Error Estimation Methods for the Finite-Difference Solution for Poisson’s Equation - Omar, Haji Omar (Eastern Mediterranean University (EMU) - Doğu Akdeniz Üniversitesi (DAÜ), 2015)
The finite-difference method is universally used for the approximation of differential equations. In this thesis two different approaches are reviewed for the error estimation of the approximation of the Dirichlet problem for elliptic equations, specifically Poisson’s and Laplace’s equations using various finite-difference schemes. The first approach is based on the difference analogue of the maximum principle. Applying Gerschgorin’s majorant method to the analysis , also the order of accuracy of the proposed scheme is obtained. The second approach uses the difference analogue of Green’s function and Green’s third identity. In order to obtain an order of approximation, Gerschgorin’s majorant method is applied in this approach also. Both methods gave similar approximations. Keywords: Finite-difference, maximum principle, Gerschgorin’s majorant method, Green’s function, Green’s third identity.

ÖZ: Sonlu-farklar metodu, yakınsak çözümlemeler için evrensel olarak kullanılan bir metoddur. Bu tezde, Poisson denklemi için Dirichlet probleminin sonlu-farklar analogu, iki farklı hata analizi yöntemi ile gözden geçirilmiştir. Birinci yöntem, maksimum ilkesine (maximum principle) bağlıdır. Gerschgorin’in majorant metodunun da uygulanması ile sonlu farklar metodu analiz edilmiştir. İkinci yöntemde ise, Green fonksiyonunun sonlu-farklar analogu, ve Green’in 3. denklemi analogu kullanılmıştır. Yakınsaklık derecesinin elde edilmesi için, Gerschgorin’in majorant metodu da kullanılmıştır. İki yöntem de benzer sonuçlar vermiştir. Anahtar kelimeler: sonlu farklar, maksimum ilkesi, Gerschgorin majorant metodu, Green fonksiyonu.
Master of Science in Mathematics. Thesis (M.S.)--Eastern Mediterranean University, Faculty of Arts and Sciences, Dept. of Mathematics, 2015. Supervisor: Prof. Dr. Adiguzel Dosiyev.
Keywords: Mathematics; Finite differences - Difference equations; Differential equations - Numerical solutions; Finite-difference; maximum principle; Gerschgorin’s majorant method; Green’s function; Green’s third identity