# Multigrid Methods for Anisotropic Edge Refinement

@article{Apel2002MultigridMF, title={Multigrid Methods for Anisotropic Edge Refinement}, author={Thomas Apel and Joachim Sch{\"o}berl}, journal={SIAM J. Numer. Anal.}, year={2002}, volume={40}, pages={1993-2006} }

A finite element method with optimal convergence on nonsmooth three dimensional domains requires anisotropic mesh refinement towards the edges. Multigrid methods for anisotropic tensor product meshes are available and are based either on line (or plane) smoothers or on semicoarsening strategies. In this paper we suggest and analyze a new multigrid scheme combining semicoarsening and line smoothers to obtain a solver of optimal algorithmic complexity for anisotropic meshes along edges.

## 23 Citations

### A new methodology for anisotropic mesh refinement based upon error gradients

- Computer Science
- 2004

### An anisotropic finite element method on polyhedral domains: interpolation error analysis

- Computer Science, MathematicsMath. Comput.
- 2018

This work develops interpolation error estimates in suitable weighted spaces for the anisotropic mesh, especially for the tetrahedra violating the maximum angle condition, which can be used to design optimal finite element methods approximating singular solutions.

### Uniform convergence of the multigrid V‐cycle on graded meshes for corner singularities

- MathematicsNumer. Linear Algebra Appl.
- 2008

It is proved that the MG V-cycle with standard smoothers and piecewise linear interpolation converges uniformly for the linear systems obtained by finite element discretization of the Poisson equation on graded meshes.

### On the robustness of a multigrid method for anisotropic reaction-diffusion problems

- MathematicsComputing
- 2007

This paper analyzes the convergence of a multigrid method with a robust (line) smoother and derives contraction number bounds smaller than one uniform with respect to the mesh size and the parameters ε and μ.

### On a nested refinement of anisotropic tetrahedral grids under Hessian metrics

- Computer Science

This paper will show that, given an arbitrary set of mid-edge points, for example, produced by a Hessian metric, a tetrahedral grid can be nestedly refined without any other new points introduced.

### Optimal anisotropic meshes for minimizing interpolation errors in Lp-norm

- Mathematics, Computer ScienceMath. Comput.
- 2007

This paper presents a new optimal interpolation error estimate in L p norm (1 < p ≤ ∞) for finite element simplicial meshes in any spatial dimension and gives new functionals for the global moving mesh method.

### A Priori Analysis of an Anisotropic Finite Element Method for Elliptic Equations in Polyhedral Domains

- MathematicsComput. Methods Appl. Math.
- 2021

A by-product of the result is to extend the application of these anisotropic meshes to broader practical computations with the price to have “smoother” interior data.

### Multigrid Methods for Linear Elliptic Optimal Control Problems

- Computer Science
- 2008

Multigrid optimization schemes that solve linear elliptic optimal control problems are discussed and a comparison is made between the multigrid for optimization (MGOPT) method and the collective smoothing multigrids (CSMG) method.

### Multilevel methods for nonuniformly elliptic operators and fractional diffusion

- MathematicsMath. Comput.
- 2016

A multilevel method with line smoothers and a nearly uniform convergence result on anisotropic meshes is presented and the so-called Xu-Zikatanov (XZ) identity is derived under the assumption that the underlying mesh is quasi-uniform.

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