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The finite element method (FEM, for example, Bulgarelli B et al 1999) is a discretization technique of the radiative transfer equation (RTE), like the discrete ordinate method (DOM) and the spherical-harmonics method (SHM).
In the finite element approach to solving the RTE, the dependence of radiance over the azimuth angle is decoupled by expanding the phase function (or the phase matrix) in Legendre polynomials (for example, http://mathworld.wolfram.com/LegendrePolynomial.html, and the radiance (or the Stokes vector) in a Fourier cosine series (for example, http://mathworld.wolfram.com/FourierCosineSeries.html) of the azimuth angle. This enables the analytical integration with respect to the azimuth angle. The remaining integrals are approximated by projecting each radiance harmonic onto a set of basis functions, each different from zero only in a finite interval, hence, the "finite element" designation of this method. The basis functions can be chosen arbitrarily, provided only that the sum of the intervals where they differ from zero, is equal to the whole interval where the solution is defined. The resulting system of matricial equations is solved analytically, imposing the necessary boundary conditions. Vertically inhomogeneous media can be divided into a large number of homogeneous sub-layers. Continuity of the solution at the boundaries between sub-layers is required.
The method is advantageous for two main reasons. First, since the FEM basis functions are finite, they generate a piecewise approximate solution which may converge faster than the global approximation adopted in SHM. Secondly, this method provides the exact flux conservation without any conditions, in contrast to DOM where the flux is conserved only if the number Gaussian quadrature (for example, http://mathworld.wolfram.com/GaussianQuadrature.html) points used to solve the integral is equal (or higher) to the order of polynomials for the phase function. Hence, the FEM approach is particularly suitable for highly asymmetric phase functions.
See also radiative transfer equation, solution methods, numerical, finite element and radiative transfer equation, solution methods, numerical for additional references.
Bulgarelli B. 2008. Finite element method (www.tpdsci.com/Tpc/RTESolNumFe.php). In: Radiative transfer equation (RTE): Numerical solution methods. Kokhanovsky A. A. (ed.) (www.tpdsci.com/Tpc/RTESolNumIntro.php). Top. Part. Disp. Sci. (www.tpdsci.com).