A Note on why Enforcing Discrete Maximum Principles by a simple a Posteriori Cutoff is a Good Idea
Provides a simple, theoretically justified method to enforce discrete maximum principles that also enhances accuracy, benefiting numerical analysts and practitioners using finite element methods.
The paper shows that enforcing discrete maximum principles via a simple a posteriori cutoff improves approximation accuracy in the energy norm for various PDE approximations, including hp-finite elements, without geometric restrictions.
Discrete maximum principles in the approximation of partial differential equations are crucial for the preservation of qualitative properties of physical models. In this work we enforce the discrete maximum principle by performing a simple cutoff. We show that for many problems this a posteriori procedure even improves the approximation in the natural energy norm. The results apply to many different kinds of approximations including conforming higher order and $hp$-finite elements. Moreover in the case of finite element approximations there is no geometrical restriction on the partition of the domain.