A Posteriori Error Estimates On Anisotropic Meshes

In this dissertation, we discuss a posteriori error estimators on anisotropic meshes for nonconforming finite element method and conforming mixed finite element method approximations of second order elliptic equations, and Stokes equation.Firstly, we discuss a posteriori error estimator for nonconforming finite element ap-proximations of Poisson problem on anisotropic meshes. The lower error bound is ob-tained by means of bubble functions and the corresponding anisotropic inverse inequali-ties. In order to prove the upper error bound, it is vital that an anisotropic mesh is well aligned with the anisotropic solution under consideration. To measure this alignment, a matching function is defined. For nonconforming finite elements, our technique is based on a Helmholtz decomposition with some orthogonality relations for the error. A numer-ical example supports the anisotropic error estimators.Secondly, we discuss a posteriori residual error estimator for the new mixed element schemes for second order elliptic problem on anisotropic meshes. The mixed element schemes for second order elliptic problem was first proposed by Raviart-Thomas, which is still widely used. But in this format required the flux space belongs to H(div;Q), and in order to ensure the convergence of the solution, the mixed finite element spaces need to satisfy the famous B-B conditions, which makes the finite element spaces construct-ing more complex and difficult. Recently, Chen etal., present a new mixed variational form for second order elliptic problem by Green formula. In this formulation, the flux space belongs to (L2(Ω))2, which avoid the trouble involving in the divergence opera-tor. In addition, when the gradient approximation space belongs to the original variable flux approximation space, B-B condition is automatically satisfied, which makes stable mixed finite element spaces are easy to construct. A posteriori error estimates about this new format has not been studied. In this paper, we present two kinds of residual error estimators for the new mixed element schemes on anisotropic meshes. The reliability and efficiency of our estimators are established without any regularity assumptions on the mesh, respectively.Finally, we study a posteriori error estimates for conforming approximations to the stationary Stokes problem on anisotropic meshes. Based on the stretching ratios of mesh elements, we improve a posteriori error estimates developed by Hannukainen et al., their estimators are reliability and efficiency only on isotropic mesh, but not suitable for anisotropic mesh. Our estimates are based on the usual H(div;Ω)-conforming, locally conservative flux reconstruction in the lowest-order Raviart-Thomas-Nedelec space on a dual mesh associated with the original anisotropic simplex one. Numerical experiments in 2D confirm that our estimates are reliable and efficient on anisotropic meshes.