate>Mon, 01 Nov 2021 00:00:00 +0000</pubDate>
<guid>https://pmatalon.github.io/publication/2021-phd-thesis/</guid>
<description></description>
</item>
<item>
<title>samurai</title>
<link>https://pmatalon.github.io/software/samurai/</link>
<pubDate>Wed, 01 Sep 2021 00:00:00 +0000</pubDate>
<guid>https://pmatalon.github.io/software/samurai/</guid>
<description></description>
</item>
<item>
<title>Towards robust, fast solutions of elliptic equations on complex domains through hybrid high‐order discretizations and non‐nested multigrid methods</title>
<link>https://pmatalon.github.io/publication/2021-non-nested-mg-for-hho/</link>
<pubDate>Sat, 01 May 2021 00:00:00 +0000</pubDate>
<guid>https://pmatalon.github.io/publication/2021-non-nested-mg-for-hho/</guid>
<description></description>
</item>
<item>
<title>An h-multigrid method for Hybrid High-Order discretizations</title>
<link>https://pmatalon.github.io/publication/2020-mg-for-hho/</link>
<pubDate>Thu, 01 Apr 2021 00:00:00 +0000</pubDate>
<guid>https://pmatalon.github.io/publication/2020-mg-for-hho/</guid>
<description><h2 id="reproduction-of-the-numerical-experiments">Reproduction of the numerical experiments</h2>
<p>The experiments can be reproduced with the open-source code <a href="https://pmatalon.github.io/software/fhhos4/">fhhos4</a>, release 1.0.</p>
<h3 id="figure-41">Figure 4.1</h3>
<pre><code class="language-bash"># 2D cart
&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,1,1 -k 0 -n {32|64|128|256|512|1024}
&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,1,1 -k 1 -n {32|64|128|256|512|1024}
&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,1,1 -k 2 -n {32|64|128|256|512|1024}
&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,1,1 -k 3 -n {32|64|128|256|512|1024}
# 2D tri
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -s mg -cycle V,1,1 -k 0 -n {32|64|128|256|512|1024|2048}
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -s mg -cycle V,1,1 -k 1 -n {32|64|128|256|512|1024}
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -s mg -cycle V,1,1 -k 2 -n {32|64|128|256|512}
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -s mg -cycle V,1,1 -k 3 -n {32|64|128|256|512}
# 3D cart
&gt; fhhos4 -geo cube -mesh cart -mesher inhouse -s mg -cycle V,1,1 -k 0 -n {16|32|64|128}
&gt; fhhos4 -geo cube -mesh cart -mesher inhouse -s mg -cycle V,1,1 -k 1 -n {16|32|64|128}
&gt; fhhos4 -geo cube -mesh cart -mesher inhouse -s mg -cycle V,1,1 -k 2 -n {8|16|32|64}
&gt; fhhos4 -geo cube -mesh cart -mesher inhouse -s mg -cycle V,1,1 -k 3 -n {8|16|32|64}
# 3D tetra
&gt; fhhos4 -geo cube -mesh stetra -mesher inhouse -s mg -cycle V,2,2 -k 0 -n 16 # diverging
&gt; fhhos4 -geo cube -mesh stetra -mesher inhouse -s mg -cycle V,2,2 -k 1 -n {8|16|32|64}
&gt; fhhos4 -geo cube -mesh stetra -mesher inhouse -s mg -cycle V,2,2 -k 2 -n {8|16|32}
&gt; fhhos4 -geo cube -mesh stetra -mesher inhouse -s mg -cycle V,2,2 -k 3 -n {8|16|32}
</code></pre>
<h3 id="figure-42">Figure 4.2</h3>
<p>Replace the * characters with numerical values in</p>
<pre><code class="language-bash">&gt; fhhos4 -geo square -mesh stri -mesher inhouse -k 1 -n 512 -s mg -cycle V,*,*
