Convergence And Supercloseness Analysis Of NIPG Method On Layer-Adapted Meshes

In this thesis,we develop and analyze a nonsymmetric interior penalty Galerkin（NIPG）method on layer-adapted meshes for solving singularly perturbed convection-diffusion problems.With the continuous development of science and technology,singularly perturbed problems have been widely used in many applications such as fluid mechanics,chemical dynamics and system control.Therefore,the numerical analysis of this kind of problems is a very active research field,which has attracted extensive attention in the scientific community.As we all know,a typical characteristic of singularly perturbed problems is that one or more regions with sharp changes usually appear in their solutions,which we call layers.At this time,the classical numerical methods are no longer applicable.In order to solve this problem,many numerical strategies have been developed,and layer-adapted meshes are one of them.Among layer-adapted meshes,the most popular are Shishkin-type meshes and Bakhvalov-type meshes.However,due to the lack of powerful tools,there are few theses on the supercloseness analysis of the NIPG method for singularly perturbed problems on layer-adapted meshes.In this thesis,we present the main analysis difficulties of the NIPG method for singularly perturbed convection-diffusion problems for the first time—the convective term outside the layer and the diffusion term inside the layer.On that basis,some special interpolations are designed to discuss the convergence and supercloseness of the NIPG method on layer-adapted meshes.The main research can be summarized as follows:（1）For a one-dimensional singularly perturbed convection-diffusion problem,we introduce a new composite interpolation according to the characteristics of the solution,mesh and numerical scheme.Briefly,Gau?Radau projection is used outside the layer;Gau?Lobatto projection is used inside the layer.On the basis of that,by selecting the penalty parameters at different mesh points,the supercloseness of almost k+1 order on a Shishkin mesh is obtained.Here k is the degree of piecewise polynomial.Then,combined with this supercloseness result,we show how to improve the accuracy of numerical solutions by constructing a post-processing operator.A new stability analysis method is proposed for this operator.Finally,the superconvergence of NIPG method in a discrete energy norm is proved.（2）For the same one-dimensional singularly perturbed problem,we first establish the convergence and supercloseness analysis on a Bakhvalov-type mesh by using the NIPG method.In this process,by making some minor modifications to the new interpolation defined above,we solve the core difficulties of convergence analysis on Bakhvalov-type meshes,and obtain the supercloseness of k+₂¹order.（3）For a two-dimensional singularly perturbed convection-diffusion problem,we obtain the supercloseness of almost k+₂¹by redesigning a new composite interpolation and choosing different penalty parameters on the edges between different types of elements.Numerical experiments support our theoretical conclusions.Furthermore,through these numerical results,we find that for high-order methods,even for solving one-dimensional singularly perturbed problems,convergence degradation may occur.This means that the application of high-order numerical algorithms needs in-depth research,especially on the iterative solver of ill-conditioned linear systems.