Numerical quadrature schemes of a non-conforming finite element method for general second order elliptic problems in two dimensional (2-D) and three dimensional (3-D) space are discussed in this paper. We present and analyze some optimal numerical quadrature schemes. One of the schemes contains only three sampling points, which greatly improves the efficiency of numerical computations. The optimal error estimates are derived by using some traditional approaches and techniques. Lastly, some numerical results are provided to verify our theoretical analysis.
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.
