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Magnetic helicity is a conserved quantity of ideal magneto-hydrodynamicscharacterized by an inverse turbulent cascade. Accordingly, it is often invokedas one of the basic physical quantities driving the generation and structuringof magnetic fields in a variety of astrophysical and laboratory plasmas. Weprovide here the first systematic comparison of six existing methods for theestimation of the helicity of magnetic fields known in a finite volume. Allsuch methods are reviewed, benchmarked, and compared with each other, andspecifically tested for accuracy and sensitivity to errors. To that purpose, weconsider four groups of numerical tests, ranging from solutions of thethree-dimensional, force-free equilibrium, to magneto-hydrodynamical numericalsimulations. Almost all methods are found to produce the same value of magnetichelicity within few percent in all tests. In the more solar-relevant andrealistic of the tests employed here, the simulation of an eruptive flux rope,the spread in the computed values obtained by all but one method is only 3%,indicating the reliability and mutual consistency of such methods inappropriate parameter ranges. However, methods show differences in thesensitivity to numerical resolution and to errors in the solenoidal property ofthe input fields. In addition to finite volume methods, we also briefly discussa method that estimates helicity from the field lines' twist, and one thatexploits the field's value at one boundary and a coronal minimal connectivityinstead of a pre-defined three-dimensional magnetic-field solution.

Magnetic helicity estimations in models and observations of the solar magnetic field. Part I: Finite volume methods