Adaptive Moving Mesh Iterative Methods For Singularly Perturbed Problems

For singularly perturbed problems, most numerical methods can't give a satisfactory solution on an even mesh. Lately, to approximate them effectively, there has been much interests in the study of moving mesh methods which are used to generate an adaptive mesh, and are widely applied to fluid dynamics, semiconductor device modeling and material science, and so on. This paper is mainly to investigate a moving mesh iterative algorithm for solving singularly perturbed problems (I) (II) by equidistributing an arc-length monitor function, make convergence analysis and conduct some numerical experiments.The article is composed of two parts. In the first part, we give a full dis-cretised scheme for our problem (I). It is a nonlinear algebra problem, which is difficult to be solved and to be made convergence analysis. So we aim to construct a moving mesh iterative algorithm, which chooses a suit monitor function, generates a nonuniform mesh adapted to the boundary layer based on equidistribution and obtains the numerical solution. Furthermore, we make error analysis about our algorithm using the results in [7]. Uniform convergence is obtained. Numerical experiments are verified. It is urgent for us to seek more accurate numerical methods to meet the demand of accuracy. Defect correct method and Richardson extrapolant method are advised. We find they are almost second-order uniformly convergent, which greatly decrease the error of numerical solutions. But regretly, we only present some numerical results, not theoretical analysis. In the second part, we give an analysis different from [7] about our algorithm for problem (II). We first derive stability properties of differential operators, then develop a posteriori error estimate on an arbitrary mesh , and get e-uniform convergence of order 1 using discrete Green's function when standard upwind scheme is applied in our algorithm.Finally, we sum up the main ideas of this paper and make some comments on the prospect of moving mesh methods for singularly perturbed problems.