Accelerating the Convergence of Algebraic Multigrid for Quadratic Finite Element Method by Using Grid Information and p-Multigrid

This paper proposes a new multigrid method to efficiently solve the finite element approximation of static or quasi-static electromagnetic problems with quadratic nodal finite elements on unstructured grids in 2D and 3D by using grid information and p-multigrid. Unlike the traditional algebraic multigrid (AMG), the new method constructs the first level coarse grid from the grid information of the finest grid directly, by using the natural geometrical coarsening relationship between the P2 and P1 elements. Then, the algebraic equations of the first level coarse grid are constructed based on the special relationship between the basis functions of quadratic finite element and its linear counterpart. At last, the traditional AMG is applied to solve the algebraic equations of the first coarse level rather than that of the finest grid. Several techniques of convenient and economic implementation are discussed. For the problems tested, the proposed method is much more efficient than the conjugate gradient method with incomplete Cholesky preconditioning; in addition, compared with traditional Krylov subspace accelerated AMG, the new method may save about 30% to 40% CPU time while achieving the same accuracy in practical computations.