Between these four different meshes for the same shape, compare them whilst considering the different results for displacement and stress. Moreover, the difference in computational cost and accuracy of the meshes.
If you put a Lagrange linear element on the first mesh (upper left with non-convex elements), you're probably going to fail to have an invertable Jacobian, and so you'll fail to be able to compute a stiffness matrix for each element, assemble the system, and compute a result altogether. Given your comment, you've probably noticed this already. Otherwise, these meshes are very different and you don't specify what function space(s) you draw your trial and test functions from, so the effective element size and discretization properties are hard for us to comment on. Assuming the lowest order conforming spaces that lead to solutions, I'd guess that the lower left and upper right meshes are a toss-up for cost and accuracy. They have the same number of degrees of freedom. None of these meshes have enough elements to make accuracy worth thinking about in too much detail. A real problem is probably going to want substantially more elements to give you a good solution.
The inner corners of the star could lead to singularities in the solution or its derivatives (like the classic L-shaped domain for the Laplacian/Poisson problem) depending on the boundary conditions which might indicate that the leading constant in the error estimate is large and that low accuracy is to be expected on such coarse meshes. Have you tried uniformly refining these meshes until you reach the asymptotic regime for the problem?