Numerical Analysis RIT

Focus of the RIT in Fall 2006: Adaptive Multilevel FEM for PDE
Meeting schedule: Monday 4:30 -- 5:30, Math 1313
Other RIT

Week 10 (Dec 11)

Chensong Zhang. Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities.

ABSTRACT: General subspace correction algorithms are proposed for a constrained convex minimization problem by Tai. General results on the convergence rates of these algorithms are derived. We discuss the application of this general framework to the solution of obstacle problems by multilevel subspace decompostions.

Week 9 (Dec 4)

Manel Cascon. The completion of newest vertex bisection

Week 7,8 (Nov 20,27)

Manel Cascon. Convergence and Optimality of an AFEM for General Second Order Linear Elliptic PDE

ABSTRACT: Despite the overwhelming computational evidence that AFEM lead to optimal meshes, the mathematical theory started very recently with Binev et al [1] and Stevenson [3]. Paper [1] discusses geometric properties of bisection which turn out to be crucial for optimality; The AFEM in [1]
includes a coarsening step which is not necessary in practise for linear elliptic PDE. Stevenson presents in [3] an algorithm with optimal complexity without coarsening, but it does not work with general linear elliptic operator and is unpractical.

In this work, we modify the Morin-Nochetto-Siebert algorithm [2] with a new procedure for oscillation reduction relative to error estimator. We prove that the resulting AFEM is a contraction for the total error between two consecutive adaptive loops. This strict error reduction property then leads to the optimality of AFEM, that is the total error decays in terms of number of degrees of freedom as predicted by the best approximation. This work is joint with R. H. Nochetto.

Bibliography

[1] P. Binev, W. Dahmen, and R. DeVore Adaptive finite element methods with convergence rate, Numer. Math., 97(2), pp.~219-268 (2004).

[2] P. Morin, R.H. Nochetto, and K.G. Siebert, Data Oscillation and Convergence of adaptive FEM. SIAM J. Numer. Anal. 38, pp.~466--488 (2000).

[3] R. Stevenson, Optimality of a standard adaptive finite element method Found. Comput. Math. Online. DOI 10.1007/s10208-005-0183-0, (2006).

Week 6 (Nov 13, 14)

Long Chen. Uniform Convergence of Multigrid Methods on Adaptive Grids obtained by Newest Vertex Bisection

ABSTRACT: We finished proof by proving a stable decomposition. In 2-D, we show that hierarchical basis MG is robust with respect to the jump of coefficients.

Week 5 (Nov 6)

Long Chen. Uniform Convergence of Multigrid Methods on Adaptive Grids obtained by Newest Vertex Bisection

ABSTRACT: We shall prove multigrid method for solving Poisson equation on adapted grids obtained by newest vertex bisection will converges uniformly with respect to the size of unknowns. Furthermore we shall show the computational cost is also optimal. This is a generalization of the result in \cite{Wu.H;Chen.Z2003} to higher dimensions with a simplified and more straightforward proof.

Week 4 (Oct 30)

Long Chen. New Chaeracterization on Adaptive Grids obtained by Newest Vertex Bisection

ABSTRACT: We give a decomposition of adaptive grids obtained by newest vertex bisection. This decomposition is new and crucial for developing multigrid.

Week 3 (Oct 23)

Long Chen. X-Z identity for successive subspace correction method.

ABSTRACT: We shall report a sharp estimate of the convergence rate of multgird methods. Indeed it is an identity on the contraction number for general successive subspace correction methods. The main reference is

[1] J. Xu and L. Zikatanov. The method of alternating projections and the method of subspace corrections in Hilbert space. Journal of The American Mathematical Society, 15:573–597, 2002.

We shall give a simple proof for a special but the most important case of this identity. It will be our basic tools for the analysis of multigrid on adapted grids.

Week 2 (Oct 16)

Long Chen. Subspace correction method.

ABSTRACT: We present iterative methods in the frame work of subspace correction method. We discuss in details the space decompoistion and several inner product structures introduced by the decomposition.

Week 1 (Oct 9)

Long Chen. Convergence analysis of classical iterative methods.

ABSTRACT: We give theories for the quantitative analysis of classical iterative methods for sovling algebraic system. In the analysis, the new inner product defined by the SPD matrix plays an important role.