</code></pre>
<h3 id="figure-43">Figure 4.3</h3>
<p>Replace the * characters with numerical values in</p>
<pre><code class="language-bash">&gt; fhhos4 -geo square -mesh stetra -mesher inhouse -k 1 -n 32 -s mg -cycle V,*,*
</code></pre>
<h3 id="figure-44">Figure 4.4</h3>
<p>Important parameter: <code>-smoothers bj23,bj23</code>.</p>
<p>Replace the * characters with numerical values in</p>
<pre><code class="language-bash">&gt; fhhos4 -geo square -mesh stri -mesher inhouse -k 1 -n 512 -s mg -smoothers bj23,bj23 -cycle V,*,*
</code></pre>
<h3 id="figure-45">Figure 4.5</h3>
<pre><code class="language-bash"># 2D cart
&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,0,3 -k 0 -n {32|64|128|256|512|1024}
&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,0,3 -k 1 -n {32|64|128|256|512|1024}
&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,0,3 -k 2 -n {32|64|128|256|512|1024}
&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,0,3 -k 3 -n {32|64|128|256|512|1024}
# 2D tri
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -s mg -cycle V,0,3 -k 0 -n {32|64|128|256|512|1024|2048}
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -s mg -cycle V,0,3 -k 1 -n {32|64|128|256|512|1024}
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -s mg -cycle V,0,3 -k 2 -n {32|64|128|256|512}
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -s mg -cycle V,0,3 -k 3 -n {32|64|128|256|512}
# 3D cart
&gt; fhhos4 -geo cube -mesh cart -mesher inhouse -s mg -cycle V,0,6 -k 0 -n {16|32|64|128}
&gt; fhhos4 -geo cube -mesh cart -mesher inhouse -s mg -cycle V,0,6 -k 1 -n {16|32|64|128}
&gt; fhhos4 -geo cube -mesh cart -mesher inhouse -s mg -cycle V,0,6 -k 2 -n {8|16|32|64}
&gt; fhhos4 -geo cube -mesh cart -mesher inhouse -s mg -cycle V,0,6 -k 3 -n {8|16|32|64}
# 3D tetra
&gt; fhhos4 -geo cube -mesh stetra -mesher inhouse -s mg -cycle V,0,6 -k 0 -n 16 # diverging
&gt; fhhos4 -geo cube -mesh stetra -mesher inhouse -s mg -cycle V,0,6 -k 1 -n {8|16|32|64}
&gt; fhhos4 -geo cube -mesh stetra -mesher inhouse -s mg -cycle V,0,6 -k 2 -n {8|16|32}
&gt; fhhos4 -geo cube -mesh stetra -mesher inhouse -s mg -cycle V,0,6 -k 3 -n {8|16|32}
</code></pre>
<h3 id="figure-47">Figure 4.7</h3>
<pre><code class="language-bash">&gt; fhhos4 -geo square4quadrants -tc kellogg -mesh cart -mesher inhouse -s mg -cycle V,1,1 -k 0 -n {32|64|128|256|512|1024}
&gt; fhhos4 -geo square4quadrants -tc kellogg -mesh cart -mesher inhouse -s mg -cycle V,1,1 -k 1 -n {32|64|128|256|512|1024}
&gt; fhhos4 -geo square4quadrants -tc kellogg -mesh cart -mesher inhouse -s mg -cycle V,1,1 -k 2 -n {32|64|128|256|512}
&gt; fhhos4 -geo square4quadrants -tc kellogg -mesh cart -mesher inhouse -s mg -cycle V,1,1 -k 3 -n {32|64|128|256|512}
</code></pre>
<h3 id="figure-48">Figure 4.8</h3>
<p>Important parameters: <code>-prolong {1|2} [-disable-hor]</code>.</p>
<h4 id="a-v03">(a) V(0,3)</h4>
<pre><code class="language-bash"># cell k and injection
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -k 1 -s mg -cycle V,0,3 -prolong 2 -disable-hor -n {32|64|128|256|512|1024}
# cell k and average
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -k 1 -s mg -cycle V,0,3 -prolong 1 -disable-hor -n {32|64|128|256|512|1024}
