A Posteriori Error Estimates Of Nonconforming Finite Element Methods For Convection-diffusion Problems

In this thesis,we consider residual a posteriori error estimation for nonconforming finite element approximations to convection-diffusion problem.For various stabilized finite element methods,we derive the semi-robust and robust a posteriori error estimates in a unified framework,and then extend the theoretical results to quadrilateral elements.For semi-robust a posteriori error estimates,we choose the usual energy norm to measure the error.In an abstract framework,we obtain a general decomposition on the error for finite element approximations to convection-diffusion problem,where the error is decomposed into three parts: residual error,consistency error and nonconforming error.In fact,the error decomposition is fixed for all the conforming and nonconforming approximations,of which the only consistency error needs to be estimated in different ways related to the particular discretizations and the remaining can be estimated in a unified way.Especially,for the conforming approximations the nonconforming error vanishes.We prove that the residual estimator is reliable and efficient in the usual energy norm,but the constants in the lower error bound depend on the diffusion coefficient and mesh size.The constants can be bounded for enough small mesh size comparable with the diffusion coefficient.Thus,the resulting error estimator is semi-robust in the usual energy norm.For robust a posteriori error estimates,we have to find an appropriate norm to measure the error.To this end,we add a discrete dual semi-norm of the convective derivative and jumps of the approximate solution over element edges(faces in three dimensions)with weights dependent on the mesh size to the primal energy norm.Similar to the semi-robust a posteriori estimates,we obtain a general error decomposition in the modified energy norm,and decompose the error into three parts: residual error,consistency error and nonconforming error.We prove that the residual estimator is reliable and efficient in the modified energy norm,and the constants in the upper and lower error bounds are independent of the diffusion coefficient and mesh size.Thus,the resulting error estimator is robust in the modified energy norm.All the above works are done for simplicial meshes.The resulting theory for a posteriori error estimates is applicable not only for various conforming stabilized methods,such as streamline-diffusion methods,continuous interior penalty methods,subgrid viscosity methods and so on,but also for various nonconforming stabilized methods,such as nonconforming streamline-diffusion methods,nonconforming face and interior penalty methods,and nonconforming subgrid viscosity method and so on.Then we extend the error analysis to quadrilateral elements and establish a unified framework.Under some certain conditions,the theoretical framework assures the semi-robustness of residual error estimator in the usual energy norm and the robustness in the modified energy norm,and applies to nonconforming triangular and quadrilateral elements,such as Crouzeix-Raviart element,nonconforming rotated Q1 element and constrained rotated Q1 element,etc.Based on the general error decomposition in different norms,we show that the key ingredient of a posteriori error estimation is the existence of a bounded linear operator with some elementary properties and the estimation on the consistency error related to the particular numerical scheme.At last,the numerical experiments are carried out to illustrate the reliability,efficiency and robustness of the residual error estimator.