Class Of Singular Perturbed Equations Of The Finite Element Analysis

The finite element approximations of some steady-state singular perturbation equations are investigated in this paper. For different singular perturbed equations, applying different methods with distinct elements for the sake of error estimates and numerical test, some uniform convergence results are obtained which are independent of the singular perturbation parameter.Singular perturbation problems (SPP) arise in ninny application areas, such as in physics engineering, fluid dynamics and chemical kinetics, etc. Its special character is that the solution of such problems undergo rapid changes within very thin subdomains, which is called layer, owing to the singular perturbation parameter. For the importance and indeterminacy of the true solutions, more and more scholars are interested in their numerical solutions.The numerical approximations for solving singular perturbation problems, applying the classical finite element methods, may arise severe oscillation across the layer, so one can not get the satisfying convergence. And due to the regular condition, the finite element meshes need to be refined enough, which will enlarge the computation works enormously. For conquering this, applying anisotropic mesh may be a finer choice. Now, one common kind of methods is improving the classical meshes, using the appropriately layer-adapted meshes, including the well known Bakhvalov-type and Shishkin-type meshes, and the graded meshes which is appeared recently. The finite element approximations on these meshes need anisotropic elements. Another kind of methods is the stabilized methods, adding the artificial viscosity terms and modifying the test functions, in order to improving the numerical oscillation arose by standard Galerkin methods, and these methods belong to the so called Petrov-Galerkin methods, including the well known streamline diffusion finite element methods (SDFEM) and residual free bubbles (RFB), etc. The main aim of theoretical analysis is to get the uniform convergence error estimates independent of the singular perturbation parameter in energy norm.The topic of this thesis is some steady-state singular perturbation equations, including reaction dominated reaction-diffusion type singular perturbed equations and semisingular perturbed equations, convection dominated convection-diffusion type singular per- turbed equations, and fourth order elliptic singular perturbation equations, etc. The author endeavored to improve the convergence results in many aspects, such as refining a priori meshes, using anisotropic element (conforming and nonconforming element), applying stabilized methods or higher order elements, and so on. Firstly, we give a general introduction on singular perturbation problems and some basic informations. In Chapter 2, on the appropriately graded meshes, the bilinear element approximations of the reaction dominated reaction-diffusion type singular perturbed equations and semisingular perturbed equations are considered at the same time, by using the special character of graded meshes and analyzing element by element, the global uniformly convergence results in the corresponding weighted norm are obtained. And by constructing a appropriately interpolation postprocessing operator, the superconvergence results are derived, then some numerical experiments are also given to confirm the theoretical analysis. In Chapter 3, a lower order five node nonconforming element is used to approximate the reaction dominated reaction-diffusion type singular perturbed equations and semisingular perturbed equations simultaneously, and by using the special character of graded meshes and analyzing element by element, all the above similar results are achieved. In Chapter, biquadratic element approximation of convection-diffusion-reaction equation is considered under graded meshes, overcoming some difficulties, and some higher order uniformly convergence results are obtained. By using the streamline diffusion finite element methods with a double set parameter nonconforming element constructed recently, are used for the convection dominated convection-diffusion-reaction equation in the following chapter, and convergence results are derived. In Chapter 6, a new nonconforming element constructed by double set parameter method, is applied to the fourth order elliptic singular perturbation problem with simple support boundary condition. The convergence uniformly in the perturbation parameter, is proved under the anisotropic meshes, and optimal convergence rate is obtained. Numerical results support the theoretical analysis. In the last chapter, the author give some conclusions about this paper and suggestions on future works.