Finite Element Method

A numerical method that divides a complex domain into smaller elements to approximate the solution of differential equations.

Source: Strang G, Fix GJ. An Analysis of the Finite Element Method. Prentice-Hall; 1973.

The finite element method approximates the solution of differential equations by dividing a complex domain into smaller elements and building an approximate solution piece by piece across them. Each element carries a simple local approximation, and these are assembled into a solution over the whole domain. This division of a complicated region into manageable elements is how the finite element method handles problems that would be hard to solve directly By covering an awkward region with a mesh of simple elements and stitching their local solutions together, the method handles geometries that a single global approximation could not manage.

Source: Strang G, Fix GJ. An Analysis of the Finite Element Method. Prentice-Hall; 1973.

The finite element method divides the domain into smaller elements because a complex region is difficult to treat as a whole, whereas each small element admits a simple local approximation. Breaking the domain down lets the method approximate the solution of differential equations across shapes that vary from place to place. This subdivision into elements is the defining step that gives the finite element method its flexibility with complex domains By covering an awkward region with a mesh of simple elements and stitching their local solutions together, the method handles geometries that a single global approximation could not manage.

Source: Strang G, Fix GJ. An Analysis of the Finite Element Method. Prentice-Hall; 1973.

A complex domain suits the finite element method, since it is designed to divide such a domain into smaller elements to approximate the solution of differential equations. Irregular or complicated regions that resist a single global treatment can be covered by a mesh of elements. This ability to handle complex domains by subdivision is a central reason the finite element method is used By covering an awkward region with a mesh of simple elements and stitching their local solutions together, the method handles geometries that a single global approximation could not manage.

Source: Strang G, Fix GJ. An Analysis of the Finite Element Method. Prentice-Hall; 1973.

The finite element method produces an approximate solution of the differential equations across the elements, assembling local approximations on each small element into a solution over the whole domain. Because the domain has been divided into elements, the method builds the overall answer from these pieces. This piecewise construction over the elements is what the finite element method yields By covering an awkward region with a mesh of simple elements and stitching their local solutions together, the method handles geometries that a single global approximation could not manage.

Source: Strang G, Fix GJ. An Analysis of the Finite Element Method. Prentice-Hall; 1973.

The finite element method divides a complex domain into smaller elements to approximate the solution of differential equations, while the finite difference method approximates derivatives using finite differences to solve them. Both solve differential equations numerically but discretise differently, the finite element method by subdividing the domain and the finite difference method by replacing derivatives. The two are connected as alternative numerical approaches to the same kind of problem By covering an awkward region with a mesh of simple elements and stitching their local solutions together, the method handles geometries that a single global approximation could not manage.

Source: Strang G, Fix GJ. An Analysis of the Finite Element Method. Prentice-Hall; 1973.