FINITE DIFFERENCE APPROXIMATIONS OF VARIOUS STEKLOV EIGENVALUE PROBLEMS

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2022-8-26
ÖZALP, MÜCAHİT
In this thesis, the finite difference method (FDM) is employed to numerically solve differently defined Steklov eigenvalue problems (EVPs) that are characterized by the existence of a spectral parameter on the whole or a part of the domain boundary. The FDM approximation of the Laplace EVP is also considered due to the fact that the defining differential operator in a Steklov EVP is the Laplace operator. The fundamentals of FDM are covered and their applications on some BVPs involving Laplace operator are discussed. Using Taylor's series expansions, approximation formulas for the derivatives of the functions are provided, with varying degrees of accuracy. To validate our formulation of FDM, the second and fourth order formulas are first used to approximate two test problems for which the analytical solutions are known, namely, the Poisson problem and the Laplace EVP. It is demonstrated that the solutions from the FDM agree well with the exact ones and that the results from the fourth order scheme are superior to those from the second order one. Secondly, we consider two Steklov eigenvalue problems that are distinct from each other by the associated boundary conditions. Specifically, the standard Steklov EVP with a mixed type boundary condition involving a spectral parameter is analyzed as the first problem, whereas in the second problem, the boundary of the computational domain is divided into two parts; one with Neumann type boundary condition and the other with spectral boundary condition. The discretization of the problem is performed by several orders of finite difference formulas for the first time to the best of our knowledge. The agreement between the approximate and exact eigenfunctions is shown using contour plots, and the rate of convergence of the approximate eigenvalues to the reference ones is given. It has been noted that the use of higher order finite difference approximations -of at least second order- for not only the differential equation but also the boundary conditions advances the rate of convergence. Consequently, the present study demonstrates how a second-order convergence can be acquired by the application of fourth-order finite difference formulas for both the differential operator and the accompanying boundary conditions.

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Citation Formats
M. ÖZALP, “FINITE DIFFERENCE APPROXIMATIONS OF VARIOUS STEKLOV EIGENVALUE PROBLEMS,” M.S. - Master of Science, Middle East Technical University, 2022.