# cell k+1 and injection
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -k 1 -s mg -cycle V,0,3 -prolong 2 -n {32|64|128|256|512|1024}
# cell k+1 and average (final algo)
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -k 1 -s mg -cycle V,0,3 -prolong 1 -n {32|64|128|256|512|1024}
</code></pre>
<h4 id="b-v12">(b) V(1,2)</h4>
<pre><code class="language-bash"># cell k and injection
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -k 1 -s mg -cycle V,1,2 -prolong 2 -disable-hor -n {32|64|128|256|512|1024}
# cell k and average
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -k 1 -s mg -cycle V,1,2 -prolong 1 -disable-hor -n {32|64|128|256|512|1024}
# cell k+1 and injection
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -k 1 -s mg -cycle V,1,2 -prolong 2 -n {32|64|128|256|512|1024}
# cell k+1 and average (final algo)
&gt; fhhos4 -geo square -mesh stri -mesher inhouse -k 1 -s mg -cycle V,1,2 -prolong 1 -n {32|64|128|256|512|1024}
</code></pre>
<h3 id="figure-49">Figure 4.9</h3>
<p>Modify the value of the parameter <code>-heterog</code> from 1e0 to 1e8:</p>
<h4 id="a-heterogeneous-weighting">(a) Heterogeneous weighting</h4>
<pre><code class="language-bash">&gt; fhhos4 -geo square4quadrants -mesh cart -mesher inhouse -n 64 -s mg -g 1 -cycle V,0,3 -k 0 -heterog {1e0-1e8}
&gt; fhhos4 -geo square4quadrants -mesh cart -mesher inhouse -n 64 -s mg -g 1 -cycle V,0,3 -k 1 -heterog {1e0-1e8}
&gt; fhhos4 -geo square4quadrants -mesh cart -mesher inhouse -n 64 -s mg -g 1 -cycle V,0,3 -k 2 -heterog {1e0-1e8}
&gt; fhhos4 -geo square4quadrants -mesh cart -mesher inhouse -n 64 -s mg -g 1 -cycle V,0,3 -k 3 -heterog {1e0-1e8}
</code></pre>
<h4 id="b-homogeneous-weighting">(b) Homogeneous weighting</h4>
<p>Important parameter: <code>-disable-heterog-weight</code>.</p>
<pre><code class="language-bash">&gt; fhhos4 -geo square4quadrants -mesh cart -mesher inhouse -n 64 -s mg -g 1 -cycle V,0,3 -disable-heterog-weight -k 0 -heterog {1e0-1e8}
&gt; fhhos4 -geo square4quadrants -mesh cart -mesher inhouse -n 64 -s mg -g 1 -cycle V,0,3 -disable-heterog-weight -k 1 -heterog {1e0-1e8}
&gt; fhhos4 -geo square4quadrants -mesh cart -mesher inhouse -n 64 -s mg -g 1 -cycle V,0,3 -disable-heterog-weight -k 2 -heterog {1e0-1e8}
&gt; fhhos4 -geo square4quadrants -mesh cart -mesher inhouse -n 64 -s mg -g 1 -cycle V,0,3 -disable-heterog-weight -k 3 -heterog {1e0-1e8}
</code></pre>
<h3 id="figure-410">Figure 4.10</h3>
<p>Important parameter: <code>-fcs {0|1}</code>.</p>
<h4 id="a-with-face-coarsening">(a) With face coarsening</h4>
<pre><code class="language-bash">&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,0,3 -cs s -fcs 1 -k 0 -n {32|64|128|256|512}
&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,0,3 -cs s -fcs 1 -k 1 -n {32|64|128|256|512}
&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,0,3 -cs s -fcs 1 -k 2 -n {32|64|128|256|512}
&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,0,3 -cs s -fcs 1 -k 3 -n {32|64|128|256|512}
</code></pre>
<h4 id="b-without-face-coarsening">(b) Without face coarsening</h4>
