Advances in Numerical Analysis

Volume 2015, Article ID 187604, 7 pages

http://dx.doi.org/10.1155/2015/187604

## Approximation Properties of a Gradient Recovery Operator Using a Biorthogonal System

School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia

Received 24 June 2014; Revised 15 December 2014; Accepted 31 December 2014

Academic Editor: Slimane Adjerid

Copyright © 2015 Bishnu P. Lamichhane and Adam McNeilly. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A gradient recovery operator based on projecting the discrete gradient onto the standard finite element space is considered. We use an oblique projection, where the test and trial spaces are different, and the bases of these two spaces form a biorthogonal system. Biorthogonality allows efficient computation of the recovery operator. We analyze the approximation properties of the gradient recovery operator. Numerical results are presented in the two-dimensional case.

#### 1. Introduction

The gradient reconstruction is a popular technique to develop reliable a posteriori error estimators for approximating the solution of partial differential equations using adaptive finite element methods [1–6]. The main idea of the gradient recovery error estimators is based on postprocessing the computed gradient and showing that the postprocessed gradient has a better convergence rate to the true gradient than the computed gradient. There are many ideas of gradient postprocessing. Most popular techniques are local least-squares fitting or patch recovery [1, 2, 4], weighted averaging [4], local or global -projection [4, 5, 7], and polynomial preserving recovery [8].

Recently we have presented a gradient reconstruction operator based on an oblique projection [9], where the global -projection is replaced by an oblique projection in order to gain the computational efficiency. The oblique projection operator is constructed by using a biorthogonal system. In fact, for the linear finite element in simplicial meshes, this approach reproduces the so-called gradient reconstruction scheme by the weighted averaging [4, 6, 10]. We proved that the error estimator based on the oblique projection is asymptotically exact in mildly unstructured meshes using the fact that the error estimator based on -projection is also asymptotically exact for such meshes [4, 5, 7, 11].

In this paper, we aim at analyzing the approximation property of the recovered gradient in one dimension using an oblique projection. As the approach of using an oblique projection reproduces the weighted average gradient recovery of linear finite elements [4, 6], this construction is quite useful in extending the weighted average gradient recovery of linear finite elements to quadrilaterals and hexahedras.

Let with and . Let be a partition of the interval . We define the interior of the grid, denoted by , as We also define the set of intervals in the partition as , where . Two sets and of indices are also defined as respectively. A piecewise linear interpolant of a continuous function is written as with where is the standard hat function associated with the point , . We define a discrete space, The linear interpolant of is the continuous function defined by . However, if we compute the derivative of this interpolant , the resulting function will not be continuous. To make the derivative continuous we project the derivative of the interpolant, , onto the discrete space . There are two different types of projection. One is an orthogonal projection and the other is an oblique projection. The orthogonal projection operator, , that projects onto is to find a that satisfies Since , we can represent it as an -dimensional vector: Now the requirement given in (5) is equivalent to a linear system: , where is a mass matrix, and Here the mass matrix is tridiagonal. We can reduce computation time greatly if we have a diagonal mass matrix. This can be done if we use a suitable oblique projection instead of an orthogonal projection. We consider the projection which is defined as the problem of finding such that where is another piecewise polynomial space, not orthogonal to , with ; see [12]. In fact, the projection operator is well-defined due to the following stability condition. There is a constant independent of the mesh-size such that [9, 13] In order to achieve that the mass matrix is diagonal we need to define a new set of basis functions for , , that are biorthogonal to the standard hat basis function (Figure 1) we used previously. This biorthogonality relation is defined as where is the Kronecker delta function: and is a positive scaling factor. The basis functions for are simply given by and, for , By using an oblique projection the mass matrix will be diagonal. We let the diagonal mass matrix be , so that our system is . The values are our estimates of the gradient of at the point . So, we estimate the gradient by finding , where We want to calculate the error in this approximation and find out when approximates exactly for each . As in [14, 15] we want to see if approximates exactly when is a quadratic polynomial.