Applications of local projection operators based on biorthogonality for finite element stabilisation and adaptivity

This thesis proposes that a biorthogonal system based local projection operator can be used both to stabilise finite element discretisation and improve derivative recovery techniques. Such research is important because while the finite element method is a useful numerical approach to solve partial differential equations, in some instances it requires stabilisation terms to guarantee convergence. Local projection operators are used in stabilisation techniques to stabilise a lower-order finite element method when applied to the Stokes and convection-diffusion-reaction problems. We present an application where the local projection operator is used for derivative recovery, with particular reference to gradient and Hessian recovery. We introduce a solution for the Stokes equations when using equal order elements for the velocity and pressure terms. A new efficient local projection stabilisation technique is presented, based on a biorthogonal system. Here, we neither need to enrich the finite element space for the velocity by bubble functions nor need two different finite element grids to achieve the locality of the stabilisation term. In addition, the convection-diffusion-reaction problem is explored by using the local projection for stabilising the finite element method and adaptive finite element method approaches. We demonstrate that the projection matrix can be reused when calculating an estimate for the error of the solution. We use this estimate to direct which elements are to be refined, increasing the quality of the solution while minimising computational cost associated with refining elements. We also explore the limitations of the finite element method for problems containing sharp gradients within the solution. Further investigation of applications for implementation of local projection operators involves gradient recovery and Hessian recovery. To date, few research articles address nonconforming elements, but our approach demonstrates how the projection of the gradient onto the nonconforming finite element space enhances the quality of the estimated gradient. We also show that when projecting onto the standard finite element space, the estimated gradient is further improved. Extending data recovery, our process enables recovering the Hessian from a finite element solution using local projection, even for lower order solutions such as linear finite elements. We show that the recovered Hessian can be used in the adaptive finite element method to direct which elements in the mesh are to be refined.