Multigrid algorithms with implicit extrapolation for solving finite element equations

We present a multigrid algorithm with an implicit extrapolation step for solving second-order elliptic boundary value problems in two-dimensional domains. The boundary value problem is discretized by using finite elements with piecewise linear functions and non-standard quadrature rules for the computation of the finite element stiffness matrices and the load vectors. The implicit extrapolation leads to an improvement of the accuracy of the finite element solution. We have shown that the iterates of the multigrid algorithm with extrapolation converge to that finite element solution which we would get by a discretization with piecewise quadratic functions. A numerical example illustrates this fact.