Lowest-degree robust finite element schemes for inhomogeneous bi-Laplace problems

TL;DR

This paper introduces the lowest-degree finite element schemes using the reduced rectangle Morley element for inhomogeneous bi-Laplace problems, achieving optimal convergence and verified through numerical experiments.

Contribution

It proposes novel finite element schemes with piecewise quadratic polynomials for inhomogeneous bi-Laplace problems, including stability and approximation analysis.

Findings

Optimal convergence rates established

Numerical experiments confirm theoretical results

Lowest-degree finite element schemes demonstrated

Abstract

In this paper, we study the numerical method for the bi-Laplace problems with inhomogeneous coefficients; particularly, we propose finite element schemes on rectangular grids respectively for an inhomogeneous fourth-order elliptic singular perturbation problem and for the Helmholtz transmission eigenvalue problem. The new methods use the reduced rectangle Morley (RRM for short) element space with piecewise quadratic polynomials, which are of the lowest degree possible. For the finite element space, a discrete analogue of an equality by Grisvard is proved for the stability issue and a locally-averaged interpolation operator is constructed for the approximation issue. Optimal convergence rates of the schemes are proved, and numerical experiments are given to verify the theoretical analysis.