multigrid

Desgin and Analysis of Multi-Grid Method

Discretization of partial differential equations often leads to linear algebraic equations of the form Au=f, where A is a N by N sparse matrix and f ia N by 1 vector. How to solve this algebraic equation efficiently is a basic question in numerical PDEs (and in all scientific and engineering computing). Multigrid methods are efficient iterative methods for solving algebraic systems arized from discretizations of differential equations.

• 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. Journal of Computational Mathematics, 28(6):767-789, 2010

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, M.J. Holst, J. Xu, and Y. Zhu. Local Multilevel Preconditioners for Elliptic Equations with Jump Coefficients on Bisection Grids. Submitted,2010.

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.

• L. Chen, R.H. Nochetto and C-S. Zhang. Multigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids. Accepted. Proceedings of 19th Domain Decomposition. 2010.

We develop and analyze an efficient multigrid method to solve the finite element systems from elliptic obstacle problems on two dimensional adaptive meshes.

• 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, Springer, 2009

We summarize our work in this book chapter.

• L. Chen. Deriving the X-Z Identity from Auxiliary Space Method. Accepted. Proceedings of 19th Domain Decomposition. 2010.

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

• Y. Wu, L. Chen, X. Xie, and J. Xu. Convergence analysis of V-cycle multigrid method for anisotropic problem. To be submitted, 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.