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Category# Discrete exterior calculus

Summary

The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points.
The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain.
The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then approximates a solution by minimizing an associated error function via the calculus of variations.
Studying or analyzing a phenomenon with FEM is often referred to as finite element analysis (FEA).
The subdivision of a whole domain into simpler parts has several advantages:
Accurate representation of complex geometry
Inclusion of dissimilar material properties
Easy representation of the total solution
Capture of local effects.
Typical work out of the method involves:
dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem
systematically recombining all sets of element equations into a global system of equations for the final calculation.
The global system of equations has known solution techniques and can be calculated from the initial values of the original problem to obtain a numerical answer.

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