20.500.12556/RUL-152451 Reševanje eliptičnih parcialnih diferencialnih enačb v Bernstein-Bézierjevi reprezentaciji magistrsko delo Solving elliptic partial differential equations in Bernstein-Bézier representation Magistrsko delo obravnava metodo za numerično reševanje eliptičnih robnih problemov dveh spremenljivk v šibki obliki z uporabo Bernsteinove baze. Podani so pogoji za obstoj in enoličnost rešitve in njene aproksimacije ter ocena napake. Metoda temelji na metodi končnih elementov, ki je tudi opisana v magistrskem delu. Rešitev iščemo v prostoru polinomskih zlepkov nad triangulacijami, pri čemer je zlepek nad posameznim trikotnikom namesto v običajni polinomski bazi zapisan v Bernsteinovi bazi. Opisane so lastnosti te baze, ki omogočajo, da približek iščemo kot rešitev optimizacijskega problema z dodatnimi pogoji, ki zagotovljajo izpolnjevanje robnih pogojev in dovoljšen red gladkosti. The master's thesis addresses a method for numerically solving elliptic boundary value problems of two variables in weak form using the Bernstein basis. It provides conditions for the existence and uniqueness of the solution and its approximation, along with an error estimate. The method is based on the finite element method, which is also described in the master's thesis. The solution is sought in the space of polynomial splines over triangulations, where each spline over an individual triangle is represented in the Bernstein basis rather than the typical polynomial basis. The thesis describes the properties of this basis, which enable us to seek an approximation as the solution to an optimization problem with additional constraints to ensure the satisfaction of boundary conditions and sufficient smoothness order. parcialne diferencialne enačbe metoda končnih elementov Bernsteinovi bazni polinomi prostori zlepkov partial differential equations finite element method Bernstein basis polynomials spline spaces true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2023-11-25 08:15:04 2023-11-25 08:15:08 2024-05-29 09:46:29 0000-00-00 00:00:00 2023 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.6 139578 173441539 11964.pdf 11964.pdf 1 D0E3A79C98CD0B6636231E50893D6C73 466db4ddf9379ed4a4f830c00175620a91b8bcf1230f1af3b7164b687d5bf0da 5e22adb0-8b62-11ee-a198-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=177972 Fakulteta za matematiko in fiziko 0 0 0