Preconditioning high order $H^2$ conforming finite elements on triangles

Abstract

We develop a nonoverlapping Additive Schwarz Preconditioner for the p-version finite element with $H^2$-conforming elements on triangles based on Argyris elements. With a particular choice of basis functions corresponding to the $C^2$ degrees of freedom (dofs), we give a preconditioner consisting of eliminating the interior dofs, global solves of the $C^0$ and $C^1$ vertex dofs, diagonal solves of the $C^2$ vertex dofs, and block diagonal solves of the edge dofs. We show that the condition number of the preconditioner system grows at most like ${\mathscr {O}}(1+\log ^3 p)$ independent of the mesh size h. The analysis permits the use of inexact interior solves which lends itself to a more efficient implementation while maintaining the same ${\mathscr {O}}(1 + \log ^3 p)$ asymptotic growth of the preconditioned system.

Appendices

A $H^2$ Polynomial extensions

Let $I = (0,1)$. We use the standard Sobolev spaces $H^s(I)$, $s \in {\mathbb {N}}$, where $H^0(I) = L^2(I)$. For spaces of vector fields we use ${\varvec{H}}^s(I)$. We shall also need the fractional order spaces $H^{1/2}(I)$, $H^{1/2}_{00}(I)$, $H^{3/2}(I)$, and $H^{3/2}_{00}(I)$ equipped with the norms

$$\begin{aligned} \Vert u\Vert _{H^{1/2}(I)}^2&:= \Vert u\Vert _{L^2(I)}^2 + |u|_{H^{1/2}(I)}^2 \\ |u|_{H^{1/2}(I)}^2&:= \int _{0}^{1} \int _{0}^{1} \left| \frac{u(x) - u(y)}{x-y} \right| ^2 \ dx \ dy, \\ \Vert u\Vert _{H^{1/2}_{00}(I)}^2&:= \Vert u\Vert _{H^{1/2}(I)}^2 + |u|_{H^{1/2}_{00}(I)}^2 \\ |u|_{H^{1/2}_{00}(I)}^2&:= 2\int _{0}^{1} \frac{u^2(x)}{x} \ dx + 2 \int _{0}^{1} \frac{u^2(x)}{1-x} \ dx, \\ \Vert u\Vert _{H^{3/2}(I)}^2&:= \Vert u\Vert _{L^2(I)}^2 + \Vert u'\Vert _{H^{1/2}(I)}^2 \\ \Vert u\Vert _{H^{3/2}_{00}(I)}^2&:= \Vert u\Vert _{L^2(I)}^2 + \Vert u'\Vert _{H^{1/2}_{00}(I)}^2. \\ \end{aligned}$$

The corresponding spaces $H^{s}(\gamma )$ and ${\varvec{H}}^{s}(\gamma )$ are defined on an element edge $\gamma \in \{ \gamma _1, \gamma _2, \gamma _3 \}$ in a similar fashion.

For a function f defined on $\partial K$, we denote $f_i = f|_{\gamma _i}$ where the edge $\gamma _i$ is parameterized by arc length in the tangential direction. The following norm equivalence [15] holds on $H^{1/2}(\partial K)$, the space traces of $H^1(K)$ functions,

$$\begin{aligned} \Vert f\Vert _{H^{1/2}(\partial K)}^2&:= \inf _{ \begin{array}{c} w \in H^1(K) \\ w|_{\partial K} = f \end{array}} \Vert w\Vert _{H^1(K)}^2 \approx \Vert f\Vert _{L^2(\partial K)}^2 + |f|_{H^{1/2}(\partial K)}^2 \end{aligned}$$

where

$$\begin{aligned} |f|_{H^{1/2}(\partial K)}^2&= \sum _{i=1}^{3} \left( |f_i|_{H^{1/2}(\gamma _i)}^2 + \int _{0}^{\delta } \frac{ | f_i(|\gamma _i|-t) - f_{i+1}(t)|^2 }{t} \ dt \right) , \end{aligned}$$

and $\delta = \min _{i\in \{ 1,2,3 \} }(|\gamma _i|)$. We define ${\varvec{H}}^{1/2}(\partial K)$ analogously.

