Joint work with A. PlaksIt is well known that the accuracy of Finite Element (FE) interpolation depends on the size and shape of elements. We establish a close relationship between interpolation errors and properties of certain FE matrices. While the proposed a priori error estimates are algebraic in nature, they have a clear geometric meaning in many cases.
First, we show that under fairly general assumptions the element-wise interpolation error in a given energy norm can be estimated via the maximum eigenvalue (or trace) of the stiffness matrix. Several well known geometric conditions (e.g. Zlámal’s minimum angle condition and the Synge-Babuška-Aziz maximum angle condition for first order triangular elements) in fact follow from the maximum eigenvalue analysis.
Second, for H1-approximation on first order tetrahedral nodal elements, or equivalently, for L2-approximation of conservative fields on tetrahedral edge elements, the interpolation error is shown to depend on the minimum singular value of the ‘edge shape matrix’ whose columns represent the unit edge vectors of the tetrahedron. This new singular value estimate is a precise and clear generalization of the maximum angle condition to three (and more) dimensions. Multiple links between the minimum singular value and the standard geometric measures (angles, the ratio of the radius of the inscribed sphere to the element diameter, etc.) are established.
Test examples and engineering problems of electromagnetic field analysis are presented to illustrate the theoretical results.