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Abstract: In this paper we discuss constrained approximation with arbitrary-level hanging nodes in adaptive higher-order finite element methods ( -FEM) for three-dimensional problems. This technique enables using highly irregular meshes, and it greatly simplifies the design of adaptive algorithms as it prevents refinements from propagating recursively through the finite element mesh. The technique makes it possible to design efficient adaptive algorithms for purely hexahedral meshes. We present a detailed mathematical description of the method and illustrate it with numerical examples. [Copyright &y& Elsevier]

Abstract: A semi-linear elliptic problem with variable coefficient is approximated by expanded mixed formulation based on the RTN and BDM mixed finite elements. In order to solve the nonlinear approximation system of equations efficiently, a two-grid algorithm is considered and discussed in this paper. The work includes a small nonlinear system on a coarse grid space with mesh size and a linear system on a fine grid space with mesh size . It follows from error estimates that asymptotically optimal accuracy can be obtained as long as the mesh sizes satisfy in the norm. Some numerical examples are presented to illustrate the theoretical results. [Copyright &y& Elsevier]