# Superconvergence of $C^0$-$Q^k$ finite element method for elliptic equations with approximated coefficients

## TL;DR

This paper proves that superconvergence properties of the $C^0$-$Q^k$ finite element method for elliptic equations remain valid when variable coefficients are approximated by their interpolants at Gauss Lobatto points, enabling high-order schemes.

## Contribution

It establishes superconvergence results for $C^0$-$Q^k$ finite element methods with coefficient approximation, leading to high-order finite difference schemes.

## Key findings

- Superconvergence persists with coefficient approximation at Gauss Lobatto points.
- A fourth order finite difference scheme is constructed using $C^0$-$Q^2$ elements.
- The method maintains accuracy despite coefficient approximation.

## Abstract

We prove that the superconvergence of $C^0$-$Q^k$ finite element method at the Gauss Lobatto quadrature points still holds if variable coefficients in an elliptic problem are replaced by their piecewise $Q^k$ Lagrange interpolant at the Gauss Lobatto points in each rectangular cell. In particular, a fourth order finite difference type scheme can be constructed using $C^0$-$Q^2$ finite element method with $Q^2$ approximated coefficients.