(국문발표 20171205) 최용호 | 3차원 표면 위에서 편미분 방정식의 수치적인 풀이
The primary purpose of this dissertation is to study the various partial differential equations on the non-flat surfaces. To solve the partial differential equations (the Allen–Cahn (AC) equation, conservative Allen–Cahn (CAC) equation, and Lengyel–Epstein (LE) equations) on surfaces, we first discuss the AC, CAC, LE equations on the general domain. And then, we describe the surface reconstruction algorithm by using modified AC equation which source data are cloud points, and slice data (CT, MRI, X-ray). That is, reconstruction from two-dimensional data to three-dimensional surface data. Next, we construct computational domain which is defined by narrow band domain and quasi-Neumann boundary condition which applied closest points method. We finally consider that solving the AC, CAC, and LE equations on the various surfaces. We present the fast, efficient, and robust numerical method. By using narrow band domain and quasi-Neumann boundary condition, we can use the standard Laplacian operator instead of Laplace–Beltrami operator. Overall numerical simulations, we use operator splitting method. The multigrid method and explicit method are used in some cases for fast solution. Various numerical results demonstrate that the proposed methods are fast and accurate.