<pre><code class="language-bash">&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,0,3 -cs s -fcs 0 -k 0 -n {32|64|128|256|512}
&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,0,3 -cs s -fcs 0 -k 1 -n {32|64|128|256|512}
&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,0,3 -cs s -fcs 0 -k 2 -n {32|64|128|256|512}
&gt; fhhos4 -geo square -mesh cart -mesher inhouse -s mg -cycle V,0,3 -cs s -fcs 0 -k 3 -n {32|64|128|256|512}
</code></pre>
<h3 id="figure-411">Figure 4.11</h3>
<p>Important parameter: <code>-cs b</code>.</p>
<pre><code class="language-bash"># Custom Bey's refinement, V(0,6)
&gt; fhhos4 -geo cube -mesh tetra -mesher inhouse -k 1 -s mg -cs b -cycle V,0,6 -n {8|16|32|64}
# Custom Bey's refinement, V(0,8)
&gt; fhhos4 -geo cube -mesh tetra -mesher inhouse -k 1 -s mg -cs b -cycle V,0,8 -n {8|16|32|64}
# Custom Bey's refinement, V(0,10)
&gt; fhhos4 -geo cube -mesh tetra -mesher inhouse -k 1 -s mg -cs b -cycle V,0,10 -n {8|16|32|64}
# Cartesian tet. refinement, V(0,6)
&gt; fhhos4 -geo cube -mesh stetra -mesher inhouse -k 1 -s mg -cs s -cycle V,0,6 -n {8|16|32|64}
</code></pre>
<h3 id="figure-412">Figure 4.12</h3>
<pre><code class="language-bash">&gt; fhhos4 -geo platewith4holes -s mg -cs b -cycle V,0,10 -k 0 -n {8|16|32} # diverges at n=32
&gt; fhhos4 -geo platewith4holes -s mg -cs b -cycle V,0,10 -k 1 -n {8|16|32}
&gt; fhhos4 -geo platewith4holes -s mg -cs b -cycle V,0,10 -k 2 -n {8|16|32}
&gt; fhhos4 -geo platewith4holes -s mg -cs b -cycle V,0,10 -k 3 -n {8|16|32}
</code></pre>
<h3 id="figure-415">Figure 4.15</h3>
<p>Important parameter: <code>-cs r</code>.</p>
<pre><code class="language-bash">&gt; fhhos4 -geo squarecircle -s mg -cs r -cycle V,0,3 -k 0 -n {32|64|128|256|512|1024}
&gt; fhhos4 -geo squarecircle -s mg -cs r -cycle V,0,3 -k 1 -n {32|64|128|256|512|1024}
&gt; fhhos4 -geo squarecircle -s mg -cs r -cycle V,0,3 -k 2 -n {32|64|128|256|512}
&gt; fhhos4 -geo squarecircle -s mg -cs r -cycle V,0,3 -k 3 -n {32|64|128|256|512}
</code></pre>
</description>
</item>
<item>
<title>fhhos4</title>
<link>https://pmatalon.github.io/software/fhhos4/</link>
<pubDate>Sat, 01 Sep 2018 00:00:00 +0000</pubDate>
<guid>https://pmatalon.github.io/software/fhhos4/</guid>
<description><p>fhhos4 (pronounce &lsquo;phosphore&rsquo;) stands for Fast HHO Solver for Research. It is an open-source code that solves diffusion problems with the Hybrid High-Order (HHO) discretizations.</p>
<p>Features:</p>
<ul>
<li>solves diffusion problems with piecewise constant tensors</li>
<li>HHO and DG discretizations</li>
<li>high-order solutions</li>
<li>handles unstructured meshes</li>
<li>binded to the GMSH mesher</li>
<li>implements geometric and algebraic multigrid methods</li>
<li>exports linear systems</li>
<li>shared-memory parallelism</li>
<li>many parameters (various choices of polynomial bases, multigrid cycles, smoothers, coarsening strategies&hellip;)</li>
</ul>
</description>
</item>
<item>
<title></title>
<link>https://pmatalon.github.io/admin/config.yml</link>
<pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
<guid>https://pmatalon.github.io/admin/config.yml</guid>
<description></description>
</item>
</channel>
</rss>