Recently, polygonal finite element methods have received considerable attention. This is because general meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and  non-matching interfaces. 

In this talk,  a new computational paradigm for discretizing PDEs  is presented via the staggered Galerkin method on general meshes.
First, a class of locally conservative, lowest order staggered discontinuous Galerkin method on general quadrilateral/polygonal
meshes for elliptic problems are proposed.  The method can be flexibly applied to rough grids such as highly distorted  meshes.
Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging
nodes. We derive a simple residual-type error estimator. Numerical results indicate that optimal convergence can be achieved for both
the potential  and vector variables, and the singularity can be well-captured by the proposed error estimator. Then, some
applications to diffusion equations, Stokes equations, and linear elasticity equations are considered. Finally, we extend this
approach to high-order polynomial approximations on general meshes.