<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><br class=""><div><br class=""><blockquote type="cite" class=""><div class="">On Mar 22, 2022, at 9:55 AM, Marco Cisternino <<a href="mailto:marco.cisternino@optimad.it" class="">marco.cisternino@optimad.it</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><meta charset="UTF-8" class=""><div class="WordSection1" style="page: WordSection1; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 18px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;"><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">Thank you Barry!<br class="">No, no reason for FGMRES (some old tests showed shorter wall-times relative to GMRES), I’m going to use GMRES.<br class="">I tried GMRES with GAMG using PCSVD on the coarser level on real cases, larger than the toy case, I cannot get machine epsilon mean value of the solution, as in the toy case but about 10e-5, which is much better than before. The rest of GAMG is on default configuration. The mesh is much more complicated being an octree with immersed geometries. I get the same behaviour using GMRES+ASM+ILU as well.<o:p class=""></o:p></span></div><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">I cannot really get why the solution is not at zero mean value. I’m using a rtol~1.0e-8 but I get the same mean value using rtol~1.o-6, I tried 1.0-12 too, but no improvement in the constant, definitely tiny but not zero.<o:p class=""></o:p></span></div><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">Thanks.</span></div></div></div></blockquote><div><br class=""></div> Are you using GMRES in the smoother or Chebyshev? Recommend using Chebyshev. <br class=""><div><br class=""></div> I have no explanation why the average is not much closer to zero.</div><div><br class=""></div><div> Here is one thing to try. Use KSPMonitorSet() to provide a monitor that calls KSPBuildSolution() and then computes the average and prints it. Is the average always around 1e-5 from the first iteration or does it start close to 1e-12 and get larger with more iterations? </div><div><br class=""></div><div><br class=""><blockquote type="cite" class=""><div class=""><div class="WordSection1" style="page: WordSection1; caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 18px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;"><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class=""><o:p class=""></o:p></span></div><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class=""><o:p class=""> </o:p></span></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span lang="IT" style="font-family: "Mulish Medium";" class="">Marco Cisternino</span><span lang="IT" class=""><o:p class=""></o:p></span></div></div><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class=""><o:p class=""> </o:p></span></div><div class=""><div style="border-style: solid none none; border-top-width: 1pt; border-top-color: rgb(225, 225, 225); padding: 3pt 0cm 0cm;" class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><b class=""><span lang="EN-US" class="">From:</span></b><span lang="EN-US" class=""><span class="Apple-converted-space"> </span>Barry Smith <<a href="mailto:bsmith@petsc.dev" style="color: blue; text-decoration: underline;" class="">bsmith@petsc.dev</a>><span class="Apple-converted-space"> </span><br class=""><b class="">Sent:</b><span class="Apple-converted-space"> </span>lunedì 21 marzo 2022 19:49<br class=""><b class="">To:</b><span class="Apple-converted-space"> </span>Marco Cisternino <<a href="mailto:marco.cisternino@optimad.it" style="color: blue; text-decoration: underline;" class="">marco.cisternino@optimad.it</a>><br class=""><b class="">Cc:</b><span class="Apple-converted-space"> </span>Mark Adams <<a href="mailto:mfadams@lbl.gov" style="color: blue; text-decoration: underline;" class="">mfadams@lbl.gov</a>>;<span class="Apple-converted-space"> </span><a href="mailto:petsc-users@mcs.anl.gov" style="color: blue; text-decoration: underline;" class="">petsc-users@mcs.anl.gov</a><br class=""><b class="">Subject:</b><span class="Apple-converted-space"> </span>Re: [petsc-users] Null space and preconditioners<o:p class=""></o:p></span></div></div></div><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div></div><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> Marco,<o:p class=""></o:p></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> I have confirmed your results. <o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> Urgg, it appears we do not have something well documented. The removal of the null space only works for left preconditioned solvers and FGMRES only works with right preconditioning. Here is the reasoning. <o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> The Krylov space for left preconditioning is built from [r, BAr, (BA)^2 r, ...] and the solution space is built from this basis. If A has a null space of n then the left preconditioned Krylov methods simply remove n from the "full" Krylov space after applying each B preconditioner and the resulting "reduced" Krylov space has no components in the n directions hence the solution built by GMRES naturally has no component in the n. <o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> But with right preconditioning the Krylov