On the solution of second-order linear elliptic equations

A. V. Shilkov

Matematičeskoe modelirovanie, Tome 31 (2019) no. 6, pp. 55-81.

Résumé

A method for solving interior boundary value problems for second-order linear elliptic equations by introducing ray variables is described. The region is divided into cells, within which the coefficients and sources of the equations have the smoothness and continuity properties necessary for the existence of regular solutions in the cell. The finite discontinuities of the coefficients (if any) pass along the cell boundaries. The regular solution in a cell is sought in the form of a superposition of the contributions made by volume and boundary sources placed on rays arriving at a given point from the cell boundaries. Next, a finite-analytic scheme for the numerical solution of boundary value problems in a domain with discontinuous coefficients and sources is constructed by matching the regular solutions emerging from cells at the cell boundaries. The scheme does not exhibit the rigid dependence of the accuracy of approximation on the sizes and shape of the cells, which is inherent in finite-difference schemes.

Détail

Mots-clés : elliptic equations
Keywords: boundary value problem, method of ray variables, numerical methods, finite-analytic scheme.

Bibliographie

[1] Vladimirov V. S., “Chislennoe reshenie kineticheskogo uravneniia dlia sfery”, Vych. matematika, 3, ed. A.A. Abramov, Izd-vo AN SSSR, M., 1958, 3–33

[2] Tikhonov A. N., Samarskii A. A., “Homogeneous difference schemes of a high degree of accuracy on non-uniform nets”, USSR Comput. mathematics and math. physics, 1:3 (1962), 465–486 | DOI | MR | Zbl

[3] Samarskii A. A., Vvedenie v teoriiu raznostnykh skhem, Nauka, M., 1971

[4] Il'in A.M., “Differencing scheme for a differential equation with a small parameter affecting the highest derivative”, Math. notes of the Acad. of sciences of the USSR, 6:2 (1969), 596–602 | DOI | Zbl

[5] Emelyanov K. V., “Raznostnaia skhema dlia trekhmernogo ellipticheskogo uravneniia s malym parametrom pri starshikh proizvodnykh”, Trudy In-ta matem. i mekhaniki Uralskogo Nauchn. Tsentra AN SSSR, 11, Sverdlovsk, 1973, 30–42

[6] Kulikov Yu.N., Chetverushkin B. N., “An implicit difference method for temperature determination in problems of radiation gas dynamics”, USSR Comput. mathematics and math. physics, 13:1 (1974), 174–186 | DOI | MR | Zbl

[7] Charakhchyan A. A., “Raschet nestatsionarnykh, sfericheski simmetrichnykh techenii izluchaiushchego serogo gaza”, Dinamika izluchaiushchego gaza, 1, ed. V.V. Aleksandrov, VTs AN SSSR, M., 1974, 54–74

[8] Chen C. J., Naseri-Nashet H., Ho K. S., “Finite analytic numerical solution of heat transfer in two dimensional cavity flow”, J. Numer. heat transfer, 4:2 (1981), 179–197

[9] Sun Y., Militzer J., “The piecewise parabolic finite analytic method. Part I: Theory”, Appl. math. modelling, 16 (1992), 576–588 | DOI | MR | Zbl

[10] Chen C. J., Bernatz R. A., Carlson K. D., Lin W., de Vahl D. G., “Finite analytic method in flows and heat transfer”, Appl. mech. rev., 55:2 (2002), B34–B34 | DOI

[11] Pontaza J. P., Chen H. C., Reddy J. N., “A local-analytic-based discretization procedure for the numerical solution of incompressible flows”, Num. methods in fluids, 49:6 (2005), 657–699 | DOI | MR | Zbl

[12] Shilkov A. V., “Even and odd parity kinetic equations of particle transport. 2: A finite analytic characteristic scheme for one dimensional problems”, Math. models and computer simulations, 7:1 (2015), 36–50 | DOI | MR | Zbl

[13] Shilkov A. V., “Even and odd parity kinetic equations of particle transport. 3: Finite analytic schemes on tetrahedral”, Math. models and computer simul., 7:5 (2015), 409–429 | DOI | MR | Zbl

[14] Basko M. M., Tsygvintsev I. P., “A hybrid model of laser energy deposition for multidimensional simulations of plasmas and metals”, Comp. physics communications, 214 (2017), 59–70 | DOI | MR | Zbl

[15] Shilkov A. V., “Reshenie ellipticheskikh uravnenii metodom luchevykh peremennykh”, Keldysh Institute preprints, 2017, 119, 36 pp.

[16] Koshlyakov N. S., Gliner E. B., Smirnov M. M., Uravneniia v chastnykh proizvodnykh matematicheskoi fiziki, Vysshaia shkola, M., 1970

[17] Babich V. M., Kapilevich M. B., Mikhlin S. G., Lineinye uravneniia matematicheskoi fiziki, Spravochnik, Nauka, M., 1964

[18] Miranda C., Partial differential equations of elliptic type, Transl. from the ital., Springer, Berlin–Heidelberg, 1970 | MR | Zbl

[19] Courant R., Hilbert D., Methods of mathematical physics, v. 2, Partial differential equations, Wiley Interscience, New York, 1962 | MR | Zbl

[20] Landis E. M., Second order partial differential equations of elliptic and parabolic type, Transl. of math. monographs, 171, AMS, 1998 | MR

[21] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniia ellipticheskogo tipa, Nauka, M., 1973

[22] Mikhlin S. G., Linear equations of mathematical physics, Holt, Rinehart and Winston, 1967 | MR | Zbl

[23] Budak B. M., Samarskii A. A., Tikhonov A. N., A Collection of problems in mathematical physics, Pergamon Press, Oxford, 1964 | MR

[24] Polyanin A. D., Nazaikinskii V. E., Linear partial differential equations for engineers and scientists, 2nd ed., CRC Press, Boca Raton, 2016 | MR | Zbl

[25] Burago Iu. D., Zalgaller V. A., Vvedenie v rimanovu geometriiu, Nauka, SPb, 1994

[26] Rashevsky P. K., Vvedenie v rimanovu geometriiu, Nauka, M., 1967

[27] Demmel J. W., Applied numerical linear algebra, SIAM, Philadelphia, PA, USA, 1997 | MR | Zbl