The Adaptive Mesh Algorithm Of Singularly Perturbed Problems

The solutions of singularly perturbed problems make a sharp change in localdomain, which makes the precision of the numerical solution low under the uniformmesh. However, the adaptive mesh method can effectively and automatically gather thegrid points in the solution changing dramatically, so that this method leads to a betternumerical solution in precision and effectiveness.In this thesis, two kinds of generally differential equation are considered using thismethod. One kind is generally convection diffusion equation in one dimension, the otherkind is the differential equation in one spacial dimension with the time dependence. Themain work of this thesis is introduced：1. Several common finite difference schemes for convection-diffusion problems inone dimension are showed. We reduce compact difference scheme with high order localtruncation error. Under the principle equidistribution, the mesh is constructed adaptivelyby a mesh density function based on the arc-length of interpolation of numericalsolution. We combine new mesh points and difference scheme to lead new numericalsolution. In addition, we prove that the solution of the problem is unique and existentunder the equidistributing meshes, then prove the stability and convergence ofcontinuous operator and discrete operator, using the skills of comparison principle andGreen function. Finally, through a series of numerical examples under adaptive meshes,the relative convergence rate is given.2. For convection-diffusion problems in one dimension, we modify the adaptivemesh algorithm, two kinds of high precision algorithm are introduced. We get almostthe second order ε-uniform convergence precision. Under different initial meshes, wecontrast the different operation time of numerical solution.3. We simply introduce posteriori error estimation of the piecewise linearinterpolation of the numerical solution in different solutions, including convergenceproperties in arbitrary non-uniform meshes, convergence properties in theequidistributing meshes, convergence properties of the final generating meshes whenthe algorithm terminates. In addition, we also introduce posteriori error estimation under piecewise quadratic and cubic spline interpolation for the general convection-diffusionequation.4. The perturbation differential equation is discussed in one spacial dimension withtime dependence. We get finite difference scheme on a fixed mesh and adaptive movingmesh. Then, we give discrete schemes of mesh density function on adaptive meshes.Under different norm, we give the different interpolation errors estimates. By differentmethods, smoothing of mesh density functions and smoothed adaptive mesh equationsare given.