space is [s ABs (AB)^2 s, ....] We would need to remove B^-1 n from the Krylov space so that (A B) B^-1 n = 0 In general we don't have any way of applying B^-1 to a vector so we cannot create the appropriate "reduced" Krylov space. <o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> If I run with GMRES (which defaults to left preconditioner) and the options <span style="font-size: 10.5pt; font-family: Menlo, serif;" class="">./testPreconditioners -pc_type gamg -ksp_type gmres -ksp_monitor_true_residual -ksp_rtol 1.e-12 -ksp_view -mg_coarse_pc_type svd</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> Then it handles the null space correctly and the solution has <span style="font-size: 10.5pt; font-family: Menlo, serif;" class="">Solution mean = 4.51028e-17</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class="">Is there any reason to use FGMRES instead of GMRES? You just cannot use GMRES as the smoother inside GAMG if you use GMRES on the outside, but for pressure equations you don't want use such a strong smoother anyways.<o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> Barry<o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> I feel we should add some information to the documentation on the removal of the null space to the user's manual when using right preconditioning and maybe even have an error check in the code so that people don't fall into this trap. But I am not sure exactly what to do. When the A and B are both symmetric I think special stuff happens that doesn't require providing a null space; but I am not sure. <o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><br class=""><br class=""><o:p class=""></o:p></div><blockquote style="margin-top: 5pt; margin-bottom: 5pt;" class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class="">On Mar 21, 2022, at 12:41 PM, Marco Cisternino <<a href="mailto:marco.cisternino@optimad.it" style="color: blue; text-decoration: underline;" class="">marco.cisternino@optimad.it</a>> wrote:<o:p class=""></o:p></div></div><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><o:p class=""> </o:p></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">Thank you, Mark.<br class="">However, doing this with my toy code</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">mpirun -n 1 ./testpreconditioner -pc_type gamg -pc_gamg_use_parallel_coarse_grid_solver -mg_coarse_pc_type jacobi -mg_coarse_ksp_type cg</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class=""> </span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">I get 16 inf elements. Do I miss anything?</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class=""> </span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">Thanks again</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class=""> </span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span lang="IT" style="font-family: "Mulish Medium";" class="">Marco Cisternino</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span lang="IT" class=""> </span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class=""> </span><o:p class=""></o:p></div></div><div style="border-style: solid none none; border-top-width: 1pt; border-top-color: rgb(225, 225, 225); padding: 3pt 0cm 0cm;" class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><b class=""><span lang="EN-US" class="">From:</span></b><span class="apple-converted-space"><span lang="EN-US" class=""> </span></span><span lang="EN-US" class="">Mark Adams <<a href="mailto:mfadams@lbl.gov" style="color: blue; text-decoration: underline;" class="">mfadams@lbl.gov</a>><span class="apple-converted-space"> </span><br class=""><b class="">Sent:</b><span class="apple-converted-space"> </span>lunedì 21 marzo 2022 17:31<br class=""><b class="">To:</b><span class="apple-converted-space"> </span>Marco Cisternino <<a href="mailto:marco.cisternino@optimad.it" style="color: blue; text-decoration: underline;" class="">marco.cisternino@optimad.it</a>><br class=""><b class="">Cc:</b><span class="apple-converted-space"> </span><a href="mailto:petsc-users@mcs.anl.gov" style="color: blue; text-decoration: underline;" class="">petsc-users@mcs.anl.gov</a><br class=""><b class="">Subject:</b><span class="apple-converted-space"> </span>Re: [petsc-users] Null space and preconditioners</span><o:p class=""></o:p></div></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> <o:p class=""></o:p></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class="">And for GAMG you can use:<o:p class=""></o:p></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> <o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class="">-pc_gamg_use_parallel_coarse_grid_solver -mg_coarse_pc_type jacobi -mg_coarse_ksp_type cg<o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> <o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class="">Note if you are using more that one MPI process you can use 'lu' instead of 'jacobi'<o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> <o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class="">If GAMG converges fast enough it can solve before the constant creeps in and works without cleaning in the KSP method.