This paper studies convergence analysis of an adaptive finite element algorithm for numerical estimation of some unknown distributed flux in a stationary heat conduction system,namely recovering the unknown Neumann data on interior inaccessible boundary using Dirichlet measurement data on outer accessible boundary.Besides global upper and lower bounds established in[23],a posteriori local upper bounds and quasi-orthogonality results concerning the discretization errors of the state and adjoint variables are derived.Convergence and quasi-optimality of the proposed adaptive algorithm are rigorously proved.Numerical results are presented to illustrate the quasi-optimality of the proposed adaptive method.
This paper is to study the convergence and superconvergence of rectangular finite elements under anisotropic meshes. By using of the orthogonal expansion method, an anisotropic Lagrange interpolation is presented. The family of Lagrange rectangular elements with all the possible shape function spaces are considered, which cover the Intermediate families, Tensor-product families and Serendipity families. It is shown that the anisotropic interpolation error estimates hold for any order Sobolev norm. We extend the convergence and superconvergence result of rectangular finite elements to arbitrary rectangular meshes in a unified way.
