Multigrid Methods for Hellan-Herrmann-Johnson Mixed Method of Kirchhoff Plate Bending Problems

A V-cycle multigrid method for the Hellan-Herrmann-Johnson (HHJ) discretization of the Kirchhoff plate bending problems is developed in this paper. It is shown that the contraction number of the V-cycle multigrid HHJ mixed method is bounded away from one uniformly with respect to the mesh size. The uniform convergence is achieved for the V-cycle multigrid method with only one smoothing step and without full elliptic regularity. The key is a stable decomposition of the kernel space which is derived from an exact sequence of the HHJ mixed method, and the strengthened Cauchy Schwarz inequality. Some numerical experiments are provided to confirm the proposed V-cycle multigrid method. The exact sequences of the HHJ mixed method and the corresponding commutative diagram is of some interest independent of the current context.Comment: 23 page

Stabilized mixed finite element methods for linear elasticity on simplicial grids in Rn\mathbb{R}^{n}

In this paper, we design two classes of stabilized mixed finite element methods for linear elasticity on simplicial grids. In the first class of elements, we use $\boldsymbol{H}(\mathbf{div}, \Omega; \mathbb{S})$-$P_k$ and $\boldsymbol{L}^2(\Omega; \mathbb{R}^n)$-$P_{k-1}$ to approximate the stress and displacement spaces, respectively, for $1\leq k\leq n$, and employ a stabilization technique in terms of the jump of the discrete displacement over the faces of the triangulation under consideration; in the second class of elements, we use $\boldsymbol{H}_0^1(\Omega; \mathbb{R}^n)$-$P_{k}$ to approximate the displacement space for $1\leq k\leq n$, and adopt the stabilization technique suggested by Brezzi, Fortin, and Marini. We establish the discrete inf-sup conditions, and consequently present the a priori error analysis for them. The main ingredient for the analysis is two special interpolation operators, which can be constructed using a crucial $\boldsymbol{H}(\mathbf{div})$ bubble function space of polynomials on each element. The feature of these methods is the low number of global degrees of freedom in the lowest order case. We present some numerical results to demonstrate the theoretical estimates.Comment: 16 pages, 1 figur

CO J=2-1 and CO J=3-2 observations toward the high-mass protostellar candidate IRAS 20188+3928

We have carried out 12CO J=2-1 and 12CO J=3-2 observations toward the high-mass protostellar candidate IRAS 20188+3928. Compared with previous observations, the 12CO J=2-1 and 12CO J=3-2 lines both have asymmetric profiles with an absorption dip. The velocity of the absorption dip is 1.0 km/s. The spectral shape may be caused by rotation. The velocity-integrated intensity map and position-velocity diagram of the 12CO J=2-1 line present an obvious bipolar component, further verifying that this region has an outflow motion. This region is also associated with an HII region, an IRAS source, and an H2O maser. The H2O maser has the velocity of 1.1 km/s. Compared with the components of the outflow, we find that the H2O maser is not associated with the outflow. Using the large velocity gradient model, we concluded that possible averaged gas densities of the blueshifted lobe and redshifted lobe are 1.0*10^{5}$cm^{-3} and 2.0*10$^{4} cm^{-3}, while kinetic temperatures are 26.9 K and 52.9 K, respectively. Additionally, the outflow has {a} higher integrated intensity ratio (I_{CO J=3-2}/I_{CO J=2-1}).Comment: 8 pages, 5 figures, Accepted by Research in Astronomy and Astrophysic