Numerička aproksimacija dvodimenzionih paraboličkih problema sa delta funkcijom

Abstract

Granični problemi za parcijalne diferencijalne jednačine predsta- vljaju matematičke modele najraznovrsnijih pojava, kao na primer pro- voea toplote, mehanike fluida, procesa atomske fizike itd. Samo u retkim slučajevima ovi zadaci se mogu rexiti klasiqnim metodama ma- tematičke analize, dok se u svim ostalim mora pribegavati priblinim metodama. Metoda konaqnih razlika je jedan od najčešće primeiva- nih metoda za numeričko rešavanje graničnih problema za parcijalne diferencijalne jednačine. U okviru metode konačnih razlika, jedan od glavnih problema je dokazivanje konvergencije diferencijskih shema koje aproksimiraju granične probleme. Od posebnog interesa su ocene brzine konvergencije saglasne sa glatkošću koeficijenata i rešenja početnog problema. Prilikom numeričke aproksimacije poqetno-graničnih paraboliqkih problema sa generalisanim rešenjima javljaju se i neki dodatni pro- blemi: koeficijenti nisu neprekidne funkcije, promenljivi koefici- jenti mogu biti i vremenski zavisni, koeficijenti i rešenje pripadaju nestandardnim anizotropnim prostorima Soboljeva itd. Ova disertacija se upravo bavi tim problemima.

Boundary problems for partial differential equations represent mathema- tical models of the most diverse phenomena, such as heat transfer, uid me- chanics, atomic physics, etc. Only in rare cases, these tasks can be solved by classical methods of mathematical analysis, while in all other must be resort to approximate methods. Finite-difference method is one of the most commo- nly used methods for the numerical solution of boundary value problems for partial differential equations. In the context of nite-difference method, one of the main problems is proving convergence of difference schemes which appro- ximating boundary problems. Of particular interest are the estimates of the rate of convergence compatible with the smoothness of the coefficients and solution. When numerical approximations parabolic initial-boundary problems with generalized solutions, there are also some additional problems: the coefficients are not continuous functions, variable coefficients can be time-dependent coe- fficients and the solution belong to nonstandard anisotropic Sobolev spaces, etc. This dissertation is concerned with precisely these problems. The dissertation is considered a two-dimensional parabolic initial-boundary problem with concentrated capacity, that problem contains Dirac delta functi- on as the coefficient of the derivative by time. A further problem, in the case boundary problems with delta function as the coefficient, is that solution not in standard Sobolev spaces. The paper demonstrated a priori estimates of the corresponding non-standard norms. Assuming that the coefficients belong to anisotropic Sobolev spaces have been constructed the difference schemes with averaged right-hand side. The estimates of the rate of convergence in the spe- cial discrete fW2; 1 2 and fW1; 1=2 2 norms, is proved. The estimates of the rate of convergence compatible with the smoothness of the coefficients and solution, are obtained.