On 4/26/19 2:50 AM, Doug wrote: > > 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.
I don't know enough about DG theory to tell what this is from. But this is true: * for your example with the sine-transformed domain, if you refine the mesh sufficiently many times, each cell will be getting closer and closer to a parallelogram * the same is the case for your initially-distorted then refined mesh. As a matter of fact, that's a general theorem: Start with some mesh and refine it sufficiently often, and all cells with tend to parallelograms. Why does this matter? Because the mapping from the reference cell to a parallelogram is linear, and consequently the derivative of the mapping (which shows up in the convergence proofs of the finite element method) will be a constant on every cell in the limit of h->0. On the other hand, if you refine a mesh *and then distort them randomly*, then the mapping will *never* be linear, and its gradient never be constant on cells. It would not surprise me if this yielded issues in the convergence theory that destroy your convergence order. Indeed, I have seen statements like this in the literature, and it can be solved by not using a bilinear mapping from the reference cell to the real cell but just a linear mapping determined by 3 of the 4 vertices of the cell. This would be bad for continuous finite elements because these functions would not be continuous along two of the four edges, but that really doesn't matter for DG methods :-) I'm not enough of a theorist to tell you where to look for these kinds of statements, but I'm not surprised by the behavior you describe either. Best W. -- ------------------------------------------------------------------------ Wolfgang Bangerth email: [email protected] 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 the Google Groups "deal.II User Group" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. For more options, visit https://groups.google.com/d/optout.
