Geodesic
Computation of the attractive force of an ellipsoid
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 12, pp. 2063-2071

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A numerical method for computing the attractive force of an ellipsoid is proposed that does not involve separating subdomains with singularities. The sought function is represented as a triple integral such as the inner integral of the kernel can be evaluated analytically with the kernel treated as a weight function. The inner integral is approximated by a quadrature for the product of functions, of which one has an integrable singularity. As a result, the integrand obtained before the second integration has only a weak logarithmic singularity. The subsequent change of variables yields an integrand without singularities. Based on this approach, at each stage of integral evaluation with respect to a single variable, quadrature formulas are derived that do not have singularities at integration nodes and do not take large values at these nodes. For numerical experiments, a rather complicated test function is constructed that is the exact attractive force of an ellipsoid of revolution with an elliptic density distribution. 
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     author = {A. O. Savchenko},
     title = {Computation of the attractive force of an ellipsoid},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {2063--2071},
     publisher = {mathdoc},
     volume = {53},
     number = {12},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_12_a10/}
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A. O. Savchenko. Computation of the attractive force of an ellipsoid. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 12, pp. 2063-2071. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_12_a10/