Geodesic
Calculation of the volume potential for ellipsoidal bodies
Sibirskij žurnal industrialʹnoj matematiki, Tome 15 (2012) no. 1, pp. 123-131.

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We consider quadrature formulas for calculating the integral of the product of functions, one of which has an integrable singularity admitting exact calculation of the integral. Basing on these formulas, we suggest a method for calculating numerically the potential of the ellipsoid without cutting out a region near the singularity. We approximate the inner integral by a function with a weak logarithmic singularity, and a subsequent change of variables enables us to perform further numerical integration without singularities in the integrand. For simulations we construct a quite complicated test function amounting to the exact potential of an ellipsoid of revolution with an elliptic density distribuion.
Keywords: volume potential, ellipsoid.
Mots-clés : simulation, quadrature formulas 
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A. O. Savchenko. Calculation of the volume potential for ellipsoidal bodies. Sibirskij žurnal industrialʹnoj matematiki, Tome 15 (2012) no. 1, pp. 123-131. http://geodesic.mathdoc.fr/item/SJIM_2012_15_1_a11/

[1] Kondratev B. P., Teoriya potentsiala. Novye metody i zadachi s resheniyami, Mir, M., 2007

[2] Muratov R. Z., Potentsialy ellipsoida, Atomizdat, M., 1976

[3] Nikiforov A. F., Uvarov V. B., Spetsialnye funktsii matematicheskoi fiziki, Nauka, M., 1978 | MR

[4] Khokni R., Istvud Dzh., Chislennoe modelirovanie metodom chastits, Mir, M., 1987

[5] Beylkin G., Cramer R., Fann G., Harrison R. J., “Multiresolution separated representations of singular and weakly singular operators”, Appl. Comput. Harmon. Anal., 23 (2007), 235–253 | DOI | MR | Zbl

[6] Hackbuch W., “Efficient convolution with Newton potential in $d$ dimensions”, Numer. Math., 110 (2008), 449–489 | DOI | MR

[7] Maz'ya V., Schmidt G., Approximate Approximations, Math. Surveys and Monographs, 141, AMS, N.Y., 2007 | MR

[8] Lanzara F., Maz'ya V., Schmidt G., “Approximate approximations from scattered data”, J. Approx. Theory, 145:2 (2007), 141–170 | DOI | MR

[9] Lanzara F., Maz'ya V., Schmidt G., “On the fast computation of high dimensional volume potentials”, Math. Comp., 80 (2011), 887–904 | DOI | MR | Zbl

[10] Ivanov T., Maz'ya V., Schmidt G., “Boundary layer approximate approximations and cubature of potentials in domains”, Adv. Comput. Math., 10:3–4 (1999), 311–342 | DOI | MR | Zbl

[11] Bakhvalov N. S., Chislennye metody, Nauka, M., 1973 | MR | Zbl

[12] Titchmarsh E., Teoriya funktsii, Nauka, M., 1980 | MR | Zbl

[13] Ilin V. A., Poznyak E. G., Osnovy matematicheskogo analiza, Ch. I, Nauka, M., 1971 | MR

[14] Duboshin G. N., Nebesnaya mekhanika. Osnovnye zadachi i metody, Nauka, M., 1968 | Zbl