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Solutions of scalar elliptic or parabolic boundary value problems satisfy
under certain conditions maximum principles. These principles reflect
important properties of the solution of the physical process that is
modeled with the boundary value problem. In many applications, it is
essential that also numerical approximations to this solution possess
these properties. The mathematical formulation of this feature is the
discrete maximum principle (DMP).

The fundamental idea and derivation of finite element methods (FEMs) is purely
based on mathematical arguments, without taking into consideration
physical properties. Nevertheless, there are FEMs that satisfy DMPs under
certain conditions, e.g., on the underlying mesh. This talk presents an
introduction of the topic and a survey of such methods.