Analysis of a combined NC1-C2 method for elliptic problem

TL;DR

This paper introduces a new combined NC1-C2 finite element method for elliptic problems, demonstrating its construction, basic properties, and effectiveness through theoretical analysis and numerical validation.

Contribution

It presents a novel combined finite element approach using non-conforming and conforming polynomials for elliptic problems, with detailed construction and error analysis.

Findings

The new element behaves like a P^1 non-conforming element.

Theoretical error estimates are confirmed by numerical results.

The method simplifies the construction of finite elements for elliptic problems.

Abstract

It is shown in this paper that non-conforming finite elements on the triangle using $P^{1}$-nonconforming polynomials and $P^{2}$ -conforming polynomials can be easily built and used.They appear as an 'enriched' version of the standard piecewise quadratic six-node element.This work is divided into two parts.In the first we present the basic- property of the element,namely how it can be built and basic error estimates.We have observed that this new element behaves like $P^{1}$ non-conforming element.In the second part we have applied our element to the elliptic problem and the theoretical estimate has been guaranteed by numerical result.