Mimetic

# Mimetic

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In mathematics, mimesis is the quality of a numerical method which imitates some properties of the continuum problem. The goal of numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

is to approximate the continuum, so instead of solving a partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

one aims to solve a discrete version of the continuum problem. Properties of the continuum problem commonly imitated by numerical methods are conservation law
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....

s, solution symmetries
Symmetry in physics
In physics, symmetry includes all features of a physical system that exhibit the property of symmetryâ€”that is, under certain transformations, aspects of these systems are "unchanged", according to a particular observation...

, and fundamental identities and theorems of vector and tensor calculus like the divergence theorem
Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gaussâ€“Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...

.
Both finite difference
Finite difference
A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...

or finite element method
Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...

can be mimetic; it depends on the properties that the method has.

For example, a mixed finite element method applied to Darcy flows
Darcy's law
Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand...

strictly conserves the mass
Conservation of mass
The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

of the flowing fluid.

The term geometric integration
Geometric integrator
In the mathematical field of numerical ordinary differential equations, a geometric integrator is a numerical method that preserves geometric properties of the exact flow of a differential equation.-Pendulum example:...

denotes the same philosophy.