# Finite element results by different meshes

There are some technique to generate mesh in a domain. My qustion is that: Is there any difference between the results using different techniques for mesh generation? If yes which one is better. For example If I solve an equation by regular triangle mesh and by criss-cross mesh wich one must be better (especially in adaptive fem)? I think maybe criss-cross mesh is better because of the number of elements. But using more elements leads much computaitional cost!!

• A properly formulated/implemented FEM should converge to the same solution, regardless of the mesh that it is solved on, on a sufficiently refined mesh. That said, some mesh generation methods do work better if you have some information about the solution ahead of time (eg, boundary layer for fluid flow). Can you give more details about your problem? Have you noticed significant differences between meshes? Jan 29, 2016 at 13:18
• @TylerOlsen, thanks for your comment, I am so new in finite element and these days just try to undrestand fem better and then programming. If there is a refrence to show differences between meshes, please let me know.
– Rosa
Jan 29, 2016 at 13:24
• There are plenty of references out there, but it sounds like you should be reading an introductory finite element method book in order to get a better handle on the basic method before delving into the details of which mesh generation technique is best-suited for adaptive (h or p?) fem. amazon.com/The-Finite-Element-Method-Fundamentals/dp/1856176339 and amazon.com/Finite-Element-Method-Mechanical-Engineering/dp/… are both good books. Jan 29, 2016 at 13:36

As a general rule, finite element solutions are more accurate on meshes with cells that (i) deviate less from the optimal shape (which for triangles are equilateral triangles and for rectangles are squares), and (ii) have local symmetries. Symmetries in a quadrilateral mesh would, for example, mean that four cells come together at every vertex where the four incoming edges form two straight lines. For triangles, this would mean that every vertex has exactly 6 adjacent cells, rather than alternating between 4 and 8 for a criss-cross mesh.

The reason this affects the solution is that (i) the deviation from the optimal shape shows up in the norm of the transformation that appears in the interpolation estimate, and equilateral triangles and squares happen to have the smallest norm; (ii) symmetries allow for Taylor expansions of the error at vertices where certain terms magically cancel, and the solution is of higher order at individual points of the mesh.

Of course, as has already been pointed out by @TylerOlsen in the comments, whatever choice of mesh you have, the solution will converge asymptotically as the maximal mesh size $h\rightarrow 0$.

• Could you please point me to the formulas which show your points (i) and (ii)? May 22, 2018 at 13:10
• I don't have particular papers in mind, but you could look through the ones by Carsten Carstensen from the 1990s for the optimal shape if you wanted to know things in more detail than you ever though. You could also look into estimates for the interpolation error from the 1970s or so. For the second point, you should look through the literature on superconvergence effects, also from the 1970s and 1980s. May 30, 2018 at 18:57

I have written an exhaustive answer on meshing here: https://engineering.stackexchange.com/questions/449/meshing-of-complex-geometrical-domains/7326#7326

The resultant mesh quality is more important than the technique used to create the mesh. Depending on the domain and type of analysis to be carried out, many different meshes can lead to the correct answer with reasonable computational cost.

To get a correct answer from analysis, you need to ensure one more thing besides mesh quality which is mesh convergence. Mesh convergence is achieved when further refinement of mesh doesn't increase the accuracy of your result by a significant margin. So practically, your result becomes independent of the mesh and the analysis results are pretty much accurate given all other conditions are taken care of.

PS: A graphic explanation of mesh convergence to be added.