Geodesic
A local algorithm for smooth approximation of approximate difference and nonsmooth variational solutions
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 1, pp. 51-60 
V. V. Smelov. A local algorithm for smooth approximation of approximate difference and nonsmooth variational solutions. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 1, pp. 51-60. http://geodesic.mathdoc.fr/item/SJVM_2001_4_1_a5/
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     title = {A~local algorithm for smooth approximation of approximate difference and nonsmooth variational solutions},
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     pages = {51--60},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2001_4_1_a5/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

A local algorithm for smooth approximation is proposed for approximate solutions of the one-dimensional problems which are obtained by difference methods or variational algorithms on the basis of piecewise smooth test functions. The algorithm is aimed at approximate solutions which are obtained with an error $O(h^\nu)$, $\nu=1,2$. The algorithm has been primordially parallelized and is utmost easy both in theoretical and in practical aspects. A result of the local approximation is a twice continuous differentiable function which keeps geometric properties of the initial approximate solution. Certain advantages of the proposed algorithm as compared to the cubic splines are shown.

[1] Streng G., Fiks Dzh., Teoriya metoda konechnykh elementov, Mir, M., 1977 | MR

[2] Marchuk G. I., Metody vychislitelnoi matematiki, Nauka, M., 1980 | MR

[3] Marchuk G. I., Agoshkov V. I., Vvedenie v proektsionno-setochnye metody, Nauka, M., 1981 | MR

[4] Marchuk G. I., Agoshkov V. I., “O vybore koordinatnykh funktsii v obobschennom metode Bubnova–Galerkina”, DAN SSSR, 232:6 (1977), 1253–1256 | MR | Zbl

[5] Smelov V. V., “O predstavlenii kusochno-gladkikh funktsii bystroskhodyaschimisya trigonometricheskimi ryadami”, Sib. zhurn. vychisl. matematiki / RAN. Sib. otd-nie. — Novosibirsk, 2:4 (1999), 385–394 | Zbl

[6] Beatson R. K., Wolkowicz H., “Post-processing piecewise cubics for monotonicity”, SIAM J. Numer. Anal., 26 (1989), 480–502 | DOI | MR | Zbl

[7] Kvasov B. I., “Algoritmy izogeometricheskoi approksimatsii obobschennymi kubicheskimi splainami”, Zhurn. vychisl. matem. i mat. fiziki, 36:12 (1996), 3–22 | MR

[8] Eskhoff Knut S., “On a high order numerical method for solving partial differential equations in complex geometries”, J. of Scientific Computing, 12:2 (1997), 119–138 | DOI | MR

[9] Fikhtengolts G. M., Kurs differentsialnogo i integralnogo ischisleniya, v. 1, OGIZ, M., L., 1947

[10] Smelov V. V., Zadachi Shturma–Liuvillya i razlozheniya funktsii v bystroskhodyaschiesya ryady, Izd-vo SO RAN, Novosibirsk, 2000