Can we extend the FEM to general meshes while maintaining the ease of implementation and computational cost comparable to that of FEM? The brand new Virtual Element Method (VEM) achieves that, and much more.
The Virtual Element spaces are like the usual (polynomial) finite element spaces with the addition of suitable non-polynomial functions. The novelty of the VEM approach is that it avoids expensive evaluations of the non-polynomial (virtual) functions by basing all computations on a set of carefully chosen degrees of freedom. In doing that we can easily deal with complicated element geometries and/or higher continuity conditions (like C1, C2, etc.), while maintaining the computational complexity comparable to that of standard finite element computations.
As you might expect, the choice and number of the degrees of freedom depends on the continuity requirements. If mesh flexibility is the goal, while one is ready to give up on regularity, other approaches can be considered. For instance, the discontinuous Galerkin method is naturally suited to deal with polygonal/polyhedral meshes.
Within this talk, I will give a gentle introduction to these approaches as well as present very recent results.