### Cite Details

Travis Austin, Thomas Manteuffel and Steve McCormick, "A robust multilevel approach for minimizing
H(div)-dominated functionals in an H1-conforming finite element space", * Numerical Linear Algebra *, vol. 11, pp. 115-140, 2004

### Abstract

The standard multigrid algorithm is widely known to yield
optimal convergence whenever all high frequency error components
correspond to large relative eigenvalues. This property
guarantees that smoothers like Gauss-Seidel and Jacobi will
significantly dampen all of the high frequency error components,
and thus, produce a smooth error. This has been established for
matrices generated from standard discretizations of most elliptic
equations. In this paper, we address a system of equations that
is generated from a perturbation of the non-elliptic operator
*[***I** - ∇ ∇ · ] by a negative * ε
Δ*. For *ε* near to one,
this operator is elliptic, but as *ε* approaches zero, the
operator becomes non-elliptic as it is dominated by its
non-elliptic part. Previous research on the non-elliptic part
has revealed that discretizing * [ ***I** - ∇ ∇
· ] with the proper finite
element space allows one to define a robust geometric multigrid
algorithm. The robustness of the multigrid algorithm depends on a
relaxation operator that yields a smooth error. We use this
research to assist in developing a robust discretization and
solution method for the perturbed problem. To this end, we
introduce a new finite element space for tensor product meshes
that is used in the discretization, and a relaxation operator
that succeeds in dampening all high frequency error components.
The success of the corresponding multigrid algorithm is first
demonstrated by numerical results that quantitatively imply
convergence for any *ε* is bounded by the convergence for *ε*
equal to zero. Then we prove that convergence of this multigrid
algorithm for the case of *ε* equal to zero is independent of
mesh size.

### BibTeX Entry

@article{austin-2004-robust,

author = {Travis Austin and Thomas Manteuffel and Steve McCormick},

title = {A robust multilevel approach for minimizing
H(div)-dominated functionals in an H1-conforming finite element space},

year = {2004},

urlpdf = {http://math.lanl.gov/~austint/Papers/austin-2004-robust.pdf},

journal = { Numerical Linear Algebra },

volume = {11},

pages = {115-140}

}