Section: New Results

Element metric, element quality and interpolation error metric

Participants : Paul Louis George [correspondant] , Houman Borouchaki.

The metric of a simplex of ${ℝ}^d$ 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 $K$ be a simplex and let us denote by $v_{ij}$ the vector joining vertex $i$ and vertex $j$ of $K$. The metric of $K$ can be written as:

where $v_{ij}{}^t{v}_{ij}$ is a $d\times d$ rank 1 matrix related to edge $ij$.

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 $ℳ$ and the identity using different measures based on the eigenvalues ${\lambda }_i=\frac{1}{h_I^2}$ of $ℳ$ or those of ${ℳ}^{-1}$, e.g. $h_i^2$. Notice that metric ${ℳ}^{-1}$ 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 ${ℳ}^{-1}$ 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 ${ℳ}^{-1}$ 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 ${ℳ}^{-1}$ or $ℳ$ 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 $ℰ$ 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 $n$ interpolated by a polynomial function of degree $n-1$. 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 $C$ is a constant depending on $d$ and $n$, $v_{ij}$ is the edge from the vertices of $K$ of index $i$ and $j$, $f^{\left(n\right)}(.)$ is the derivative of order $n$ of $f$ applied to a $n$-uple uniquely composed of $v_{ij}$. If we consider the case $d=2$ and $u=(x,y)$ is a vector in ${ℝ}^2$, we have

where the quadratic forms ${\mathscr{H}}_{(n-2,n-2-i,i)}$ are defined by the matrices of order 2 (with constant entries):

those matrices being the hessians of the derivatives of $f$ of order $n-2$.

This work resulted in a paper submitted in a journal and currently under revision.