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Note: On this page, we use the acronym FIEM (Fast Integral Equation Methods) to refer to the combination of integral equation formulations with fast solvers. FIEM is generally useful only for linear problems (which, of course, frequently arise as subproblems in solvers for nonlinear equations). FIEM is the most competitive for problems involving constant-coefficient partial differential operators, and no body loads. Since the integral equation formulations of such problems are defined only on the boundary of the domain, FIEM tends to be far faster than competing methods for such problems. FIEM is a good choice even for problems involving body loads and non-constant coefficients if very high accuracy and / or stability is of paramount importance. Integral equation based methods are also particularly useful for external problems, such as, e.g., scattering problems. As a practical matter, FIEM is not ideally suited for multi-physics problems, since it can in such environments be difficult to derive well-conditioned integral equation formulations. Note: Unfortunately, there exist many situations in which FIEM would in principle be an excellent choice of numerical method, but there currently do not exist available software (or even well-established theory). Such situations include
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