Lowest-degree robust finite element scheme for a fourth-order elliptic singular perturbation problem on rectangular grids

TL;DR

This paper introduces a robust, lowest-degree quadratic nonconforming finite element method on rectangular grids for a challenging fourth-order elliptic singular perturbation problem, with proven optimal approximation and verified numerical results.

Contribution

It presents the first lowest-degree quadratic nonconforming finite element scheme that is robust for singular perturbation problems, along with a new non-projective interpolation operator and a solution to a long-standing open problem.

Findings

Method is robust with respect to the perturbation parameter.

Optimal order approximation is achieved by a new interpolation operator.

Numerical results confirm theoretical convergence.

Abstract

In this paper, a piecewise quadratic nonconforming finite element method on rectangular grids for a fourth-order elliptic singular perturbation problem is presented. This proposed method is robustly convergent with respect to the perturbation parameter. Numerical results are presented to verify the theoretical findings. The new method uses piecewise quadratic polynomials, and is of the lowest degree possible. Optimal order approximation property of the finite element space is proved by means of a locally-averaged interpolation operator newly constructed. This interpolator, however, is not a projection. Indeed, we establish a general theory and show that no locally defined interpolation associated with the locally supported basis functions can be projective for the finite element space in use. Particularly, the general theory gives an answer to a long-standing open problem presented in [Demko, J. Approx. Theory, $\bf{43}$(2):151--156, 1985].