Numerical studies of serendipity and tensor product elements for eigenvalue problems

Abstract

While the use of finite element methods for the numerical approximation of eigenvalues is a well-studied problem, the use of serendipity elements for this purpose has received little attention in the literature. We show by numerical experiments that serendipity elements, which are defined on a square reference geometry, can attain the same order of accuracy as their tensor product counterparts while using dramatically fewer degrees of freedom. In some cases, the serendipity method uses only 50% as many basis functions as the tensor product method while still producing the same numerical approximation of an eigenvalue. To encourage the further use and study of serendipity elements, we provide a table of serendipity basis functions for low order cases and a Mathematica file that can be used to generate the basis functions for higher order cases.

The hot spots conjecture can be false: Some numerical examples

“…The same is true for the L-shaped domain that is given by = [0, 1] 2 −[0.5, 1] 2 . We obtain the first non-trivial interior Neumann eigenvalue 2.429 474 and compare it with the well-known value (see [14] for the approximation 2 √ 1.475 621 845 ≈ 2.429 504). As we can see in Table 4 the absolute error decreases dramatically since we have less regularity of the solution at the corner.…”