In order to show the existence of functions satisfying (3.7) and hence complete the proof of (4.2), we need the following consequence of [3, Corollary 2.3]:

Corollary A.1

Let $f,g : \partial K \rightarrow {\mathbb {R}}$ be such that $f_i \in {\mathbb {P}}_p(\gamma _i)$ and $g_i \in {\mathbb {P}}_{p-1}(\gamma _i)$ for $p \ge 1$. Assume further that f and g satisfy the compatibility conditions at the vertices:

$$\begin{aligned} f_{i+1}(n_{i+2})&= f_{i}(n_{i+2}) \\ \partial _t f_{i+1}(n_{i+2})&= -\cos \theta _{i+2} \partial _t f_{i}(n_{i+2}) - \sin \theta _{i+2} g_i(n_{i+2}) \\ g_{i+1}(n_{i+2})&= \sin \theta _{i+2} \partial _t f_i(n_{i+2}) - \cos \theta _{i+2} g_i(n_{i+2}) \\ \partial _t g_{i+1}(n_{i+2})&= \cot \theta _{i+2} \partial _{tt} f_{i+1}(n_{i+2}) -\cot \theta _{i+2} \partial _{tt} f_i(n_{i+2}) - \partial _t g_i(n_{i+2}). \end{aligned}$$

for $i \in \{1,2,3\}$. Then, there exists a polynomial $U \in {\mathbb {P}}_p(K)$ such that

• 1. U has trace (f, g): $U |_{\gamma _i} = f_i$, $\partial _n U |_{\gamma _i} = g_i$, and

• 2. U depends continuously on f and g:

  $$\begin{aligned} \begin{aligned}&h_K^{-1} \Vert U\Vert _{L^2(K)} + |U|_{H^1(K)} + h_K |U|_{H^2(K)} \\&\qquad \le C \left( h_K^{-1/2} \Vert f\Vert _{L^2(\partial K)} + h_{K}^{1/2} \Vert {\varvec{\sigma }}(f,g) \Vert _{L^2(\partial K)} + h_K|{\varvec{\sigma }}(f,g)|_{H^{1/2}(\partial K)} \right) \end{aligned} \end{aligned}$$

  (A.1)

  where ${\varvec{\sigma }}(f,g) = (\partial _t f) {\varvec{n}} - g {\varvec{t}},$ and C is independent of p and $h_K$.

Proof

The existence of a polynomial U satisfying the required conditions is given by [3, Corollary 2.3] where the estimate (A.1) is also established in the case $h_K = 1$. Employing the usual scaling argument gives the estimate (A.1). $\square$

B Auxilliary univariate polynomials

The following lemma defines some univariate functions which are later used to construct $C^2$ vertex functions satisfying the condition (3.7).

Lemma B.1

Let $m,p \in {\mathbb {N}}$, $\beta > -1$. Define $\gamma := 2m-1+\beta$ and

$$\begin{aligned} \psi _{p,m,\beta }(x) := \left( \frac{1-x}{2} \right) \left( \frac{1 - x^2}{2} \right) ^{m-1} P_{p}^{(\gamma ,\gamma )}(x). \end{aligned}$$

Then, $\psi _{p,m,\beta }^{(j)}(\pm 1) = 0$ for $j = 0, 1, \ldots , m-2$, $\psi _{p,m,\beta }^{(m-1)}(1) = 0$, and $\psi _{p,m,\beta }^{(m-1)}(-1) \ne 0$. Moreover,

$$\begin{aligned} \frac{ 1 }{ \psi _{p,m,\beta }^{(m-1)}(-1)^2 } \int _{-1}^{1} \left( 1-x^2\right) ^{\beta } \psi _{p,m,\beta }(x)^2 \ dx&\le \frac{ 2^{2m-1-2\beta } \varGamma (\gamma +1) \varGamma (\gamma ) p! }{ (m-1)! (m-1)! \varGamma (p+2\gamma +1) } \\&\sim p^{-2(2m-1+\beta )}. \end{aligned}$$

