The Study Of Multiscale Numerical Simulations For Efficiently Solving The Singularly Perturbed Problems

Scientific computation, one of the three most important research ways together with theoretical study and scientific experiment, has become more and more signif-icant. The singularly perturbed problem with very small perturbation parameter will lose its boundary condition, and it will bring the so-called boundary layers phenomena. So we are encouraged to study the efficiently numerical simulations. Multiscale methods are hot spots in modern scientific computation. Their goal is to find an ideal balance between the numerical accuracy and computational cost. We are trying to present a novel numerical method, which has high accuracy and low computation costs, to settle the complicated multiscale problems.This doctoral dissertation is organized as following:In Chapter 1, we introduce the singularly perturbed problem and the multiscale computational methods, and we elaborate a great significance for the combinations of multiscale simulation and adaptive technique. Then we provide the necessary notations, definitions and important inequalities in the dissertation.In Chapter 2, we give the corresponding theory analysis on the solution exis-tence and convergence of both standard finite element and multiscale finite element methods, respectively.In Chapter 3, for the one dimensional convection-diffusion model we study the computational performance of multiscale finite element method. Based on the differential operator we build the sub-problem for the multiscale basis functions, which can reflect the original microscopic information, then we obtain an efficient simulation through the finite element scheme. We construct adapted grids on the error estimates such as new Shishkin, Graded and Bakhvalov grids. Through the numerical experiment we demonstrate the accuracy and efficiency of our multiscale method.In Chapter 4, for the two dimensional reaction-diffusion model we study the effective Galerkin and Petrov-Galerkin multiscale finite element approximations. We prove the stability and convergence of multiscale numerical solution by theoretic analysis. Being different from the Galerkin mode, our Petrov-Galerkin mode applies the independent trial function space and test function space, so that it provides a more flexible enrichment for the multiscale basis functions. No matter for constant or inconstant coefficient model, using the multiscale method on the adapted grids we need no special techniques to eliminate the multiscale resonance error. As a result, we can acquire a greatly efficient multiscale simulation on the coarse grid, which is independent of the perturbed parameter ε and is high order uniformly-convergent.In Chapter 5, we set up the full-discretization on time for the rapidly oscillating parabolic multiscale model. The multiscale basis functions have the abilities to capture the intrinsical oscillations for the whole problem. We apply the Euler backward difference scheme to form the global system. We just need to compute on coarse grid, and the computational costs are reduced greatly.We have obtained several meaningful theoretic and numerical results in the dissertation. In the future we are interested in the post-estimates for high perfor-mance, the multiscale computation for multi-phase flow problem, and the parallel computation package and integration, etc.