Superconvergence Of Least-squares Mixed Finite Element Methods For Second Order Semilinear Elliptic Problems

In recent years,there has aroused the domestic and foreign scholars a wide interest in the least-squares mixed finite element methods.Least-squares finite element methods have become a hot topic because they can obtain symmetric algebraic systems and are not subject to the Ladyzhenskaya Babuska Brezzi (LBB) consistency requirement.The main task of this paper is to investigate the least-squares mixed finite method for the semilinear elliptic problems. By introducing the interpolation operators and an auxiliary projection, we do some analysis for the superconvergence and finally obtain the general superconvergence results.Scientific researchers have done much work on the superconvergence of the least-squares mixed finite element methods.In this paper, the resulting procedure can be spilt into two independent symmetric positive definite schemes by select-ing the least-squares functional properly, the existence and uniqueness of semilin-ear elliptic problem can be also proved by the least-squares mixed finite element methods.The primary solution u and the fluxσare approximated by using finite element spaces consisted of piecewise polynomials of degree k and r respectively. Based on interpolation operators and an auxiliary projection,superconvergent H1-error estimates of both the primary solution approximation uh and the flux approximationσh are obtained under the standard quasi-uniform assumption on the finite element partition.If Raviart-Thomas or Brezzi-Douglas-Fortin-Marini element of order r are employed with optimal error estimate of O(hr+1),the su-perconvergence indicates an accuracy of O(hr+2) for the least-squares mixed finite element approximation.