Global L2 superconvergence of the tetrahedral quadratic finite element

This paper provides the global L-2 superconvergence of the tetrahedral quadratic finite element parallel to u(h) - u(I)parallel to(0) <= ch(4)parallel to u parallel to(4), where u(h) and u(I) are the finite element approximation and the quadratic Lagrange interpolation of the exact solution u respectively. The standard analysis of the global L-2 superconvergence is combined with the weak estimate of the second type (WE2). However, the WE2 of the tetrahedral quadratic finite element has kept absent for a rather long time. To this end, we define the four-element-based uniformtetrahedral mesh, and then present the simplified weak estimate of the second type (SWE2), which simplifies the result of the WE2 and can also derive the global L-2 superconvergence. For the completeness, the proof of the WE2 can be found in the appendix. Finally, this supercloseness will be used to construct a post-processing that increases the order of approximation to the exact solution in L-2 norm. Numerical experiments are provided to illustrate our theoretical results.