  This paper proposes and analyzes an optimal preconditioner for a general
linear symmetric positive definite (SPD) system by following the basic idea of
the well-known BPX framework. The SPD system arises from a large number of
nonstandard finite element methods for diffusion problems, including the
well-known hybridized Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM) mixed
element methods, the hybridized discontinuous Galerkin (HDG) method, the Weak
Galerkin (WG) method, and the nonconforming Crouzeix-Raviart (CR) element
method. We prove that the presented preconditioner is optimal, in the sense
that the condition number of the preconditioned system is independent of the
mesh size. Numerical experiments are provided to confirm the theoretical
results.
