How would you define the finite-difference method? Over the last ten years or so, it has been realized that the finite element method is also suitable for a large class of multi-physics problems. Historically, the method was first applied to structural analysis. A method may be suitable for structural mechanics but not for electromagnetics, and vice versa. To speed things up, this structure is exploited by using a tailored numerical method. The type of solver used depends on the original physics, since each type of physics gives its unique imprint on the structure of the matrix. When the contributions from all elements are assembled you end up with a large sparse matrix equation system that can be solved by any of a number of well-known sparse matrix solvers. This gives an approximate local description of the physics by a set of simple linear (but sometimes nonlinear) equations. This is handled by approximating the fields within each element as a simple function, such as a linear or quadratic polynomial, with a finite number of degrees of freedom (DOFs).
The next step is to take a system of field equations, mathematically represented by partial differential equations (PDEs) that describe the physics you are interested in, and formulate these equations for each element. The collection of all these simple shapes constitutes the so-called finite-element mesh. The finite-element method is a computational method that subdivides a CAD model into very small but finite-sized elements of geometrically simple shapes. How would you define the finite-element method? This file type includes high-resolution graphics and schematics when applicable.
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