Error Analysis Of The Approximated Solution And Derivative For Singularly Perturbed Problems Based On Equidistribution Principle

Recently much attention has been paid to the singularly perturbed two-point boundary value problem. It can widely applied to fluid dynamics, semiconductor device modeling, material science, and so on. But it is well known that a central or upwind difference scheme on an even mesh can't give a satisfactory numerical solution. In order to obtain a reliable numerical solution, we must use an adaptive mesh.In this work, we consider the more general non-conservation form. And the artical is composed of two parts. In the first part, an upwind scheme based on the forward difference approximation for the second-order derivative will be chosed. And the mesh is constructed adaptively by equidistributing a monitor function over the domain of the problem. In this part, we set M(x)=(1+(ε^-1e^-βx/ε)²)^1/2. Thus we can obvate use the approximated solution u^N. By using the discrete Green's function, a convergent result which is independent of the perturbation parameterεis obtained. The order is O(N^-1). In a number of applications the user is more interested in the approximation of the gradient or of the flow than in the solution itself. So the error bound for the weighted derivative is established. The order is also O(N^-1). In the second part, an upwind scheme based on the central difference approximation for the second-order derivative will be chosed. This scheme is first-order consistent on arbitrary meshes. The problem is solved numerically by using an adaptive mesh. We set M(x) =(1+(D-u_i^N)²)^1/2, so the mesh is constructed adaptively by equidistributing the monitor function based on the arc-length of the approximated solution u^N. By using the discrete Green's function, a convergent result which is independent of the perturbation parameter is obtained. The order is O(N^-1). And the error bound for the weighted derivative is established. The order is also O(N^-1).Finally, we summarize the main ideas of this artical and make some comments on the prospect of the general on-conservation singularly perturbed twopoint boundary value problem.