Higher-Order Cartesian Grid Based Finite Difference Methods for Elliptic Equations on Irregular Domains and Interface Problems and their Applications

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Title: Higher-Order Cartesian Grid Based Finite Difference Methods for Elliptic Equations on Irregular Domains and Interface Problems and their Applications
Author: Kyei, Yaw
Advisors: Dr. Kazufumi Ito, Committee Chair
Dr. Zhilin Li, Committee Member
Dr. Ralph Smith, Committee Member
Dr Hien T. Tran, Committee Member
Abstract: This thesis describes higher-order finite difference methods for solving elliptic equations on irregular domains with general boundary conditions and the corresponding elliptic interface problems. We develop second and fourth order methods for two and three dimensions using uniform Cartesian grids. However, with an irregular domain we cannot apply the standard finite difference schemes directly at the grid points near the boundary and therefore some treatment is required in order to use the uniform Cartesian grids. Our approach involves modifying the standard finite difference schemes. In particular, we use the standard five-point and standard compact nine-point stencil schemes for the second and fourth order methods, respectively. That is, on the standard stencils that contain the boundary, we carry out the modification by applying the continuation of solution from the inside of domain to the outside. The method of continuation of solution uses Taylor series expansion of the solution about selected boundary points, the equation and the boundary values of the local and their nearby boundary points. Naturally, second and fourth order Taylor series expansions about the boundary points are used for the second and fourth order methods respectively. Our methods have a unified and an effective approach to deal with general boundary conditions and capture the boundary and its local geometrical properties by the level set function and the local coordinate system at the boundary points. The resulting finite difference system matrices of our methods remain symmetric positive definite and maintain the sparsity of the standard finite difference schemes. As part of our main objective, we apply our fourth order method to semi-discretize the corresponding parabolic equation in space on the irregular domain and obtain an ODE system. The validity and effectiveness of the proposed method is clearly demonstrated through the computation of eigenvalues of the associated eigenvalue problem for the circular domain and other irregular domains. As one of the essential applications of our method, we design a state feedback controller for the Dirichlet boundary control problem of the heat equation. By the standard $LQR$ theory of the optimal state-feedback design, we solve the associated Riccati equation where the ODE system resulting from our fourth order method of semi-discretization serves as the state equation. Conventional second order methods have system matrices of higher dimensions which makes the Riccatti equation almost impractical to solve numerically. But with our nine-point compact fourth order methods, we capture the essential properties of the equation through our low order system matrices and demonstrate the capability of our approach through our computations. Another important contribution of the thesis is that we extend our fourth order method to develop a nine-point compact finite difference method for the variable elliptic equation and the corresponding interface problem. For the interface problem, the method is based on the continuation of solution procedure and in this case, across the material interface from one side to the other. Specifically we develop the procedure to define the scheme near the interface through the optimization technique to preserve the M-property of finite difference schemes.
Date: 2004-05-13
Degree: PhD
Discipline: Mathematics
URI: http://www.lib.ncsu.edu/resolver/1840.16/5809


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