Optimal Quasi-diagonal Preconditioners for Pseudodifferential Operators of Order Minus Two

Abstract

We present quasi-diagonal preconditioners for piecewise polynomial discretizations of pseudodifferential operators of order minus two in any space dimension. Here, quasi-diagonal means diagonal up to a sparse transformation. Considering shape regular simplicial meshes and arbitrary fixed polynomial degrees, we prove, for dimensions larger than one, that our preconditioners are asymptotically optimal. Numerical experiments in two, three and four dimensions confirm our results. For each dimension, we report on condition numbers for piecewise constant and piecewise linear polynomials.

Appendix A. Proof of Lemma 7

We follow exactly the same ideas and lines of proof as in [1, Appendix A] adapted to our situation (with volume force but homogeneous boundary conditions) and notation.

Throughout fix $T\in \mathcal {T}$ and let $\eta \in C^\infty (\omega (T))$ denote a cut-off function with the properties

$$\begin{aligned} \eta |_T&= 1, \qquad \eta |_{\Omega \setminus \omega (T)} = 0, \end{aligned}$$

(17a)

$$\begin{aligned} \Vert D^m \eta \Vert _{L^\infty (\omega (T))}&\lesssim h_T^{-m}, \quad \text {for } m=0,1,2. \end{aligned}$$

(17b)

Let $u\in H_0^1(\Omega )$ denote the solution of (13) with datum $f\in L^2(\Omega )$. Let $s = s(\Omega )\in (1/2,1]$ denote the regularity shift. Then, by (14) we have that

$$\begin{aligned} \Vert u\Vert _{1+s,T} \le \Vert \eta u\Vert _{1+s,\omega (T)} = \Vert \eta u\Vert _{1+s} \lesssim \Vert \Delta (\eta u)\Vert _{-1+s}, \end{aligned}$$

(18)

since $\eta u|_\Gamma = 0$ and $\Delta (\eta u)\in H^{-1+s}(\Omega )$.

We consider the case where $|\partial \omega (T)\cap \Gamma | = 0$. Then, $\nabla (\eta u)\cdot {\varvec{n}}= 0$ on $\Gamma$. Let $v\in H^{1-s}(\Omega )$. Using $v_{\omega (T)} := |\omega (T)|^{-1} \int _{\omega (T)} v \,dx$ and the product rule

$$\begin{aligned} \Delta (\eta u) = u\Delta \eta + 2\nabla \eta \cdot \nabla u -\eta f, \end{aligned}$$

(19)

we infer that

$$\begin{aligned} (\Delta (\eta u),v)&= (\Delta (\eta u),v-v_{\omega (T)}) = (\Delta (\eta u),v-v_{\omega (T)})_{\omega (T)} \\&= (u\Delta \eta + 2\nabla \eta \cdot \nabla u - \eta f,v-v_{\omega (T)})_{\omega (T)}. \end{aligned}$$

Note that $\Vert v-v_{\omega (T)}\Vert _{\omega (T)} \lesssim h_T^{1-s}|v|_{1-s,\omega (T)}$ and therefore,

$$\begin{aligned} |(\Delta (\eta u),v)|&\lesssim h_T^{1-s} (\Vert \Delta \eta \Vert _{L^\infty (\omega (T))}\Vert u\Vert _{\omega (T)} + \Vert \nabla \eta \Vert _{L^\infty (\omega (T))}\Vert \nabla u\Vert _{\omega (T)}\\&\quad + \Vert \eta \Vert _{L^\infty (\omega (T))} \Vert f\Vert _{\omega (T)}) \Vert v\Vert _{1-s} \\&\lesssim (h_T^{-1-s}\Vert u\Vert _{\omega (T)} + h_T^{-s}\Vert \nabla u\Vert _{\omega (T)} + h_T^{1-s}\Vert f\Vert _{\omega (T)})\Vert v\Vert _{1-s}. \end{aligned}$$

Recall that we consider the case where $\partial \omega (T)$ does not share a boundary facet. So the same estimates hold true when we replace u by $u=u-u_{\omega (T)}$ since $\eta (u-u_{\omega (T)})|_{\Gamma } = 0$ and $\Delta (\eta (u-u_{\omega (T)}))\in H^{-1+s}(\Omega )$. Using $\Vert u-u_{\omega (T)}\Vert _{\omega (T)}\lesssim h_T \Vert \nabla u\Vert _{\omega (T)}$, dividing by $\Vert v\Vert _{1-s}$, and taking the supremum we get

$$\begin{aligned} h_T^s|\nabla u|_{s,T} \le h_T^s\Vert u-u_{\omega (T)}\Vert _{1+s,T} \lesssim h_T^s\Vert \Delta (\eta (u-u_{\omega (T)}))\Vert _{-1+s} \lesssim \Vert \nabla u\Vert _{\omega (T)} + h_T\Vert f\Vert _{\omega (T)}. \end{aligned}$$

Now we tackle the case where $\partial \omega (T)$ includes at least one boundary facet $E\in \mathcal {E}^\Gamma$. First, note that if a function w vanishes on one facet E, then

$$\begin{aligned} \Vert w\Vert _{\omega (T)} \lesssim h_T^r |w|_{r,\omega (T)}, \quad 0\le r\le 1. \end{aligned}$$

Second, recall that

$$\begin{aligned} \Vert \phi \Vert _{-1+s} = \sup _{0\ne v\in C_0^\infty (\Omega )} \frac{(\phi ,v)}{\Vert v\Vert _{1-s}}. \end{aligned}$$

Then, the product rule (19) and the properties of the cut-off function prove that

$$\begin{aligned} |(\Delta (\eta u),v)|&\lesssim (h_T^{-2}\Vert u\Vert _{\omega (T)} + h_T^{-1}\Vert \nabla u\Vert _{\omega (T)} + \Vert f\Vert _{\omega (T)})\Vert v\Vert _{\omega (T)} \quad \text {for all } v \in C_0^\infty (\Omega ). \end{aligned}$$

Using $\Vert v\Vert _{\omega (T)}\lesssim h_T^{1-s}|v|_{1-s,\omega (T)}$, $\Vert u\Vert _{\omega (T)}\lesssim h_T\Vert \nabla u\Vert _{\omega (T)}$ we further infer that

$$\begin{aligned} |(\Delta (\eta u),v)|&\lesssim h_T^{-s} (\Vert \nabla u\Vert _{\omega (T)} + h_T\Vert f\Vert _{\omega (T)})\Vert v\Vert _{1-s}. \end{aligned}$$

Dividing by $\Vert v\Vert _{1-s}$ and taking the supremum, we conclude that

$$\begin{aligned} h_T^s|\nabla u|_{s,T} \le h_T^s\Vert u\Vert _{1+s,T} \lesssim h_T^s \Vert \Delta (\eta u)\Vert _{-1+s} \lesssim \Vert \nabla u\Vert _{\omega (T)} + h_T \Vert f\Vert _{\omega (T)}. \end{aligned}$$

This finishes the proof of Lemma 7. $\square$