<o:p class=""></o:p></div></div></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> <o:p class=""></o:p></div></div><div class=""><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class="">On Mon, Mar 21, 2022 at 12:06 PM Mark Adams <<a href="mailto:mfadams@lbl.gov" style="color: blue; text-decoration: underline;" class="">mfadams@lbl.gov</a>> wrote:<o:p class=""></o:p></div></div></div><blockquote style="border-style: none none none solid; border-left-width: 1pt; border-left-color: rgb(204, 204, 204); padding: 0cm 0cm 0cm 6pt; margin: 5pt 0cm 5pt 4.8pt;" class=""><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class="">The solution for Neumann problems can "float away" if the constant is not controlled in some way because floating point errors can introduce it even if your RHS is exactly orthogonal to it.<o:p class=""></o:p></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> <o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class="">You should use a special coarse grid solver for GAMG but it seems to be working for you.<o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> <o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class="">I have lost track of the simply way to have the KSP solver clean the constant out, which is what you want.<o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> <o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class="">can someone help Marco?<o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> <o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class="">Mark<o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> <o:p class=""></o:p></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> <o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> <o:p class=""></o:p></div></div></div><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> <o:p class=""></o:p></div></div></div></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> <o:p class=""></o:p></div></div><div class=""><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class="">On Mon, Mar 21, 2022 at 8:18 AM Marco Cisternino <<a href="mailto:marco.cisternino@optimad.it" target="_blank" style="color: blue; text-decoration: underline;" class="">marco.cisternino@optimad.it</a>> wrote:<o:p class=""></o:p></div></div></div><blockquote style="border-style: none none none solid; border-left-width: 1pt; border-left-color: rgb(204, 204, 204); padding: 0cm 0cm 0cm 6pt; margin: 5pt 0cm 5pt 4.8pt;" class=""><div class=""><div class=""><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">Good morning,</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">I’m observing an unexpected (to me) behaviour of my code.</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">I tried to reduce the problem in a toy code here attached.<br class="">The toy code archive contains a small main, a matrix and a rhs.</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">The toy code solves the linear system and check the norms and the mean of the solution.</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">The problem into the matrix and the rhs is the finite volume discretization of the pressure equation of an incompressible NS solver.</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">It has been cooked as tiny as possible (16 cells!).</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">It is important to say that it is an elliptic problem with homogeneous Neumann boundary conditions only, for this reason the toy code sets a null space containing the constant.</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class=""> </span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">The unexpected (to me) behaviour is evident by launching the code using different preconditioners, using -pc-type <pctype><br class="">I tested using PCNONE (“none”), PCGAMG (“gamg”) and PCILU (“ilu”). The default solver is KSPFGMRES.</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">Using the three PC, I get 3 different solutions. It seems to me that they differ in the mean value, but GAMG is impressive.</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">PCNONE gives me the zero mean solution I expected. What about the others?</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class=""> </span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">Asking for residuals monitor, the ratio ||r||/||b|| shows convergence for PCNONE and PCILU (~10^-16), but it stalls for PCGAMG (~10^-4).</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">I cannot see why. Am I doing anything wrong or incorrectly thinking about the expected behaviour?</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class=""> </span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">Generalizing to larger mesh the behaviour is similar.</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class=""> </span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class="">Thank you for any help.</span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Light", serif;" class=""> </span><o:p class=""></o:p></div></div><div class=""><div style="margin: 0cm; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-family: "Mulish Medium";" class="">Marco Cisternino</span></div></div></div></div></blockquote></div></blockquote></div></div></blockquote></div></div></div></div></blockquote></div><br class=""></body></html>