In this talk, we shall present some local multigrid algorithms for solving the linear algebraic systems arising from the adaptive continuous and discontinuous finite element methods. The abstract Schwarz theory is applied to analyze the multigrid methods withJacobi or Gauss-Seidel smoother performed on local nodes on each level. It is shown that the local multigrid methods are optimal,which means that the convergence rates are independent of the mesh sizes and mesh levels. Furthermore, we shall explore the local multigrid methods for solving convection-diffusion equation and time-harmonic Maxwell equation.

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