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How would you define the finite-element method?
The finite-element method is a computational method that subdivides a CAD model into very small but finite-sized elements of geometrically simple shapes. The collection of all these simple shapes constitutes the so-called finite-element mesh.
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. 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). This gives an approximate local description of the physics by a set of simple linear (but sometimes nonlinear) equations. 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.
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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. To speed things up, this structure is exploited by using a tailored numerical method. A method may be suitable for structural mechanics but not for electromagnetics, and vice versa. Historically, the method was first applied to structural analysis. 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.
How would you define the finite-difference method?
The finite-difference method is the most direct approach to discretizing partial differential equations. You consider a point in space where you take the continuum representation of the equations and replace it with a set of discrete equations, called finite-difference equations. The finite-difference method is typically defined on a regular grid and this fact can be used for very efficient solution methods. The method is therefore not usually used for irregular CAD geometries, but more often for rectangular or block-shaped models.
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