Convergence Analysis For A Singularly Perturbed Problem Using A Standard Upwind Difference On Adaptive Meshes

Adaptive grid methods are becoming established as valuable computational techniques for the numerical solution of differential equations with near-singular solutions. Adaptive methods are equally effective in approximating solutions of problems with boundary layers or interior layers. In this paper, we present a analysis for a standard upwind difference scheme of a singularly perturbed twopoint boundary value problem on adaptive meshes. The meshes are generated adaptively by equdistributing a monitor function. In this paper, we choose two kinds of monitor function, and analyze their errors respectively.The article is composed of two parts. In the first part, We use a standard upwind difference scheme on adaptive meshes, which are generated by equidistributing a monitor function. The monitor function can be chosen as M(x)=(?). We prove that the scheme is uniformly convergent with the order O(N^-1 ln N). In the second part, on the same difference scheme, we propose the use of a monitor function that is a linear combination of a positive constant and a power of the second derivative of the solution. That is M(x)=α+|ε^-2e^-βx/ε|^1/m. The constant is chosen so that the resulting grid has an equal distribution of points both internal and external to boundary layers. Our analysis shows how the constant should be chosen to ensureε-uniform convergence and indicates the convergence behaviour for such grids. We prove that the scheme is uniformly convergent with the order O(N^-1). Both of the convergence analysis are achieved by decomposing the numerical and analytical solutions into smooth and singular components and then analyzing the error in the singular component on two non-overlapping regions of the domain that are naturally suggested by the grid structure.