Abstract
In this paper, we consider two methods, the Second order Central Difference Method (SCDM) and the Finite Element Method (FEM) with P1 triangular elements, for solving two dimensional general linear Elliptic Partial Differential Equations (PDE) with mixed derivatives along with Dirichlet and Neumann boundary conditions. These two methods have almost the same accuracy from theoretical aspect with regular boundaries, but generally Finite Element Method produces better approximations when the boundaries are irregular. In order to investigate which method produces better results from numerical aspect, we apply these methods into specific examples with regular boundaries with constant step-size for both of them. The results which obtained confirm, in most of the cases, the theoretical results.
Highlights
Finite Difference schemes and Finite Element Methods are widely used for solving partial differential equations [1]
Finite Element methods are more complicated than Finite Difference schemes because they use various numerical methods such as interpolation, numerical integration and numerical methods for solving large linear systems
We contacted a numerical study with Matlab R2015a. We apply these methods into specific elliptical problems, in order to test which of these methods produce better approximations when the Dirichlet and Neumann boundary conditions are imposed
Summary
Finite Difference schemes and Finite Element Methods are widely used for solving partial differential equations [1]. We will describe the Second order Central Difference Scheme and the Finite Element Method for solving general second order elliptic partial differential equations with regular boundary conditions on a rectangular domain. For both of these methods, we consider the Dirichlet and Neumann Boundary conditions, along the four sides of the rectangular area. We contacted a numerical study with Matlab R2015a We apply these methods into specific elliptical problems, in order to test which of these methods produce better approximations when the Dirichlet and Neumann boundary conditions are imposed.
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