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!!
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 $h\rightarrow 0$.
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.