# Optimal quadratic element on rectangular grids for $H^1$ problems

## TL;DR

This paper introduces a new quadratic finite element method on rectangular grids for $H^1$ problems, achieving optimal convergence and providing lower bounds for eigenvalues, with promising numerical results.

## Contribution

It presents a reduced quadratic rectangular Morley element with proven convergence rates and eigenvalue bounds, enhancing finite element analysis for $H^1$ problems.

## Key findings

- Convergence rate of $O(h^2)$ in energy norm on uniform grids.
- Lower bounds for eigenvalues are obtained.
- Numerical results demonstrate the method's potential.

## Abstract

In this paper, a piecewise quadratic finite element method on rectangular grids for the $H^1$ problems is presented. The proposed method can be viewed as a reduced rectangular Morley element. For the source problem, the convergence rate of this scheme is $O(h^2)$ in the energy norm on uniform grids. Besides, a lower bound of the $L^2$-norm error is also proved, which makes the capacity analysis of this scheme more clear. On the other hand, for the eigenvalue problem, the numerical eigenvalues by this element are shown to be the lower bounds of the exact ones. Some numerical results are presented, which show the potential of the proposed finite element.