multigrid

Desgin and Analysis of Multi-Grid Method

• L. Chen, R.H. Nochetto and J. Xu. Local Multilevel Methods on Graded Bisection Grids. To be submitted, 2009.

We present a novel decomposition of spaces based on the geometric structure of bisection grids in any dimension and use it to bridge the gap between graded and quasi-uniform grids. We show that local smoothing for a new added node needs to be performed only for three vertices.

• L. Chen and C-S. Zhang. A coarsening algorithm on adaptive grids by newest vertex bisection and its applications. Submitted,2009.

We develop an efficient and easy-to-implement coarsening algorithm to find the decomposition bisection grids. The coarsening algorithm does not require storing the binary refinement tree explicitly. Numerical experiments demonstrate that the proposed coarsening algorithm is very efficient when applied for multilevel preconditioners and mesh adaptivity for time-dependent problems.

• L. Chen, R.H. Nochetto and J. Xu. Local Multilevel methods on graded bisection grids: H(curl) and H(div) systems. In prepartion, 2008.

We design and analyze multigrid methods for H(curl) and H(div) systems on adaptive grids obtained by bisection methods. We improve the existing results by removing the regularity assumptions on the solution and the grid.

• J. Xu, L. Chen and R.H. Nochetto. Optimal Multilevel Methods for H(grad), H(curl), and H(div) Systems on Adaptive and Unstructured Grids in Multiscale, Nonlinear and Adaptive Approximatio, 2009.

We summarize our work in this book chapter.

• L. Chen, J. Xu and Y. Zhu. Local Multilevel Preconditioners for Elliptic Equations with Jump Coefficients on Bisection Grids. In prepration, 2009.

We use local mutilevel method as a preconditioner and show it is robust for symmetric elliptic problems with piecewise constant coefficients with possibly large jump.

• Y. Wu, L. Chen, X. Xie, and J. Xu. Convergence analysis of V-cycle multigrid method for anisotropic problem. In prepration, 2009.

For elliptic equations with anisotropic diffusion, we prove uniform convergence of multigrid algorithms without using "regularity assumption" in both the semi-coarsening and uniform coarsening cases.

• L. Chen. Derivation of X-Z identity from Auxiliary Space Method. Research Notes. 2009.

The X-Z identity is derived from the method of auxiliary spaces.