A comparison between two methods of adaptive mesh refinement for finite differences : one dimensional Dirichlet problems

Abstract

In the present work, we consider two methods of adaptive mesh refinement in finite differences for one dimensional Dirichlet problems. There are two approaches in the discretization: node based and centered based schemes. For each one of these schemes, the discretization equations over a non-uniform grid and the error analysis are established. The two methods of adaptive mesh refinement have different criteria. The first method considers two main criteria: the size of the cell and the average length of the gradient of the solution. On the other hand, the second method considers the error tolerance as the refinement criterion. The two methods also differ in the meshing algorithm. In fact, method 1 simply splits a cell into two cells, whereas method 2 clusters nodes together and performs the meshing process. The two methods are compared and some numerical examples are considered. As an application, a C ++ code was written to implement the node based adaptive mesh refinement for both methods, and the plotting was done in MATLAB.