Hello Gary, Since we have two independent implementation of DG, I am curious to know if you achieve optimal error convergence order of (p+1) on *distorted* grids for either linear advection, diffusion, or convection-diffusion.
Best Regards, Doug On Friday, April 26, 2019 at 11:48:58 AM UTC-4, Gary Uppal wrote: > > Prof. Bangerth, > > Thanks for the reply. I get problems on less distorted meshes as well. > For example, in the mesh with a hole shown below. I eventually need > multiple holes and an advection term. Actually, the advection seems ok and > I mainly get issues when the diffusion is large. If the distorted mesh is > the issue, is there a better way to construct the geometry? Is there not a > way to get better accuracy even with the distorted mesh? I've tried > refining the mesh and lowering the time step, but that doesn't seem to > change the results. > > Thank you, > Gary > [image: with_hole.jpg] > > On Fri, 26 Apr 2019 at 04:50, Doug <doug....@gmail.com <javascript:>> > wrote: > >> Prof. Bangerth, >> >> I am encountering a similar "issue" regarding distorted grids. I am using >> DG for linear advection with a manufactured solution to verify the code. >> >> I recover optimal convergence orders (p+1) for straight meshes when >> refining in h. I recover optimal orders (p+1) for a sine-transformed grid >> with some skewed angles up to around 45 degrees (as seen attached). >> >> However, I lose my optimal (p+1) order as soon as I apply any small >> amount of random distortion to my final refined grid (see attached). Note >> that the distortion factor of 0.15 is used, but this loss of order also >> occurs for small distortion factors down to 0.01. Instead of (p+1), I may >> recover (p) or even (p-1). This only applies when the final grid is >> randomly distorted. If I start from a coarse-ish distorted grid (say 100 >> cells), and refine globally such that its children aren't distorted, I then >> recover my optimal orders again. >> >> Since my distorted cell still have reasonable angles no matter its >> refinement level, degenerate cells wouldn't be an issue. Also, since I am >> not performing adaptive refinement, there are no hanging nodes such that >> the triangulation becomes irregular. >> >> Would you know happen to know if some other condition is violated from >> random distortions such that error estimates do not hold? >> >> Best regards, >> >> Doug >> >> On Thursday, April 25, 2019 at 11:15:31 PM UTC-4, Wolfgang Bangerth wrote: >>> >>> On 4/25/19 2:50 PM, Gary Uppal wrote: >>> > >>> > I am trying to solve the diffusion equation with Discontinuous >>> Galerkin >>> > elements. The solution looks good with a regular structured mesh, but >>> if I >>> > distort the mesh, the solution blows up and does not converge. Is >>> there an >>> > obvious reason this would happen? I later need a mesh with holes in >>> it, so I >>> > cannot always use the structured mesh. >>> > >>> > I use the interior penalty method and get the diffusion matrix using >>> > MeshWorker as in Step-39. I compute the mass matrix and solve the >>> diffusion >>> > equation with an implicit backward Euler method and am using periodic >>> boundary >>> > conditions. Snapshots of the solution with structured and distorted >>> meshes are >>> > shown below. Any help is appreciated! >>> >>> Gary -- the wrong solutions happen on cells that are very nearly >>> degenerate >>> (i.e., are almost triangular). The finite element theory says that the >>> error >>> between the exact and the numerical solution is bounded by a constant >>> times >>> some power of h, where the constant depends on the minimal and maximal >>> angles >>> at the vertices of the cells. Theory then also predicts that this >>> constant >>> goes to infinity if the maximal angle at one vertex comes close to 180 >>> degrees >>> -- which is exactly what is happening in your case. >>> >>> So choose a mesh that is less distorted and you should be fine. >>> >>> Best >>> W. >>> >>> >>> -- >>> ------------------------------------------------------------------------ >>> Wolfgang Bangerth email: bang...@colostate.edu >>> www: >>> http://www.math.colostate.edu/~bangerth/ >>> >>> -- >> The deal.II project is located at http://www.dealii.org/ >> For mailing list/forum options, see >> https://groups.google.com/d/forum/dealii?hl=en >> --- >> You received this message because you are subscribed to a topic in the >> Google Groups "deal.II User Group" group. >> To unsubscribe from this topic, visit >> https://groups.google.com/d/topic/dealii/ZSHIDbp7yfE/unsubscribe. >> To unsubscribe from this group and all its topics, send an email to >> dea...@googlegroups.com <javascript:>. >> For more options, visit https://groups.google.com/d/optout. >> > -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. To unsubscribe from this group and stop receiving emails from it, send an email to dealii+unsubscr...@googlegroups.com. For more options, visit https://groups.google.com/d/optout.
