Analysis, algebra and geometry (CVT)

$Q. Du$
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Qiang's research gallery 9

Our latest lemma - for those math-savvy readers
(see Qiang's publications for details)

Most materials related to my web presence are designed for the general public rather than professional mathematicians. It's no wonder that very little technical discussion was made available. This page serves as the exception, in case some are interested in what we have proved lately.

Be aware that we would like to showcase some examples that are simple, elegant, and perhaps, surprising. So here is my two cents worth.

A tale of two worlds - connections of algebra and geometry
appeared in analysis and computation

As mathematician, we've always been interested in making connections between the concepts in various different branches of mathematics and utilizing them for problem solving. The latest lemma is a case where we make the connection between a differential operator (ala analysis), the stiffness matrix (ala algebra) from finite element approximation (ala computation) and the geometric features of the underlying unstructured simplicial mesh (ala geometry).

Notation:

K_t: the element stiffness matrix of FEM of -Δ ;

t : any n-dim simplex element in the geometric mesh;

|t| : volume of the simplicial element t;

{A_i} : n-1 dim volume of the face of the simplicial element t;

t^: the standard reference simplex in Rⁿ;

{λ_j}: barycentric coordinates in the reference simplex t^;

{L_m}: finite element nodal basis functions depending only on {λ_j}.

The DWZ trace formula:

Lemma (DU-WANG-ZHU 2007)

where the two terms are given respectively by:

and

Remarks:

Discussions on the element stiffness matrix and the properties of the underlying mesh have been made through the development of the finite element theory and practice. Yet, to our knowledge, nothing so precise and general has been presented before.

For conforming linear element, the algebraic constant γ_dⁿ equals to 1, a well-known fact that has been widely stated in the literature. The particular 2d case has been known ever since the invention of finite element. Yet, even for general element, this constant is independent of the sub indices after the summation as long as the invariance of the basis with respect to the permutation of barycentric coordinates is assured, which is the case for most element and the natural basis. This is the key observation in proving the trace formula.

The elegant lemma tells a tale of two worlds, as the formula implies that the trace of the stiffness matrix is given by a product of two contributing factors: one is purely algebraic and depends on the nodal basis set while the other is purely geometric and depends only on the geometry of the simplex mesh.

This lemma has been applied to characterize the relation between the global stiffness matrix conditioning and the mesh geometry/quality for general finite element discretizations of general second order elliptic problems. Maybe you will find other applications of it too!

Further reading:

On Mesh Geometry and Stiffness Matrix Conditioning for General Finite Element Spaces,
by Q. Du, D. Wang and L. Zhu, SIAM J. Numer. Analy., 47, pp.1421-1444, 2009.
You may find the proof of the trace formula (if you feel that not enough details have been
disclosed in the above) as well as many more references on the subject in the paper including
the text books by Strang-Fix/Ciarlet/Axelson-Barker/Scott-Brenner.
Qiang's other publications.

Collaborators: D. Wang and L. Zhu

Contact Qiang Du 2008-07-28