Anisotropic A Posteriori Error Estimation By Local Reconstruction And Adaptive Computation

In this thesis, by local reconstruction we derive robust a posteriori error estimates for various numerical methods of diffusion problems with discontinuous coefficients and singularly perturbed reaction-diffusion problems on anisotropic meshes, and further implement anisotropic adaptive computation.For diffusion problems with discontinuous coefficients, we derive a posteriori error estimates by local reconstruction on anisotropic meshes. Based on the local conservation assumption on the equilibrated flux, an abstract upper error bound is established such that the remaining task is to construct such an equilibrated flux satisfying the given assumption. Here we present two different constructions of the equilibrated flux:the direct prescription and mixed finite element method. The direct prescription is simple and doesn’t need special treatment of the effect of mesh anisotropy, while for the mixed finite element method we have to introduce the mesh anisotropy into the construction of equilibrated flux. The resulting error estimators are robust with respect to the diffusion coefficient, and can be applied to finite volume, finite difference and. finite element methods. We show that the error estimators are reliable and efficient with respect to the error measured either as the energy norm of the difference between the exact and approximate solutions, or as a dual norm of the residual. Indeed, they are equivalent to the estimates of Vohralfk (J Sci Comput 46:397-438,2011) in the case of isotropic meshes. Therefore, in some sense we extend a posteriori error estimates developed by Vohralfk.For the finite volume method of singularly perturbed reaction-diffusion prob-lems, we derive a posteriori error estimates by local reconstruction on anisotropic meshes. Similarly, based on the local conservation assumption on the equilibrated flux, an abstract upper error bound is established in the sense of energy norm. We construct the equilibrated flux by the direct prescription. The resulting er- ror estimators are shown to be reliable, efficient and robust with respect to the reaction coefficient. Indeed, they are equivalent to the estimates of Cheddadi et al (ESAIM-Math Model Numer Anal 43:867-888,2009) in the case of isotropic meshes. Therefore, in some sense we extend a posteriori error estimates developed by Cheddadi et al.Based on the anisotropic a posteriori error estimates, we derive the corre-sponding anisotropic error indicator and further implement anisotropic adaptive refinement of the mesh. In fact, the primal anisotropic error estimator is not available for anisotropic adaptive refinement, because it cannot represent the con-tributions to the whole error in different directions and the alignment measure contains the exact solution which is unknown. In order to obtain the anisotrop-ic error indicator, we reformulate the definition of alignment measure and then approximate it by a post-processing way. Finally, we present an anisotropic adap-tive algorithm to build a right metric field by the anisotropic error indicator, such that a new anisotropic mesh is generated.The above theoretical results have been verified by the numerical experi-ments.