Multidimensional Finite Element Methods
Multidimensional partial differential equations arise in many
problems in science and engineering. Most of the numerical codes
for the approximation of partial differential equations are limited
to at most three-dimensional problems. The main reason for this is that
the number of degrees of freedom in discretization of a multidimensional
domain using finite element methods increases exponentially with its dimension.
This poses a severe restriction on the dimension of the domain to be
approximated using a computer with limited memory. Also, most of the problems
we need to approximate are one-, two- or three-dimensional, since the
domain is usually a subset of the physical space. Nevertheless,
approximations of PDEs that model processes in the phase space usually require
dimensions higher than three. One example is the approximation
of the Fokker-Planck equation describing the probability density
of the solution of a stochastically perturbed system of ordinary
differential equations, with linearly appearing "white Gaussian noise".
Examples:
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Linear oscilator
Duffing oscilator