Section:
New Results
Element metric, element quality and interpolation error metric
Participants :
Paul Louis George [correspondant] , Houman Borouchaki.
The metric of a simplex of is a metric tensor
(symmetric positive definite matrix) in which the element
is unity (regular with unit edge lengths). This notion is related to the
problem of interpolation error of a given field over a mesh.
Let be a simplex and let us denote by the
vector joining vertex and vertex of .
The metric of can be written as:
where is a rank 1 matrix related to
edge .
The metric of a simplex also characterizes the element
shape. In particular, if it is the identity, the element is
unity. Hence, to define the shape quality of an element, one can
determine the gap of the element metric and the identity using
different measures based on the eigenvalues of or those of ,
e.g. .
Notice that metric is directly related to the
geometry of the element (edge length, facet area, element volume).
The first algebraic
shape quality measure ranging from 0 to 1 is defined as the ratio of the geometric average
of the eigenvalues of and their arithmetic average:
As the geometric average is smaller than the arithmetic average, this
measure is well defined.
In addition, it is the algebraic reading of the well-known quality
measure defined by:
where the volume and the square of the edge lengths are
involved. The algebraic meaning justifies the above geometric measure.
The second algebraic
shape quality measure is defined as the ratio of the harmonic average
of the eigenvalues of and their arithmetic
average (ranging also from 0 to 1):
As above, this measure is well defined, the harmonic average being
smaller the arithmetic one.
From this measure, one can derive another well-known measure involving
the roundness and the size of an element (measure which is widely used for
convergene issues in finite element methods).
Note that these measures use the invariants of or
and thus can be evaluated from the coefficients of the
characteristical polynomial of those matrices (avoiding the effective
calculation of their eigenvalues).
Another advantage of the above algebraic shape measures is their easy
extensions in an arbitrary Euclidean space. Indeed, if
is the metric of such a space, the algebraic shape measures read:
This work has been published in a journal, [8].
Following this notion of a element metric, a natural work was done regarding how to define the element
metric so as to achieve a given accuracy for the interpolation error of a function using a finite
element approximation by means of simplices of arbitrary degree.
This is a new approach for the majoration of the
interpolation error of a polynomial function of arbitrary degree
interpolated by a polynomial function of degree . From that
results a metric, the so-called interpolation metric, which allows for
a control of the error. The method is based on the geometric and
algebraic properties of the metric of a given element, metric in
which the
element is regular and unit. The interpolation metric plays an
important role in advanced computations based on mesh adaptation.
The method relies in a Bezier reading of the functions combined with
Taylor expansions. In this way, the error in a given element is
fully controled at the time the edges of the element are controled.
It is shown that the error in bounded as
where is a constant depending on and ,
is the edge from the vertices of of index and ,
is the derivative of order of applied to a
-uple uniquely composed of .
If we consider the case and
is a vector in , we have
where
the quadratic forms are defined
by the matrices of order 2 (with constant entries):
those matrices being the hessians of the derivatives of of order .
This work resulted in a paper submitted in a journal and currently under revision.