The Hessian Discretisation Method for Fourth Order Linear Elliptic Equations

Abstract

In this paper, we propose a unified framework, the Hessian discretisation method (HDM), which is based on four discrete elements (called altogether a Hessian discretisation) and a few intrinsic indicators of accuracy, independent of the considered model. An error estimate is obtained, using only these intrinsic indicators, when the HDM framework is applied to linear fourth order problems. It is shown that HDM encompasses a large number of numerical methods for fourth order elliptic problems: finite element methods (conforming and non-conforming) as well as finite volume methods. We also use the HDM to design a novel method, based on conforming $\mathbb {P}_1$ finite element space and gradient recovery operators. Results of numerical experiments are presented for this novel scheme and for a finite volume scheme.

Appendix: Technical Results

Lemma A.1

(Poincaré inequality along an edge) Let ${\sigma }$ be an edge of a polygonal cell, $w \in H^1({\sigma })$ and assume that w vanish at a point on the edge ${\sigma }\in {\mathcal F}$. Then there exists $C > 0$ such that

$$\begin{aligned} \Vert w\Vert _{L^2({\sigma })} \le h_{{\sigma }} \Vert \partial w\Vert _{L^2({\sigma })}, \end{aligned}$$

where $\partial$ denotes the derivative along the edge and $h_{\sigma }$ is the length of the edge.

Proof

Let m denote the point on the edge ${\sigma }$ which satisfies $w(m) = 0$. For $m < x$, we get

$$\begin{aligned} w(x) = w(m) + \int _{m}^{x}\partial w(y){\,\mathrm d}y = \int _{m}^{x}\partial w(y){\,\mathrm d}y. \end{aligned}$$

A use of Cauchy–Schwarz inequality yields

$$\begin{aligned} |w(x)|&\le |x-m|^{1/2}\bigg (\int _{m}^{x}|\nabla w|^2{\,\mathrm d}y\bigg )^{1/2}\\&\le \sqrt{h_{\sigma }}\bigg (\int _{\sigma }|\partial w|^2{\,\mathrm d}y\bigg )^{1/2}. \end{aligned}$$

Squaring this yields $|w(x)|^2\le h_{\sigma }\int _{\sigma }|\partial w|^2{\,\mathrm d}y$ and integrating over the edge concludes the proof. $\square$

Lemma A.2

(Integration by parts) Let P be a fourth order tensor. For $\xi \in H^2({\Omega })^{d \times d}$ and $\phi \in H^1(\Omega )$, we have

$$\begin{aligned} \int _\Omega (\mathcal H:P\xi )\phi =-\int _\Omega \nabla \phi \cdot \mathrm{div}(P\xi ) + \int _{\partial \Omega } \mathrm{div}(P\xi \cdot n)\phi . \end{aligned}$$

For $\psi \in H^2(\Omega )$,

$$\begin{aligned} \int _\Omega P\xi :\mathcal H\psi =-\int _\Omega \nabla \psi \cdot \mathrm{div}(P\xi ) + \int _{\partial \Omega } (\mathrm{div}(P\xi n))\cdot \nabla \psi . \end{aligned}$$

For $\zeta \in H^1(\Omega )^d$,

$$\begin{aligned} \int _\Omega P\xi :\nabla \zeta = -\int _\Omega \mathrm{div}(P\xi )\cdot \zeta + \int _ {\partial \Omega }(\mathrm{div}(P\xi n))\cdot \zeta . \end{aligned}$$