The method of lines (MOL, NMOL, NUMOL[8][9][10]) is a technique for solving partial differential equations (PDEs) in which all dimensions except one are discretized.
A large number of integration routines have been developed over the years in many different programming languages, and some have been published as open source resources.
[12] The finite element method (FEM) is a numerical technique for finding approximate solutions to boundary value problems for differential equations.
Analogous to the idea that connecting many tiny straight lines can approximate a larger circle, FEM encompasses all the methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a larger domain.
Spectral methods are techniques used in applied mathematics and scientific computing to numerically solve certain differential equations, often involving the use of the fast Fourier transform.
Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the fastest possible, when the solution is smooth.
[14] Meshfree methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computing time and programming effort.
The problems on the subdomains are independent, which makes domain decomposition methods suitable for parallel computing.
In the engineering practice in the finite element method, continuity of solutions between non-matching subdomains is implemented by multiple-point constraints.
Finite element simulations of moderate size models require solving linear systems with millions of unknowns.
Multigrid (MG) methods in numerical analysis are a group of algorithms for solving differential equations using a hierarchy of discretizations.
They are an example of a class of techniques called multiresolution methods, very useful in (but not limited to) problems exhibiting multiple scales of behavior.
The typical application for multigrid is in the numerical solution of elliptic partial differential equations in two or more dimensions.