PHAML supports 5 types of grid refinement:

*h*-refinement is accomplished using *newest node bisection* of
triangles. To refine a triangle, one of the vertices, called the *peak*,
is connected to the midpoint of the opposite side, called the *base*.
The new vertex becomes the peak of the new triangles. Unlike other adaptive
refinement strategies, newest node bisection always maintains a compatible
grid with no hanging nodes. This is accomplished by always refining triangles
in pairs (except when the base is on the boundary). If two triangles share
a common base, they can be refined as a pair. If not, then after refining
one of them, one of its children will share the base of the other and they
can be refined as a pair. Nondegeneracy of the triangles is guaranteed, since
newest node bisection creates only 4 triangle shapes from a starting triangle.
The linear *h*-hierarchical basis functions are easily defined by
defining a new basis function at the new vertex while leaving all other
basis functions unchanged. High order *h*-hierarchical bases are
also easily defined, if the polynomial degree is constant throughout the grid.

Newest node bisection of triangles, and the 4 triangle shapes created.

Peaks must be assigned to the initial grid in such a way that every triangle
is paired with another triangle, or the boundary. It has been proven that
such a *perfect matching* exists for any compatible triangulation,
but finding it is, in general, an NP-hard problem. But given any initial
triangulation, a triangulation can be constructed for which the matching
is trivial. This is accomplished by pairing up as many triangles as possible
with a simple algorithm, bisecting these pairs of triangles, and trisecting
any remaining triangles by connecting the midpoint of the triangle to the
three vertices. The vertices created by this process are the peaks of the
triangles in the new initial grid.

*p*-refinement means to increase the degree of the piecewise
polynomials over a triangle. The *p*-hierarchical basis functions
consist of a linear basis function associated with each vertex, quadratic
and higher bases associated with each edge, and cubic and higher bases
associated with each triangle. In PHAML, the degree of two neighboring
triangles is restricted to differ by at most 1. The degree of the edge
between two triangles is the maximum of the degrees of the two triangles.

*hp*-adaptive refinement uses both *h-* and
*p*-refinement. The advantage is that it can achieve exponential
rates of convergence w.r.t. the number of degrees of freedom.
We have implemented 13 strategies for *hp*-adaptive refinement
in PHAML and used it to perform an experiment to compare the effectiveness
of these strategies using 21 test problems.

Exponential convergence of

**Publications**

Mitchell, W.F. and McClain, M.A., ** A Comparison of hp-Adaptive Strategies for Elliptic
Partial Differential Equations**,

Mitchell, W.F. and McClain, M.A., ** A Comparison of hp-Adaptive Strategies for Elliptic
Partial Differential Equations (long version)**,

Mitchell, W.F., ** A Collection of 2D Elliptic Problems for Testing Adaptive
Algorithms**, * NISTIR 7668*, 2010.
( pdf, 1.6M)

Mitchell, W.F. and McClain, M.A., ** A Survey of hp-Adaptive Strategies for Elliptic Partial
Differential Equations**, in Recent Advances
in Computational and Applied Mathematics (T. E. Simos, ed.), Springer,
2011, pp. 227-258.
(preprint, pdf, 16M)

Mitchell, W.F., **Adaptive refinement for arbitrary finite element
spaces with hierarchical bases**, *J. Comp. Appl. Math.* 36
(1991), pp. 65-78.

Mitchell, W.F. **A comparison of adaptive refinement techniques
for elliptic problems**. *ACM Trans. Math. Soft.* 15 (1989),
pp. 326-347.

Mitchell, W.F., **Unified multilevel adaptive finite element
methods for elliptic problems**, Ph.D. thesis, Technical report
UIUCDCS-R-88-1436, Department of Computer Science, University
of Illinois, Urbana, IL, 1988.
(
gzipped postscript, 194k)

Last change to this page: December 14, 2011

Date this page created: April 2, 2007

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