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The Finite Element Method (FEM) is a popular numerical technique used for solving boundary-valued problems in various areas of mechanical and electrical engineering.

At its heart, the FEM minimizes a variational form of the the differential equations that describe physical phenomena to obtain the unknown quantity. Typically the unknown is caused by a forcing function in the differential equation. The solution region is first subdivided into small sub-cells or elements, using CAD drawing software and meshing tools. For two-dimensional problems such as beam stress analysis and solution of waveguide field distributions, the regions are typically meshed as triangles or quads. For three-dimensional problems, tetrahedrons or quadrilateral bricks are often used. A basis function is chosen which is expected to properly represent the quantity to be solved for (electric fields, etc). These can range from piecewise-linear to quadratic or higher-order polynomial approximations inside each element.

Incorporation of this expansion function into the differential equation(s) leads to a linear system which can be easily solved for the unknown on a computer using available matrix routines.

One of the most attractive features of the FEM is that it will treat inhomogeneous regions and materials, as each element may have different material properties. Unfortunately, the FEM can only model regions of finite extent (no pun intended), and the boundaries often must be terminated with some sort of artificial absorber or boundary condition. Problems such as those of unbounded radiation cannot be easily solved with the FEM.

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