Proof

Let $\psi _p := \psi _{p,m,\beta }$. We first bound

$$\begin{aligned} 4^{\beta } \int _{-1}^{1} \left( \frac{1-x^2}{4}\right) ^{\beta } \psi _p(x)^2 \ dx&= \int _{-1}^{1} \left( \frac{1-x}{2} \right) ^{\gamma + 1} \left( \frac{1+x}{2} \right) ^{\gamma - 1} P_p^{(\gamma , \gamma )}(x)^2 \ dx \\&\le \int _{-1}^{1} \left( \frac{1-x}{2} \right) ^{\gamma } \left( \frac{1-x}{2} \right) ^{\gamma -1} P_p^{(\gamma , \gamma )}(x)^2 \ dx \\&=: Q_p. \end{aligned}$$

Now, using the identity [23, p. 4]:

$$\begin{aligned}&(2p + 2\gamma - 1) P_{p}^{(\gamma ,\gamma - 1)} = (p+2\gamma )P_p^{(\gamma ,\gamma )} + (p+\gamma )P_{p-1}^{(\gamma ,\gamma )} \end{aligned}$$

we get

$$\begin{aligned} P_p^{(\gamma ,\gamma )} = \mu _p P_p^{(\gamma ,\gamma -1)} - \nu _p P_{p-1}^{(\gamma ,\gamma )} \end{aligned}$$

with

$$\begin{aligned} \mu _p = \frac{2(p+\gamma )}{p+2\gamma }, \quad \nu _p = \frac{p+\gamma }{p+2\gamma }. \end{aligned}$$

Using this identity and orthogonality, there holds

$$\begin{aligned} Q_p&= \nu _p^2 \int _{-1}^{1} \left( \frac{1-x}{2} \right) ^{\gamma } \left( \frac{1+x}{2} \right) ^{\gamma -1} P_{p-1}^{(\gamma ,\gamma )}(x)^2 \ dx \nonumber \\&\qquad + \mu _p^2 \int _{-1}^{1} \left( \frac{1-x}{2} \right) ^{\gamma } \left( \frac{1+x}{2} \right) ^{\gamma -1} P_p^{(\gamma ,\gamma -1)}(x)^2 \ dx \nonumber \\&=: \nu _p^2 Q_{p-1} + \mu _p^2 T_p. \end{aligned}$$

(B.1)

First note that

$$\begin{aligned} T_p = \frac{1}{p+\gamma } \frac{\varGamma (p+\gamma +1) \varGamma (p+\gamma ) }{ \varGamma (p+2\gamma ) p! } = \frac{ \varGamma (p+\gamma )^2 }{ \varGamma (p+2\gamma ) p! }. \end{aligned}$$

We now claim

$$\begin{aligned} Q_p \le \frac{\varGamma (p+\gamma + 1)^2 }{ \varGamma (p+2\gamma +1)! p! } \cdot \frac{2}{\gamma }. \end{aligned}$$

(B.2)

For $p = 0$, we have

$$\begin{aligned} Q_0 = \int _{-1}^{1} \left( \frac{1-x}{2} \right) ^{\gamma } \left( \frac{1+x}{2} \right) ^{\gamma -1} \ dx = \frac{ \varGamma (\gamma )^2 }{ \varGamma (2\gamma ) p! }, \end{aligned}$$

and (B.2) holds as an equality in the case $p = 0$.

Now suppose the result holds true for $p = q-1$. Then, by (B.1),

$$\begin{aligned} (q + 2\gamma )^2 Q_q = (q + \gamma )^2 Q_{q-1} + 4(q+\gamma )^2 T_q. \end{aligned}$$

Direct computation gives

$$\begin{aligned} Q_q&\le \frac{(q+\gamma )^2}{ (q+2\gamma )^2 } \cdot \frac{\varGamma (q+\gamma )^2}{ \varGamma (q+2\gamma ) q! } \cdot \frac{2}{\gamma } \left( q + 2\gamma \right) = \frac{ \varGamma (q+\gamma +1)^2 }{ \varGamma (q+2\gamma +1) q! } \cdot \frac{2}{\gamma } \end{aligned}$$

and (B.2) follows by induction.

Again using identities from [23, p. 4]

$$\begin{aligned} \psi _p^{(m-1)}(-1) = \frac{(m-1)!}{2^{m-1}} P_p^{(\gamma ,\gamma )}(-1) = \frac{(-1)^p (m-1)! \varGamma (p+\gamma +1)}{2^{m-1} \varGamma (\gamma +1) p! }. \end{aligned}$$

Thus,

$$\begin{aligned} \frac{1}{ \psi _p^{(m-1)}(-1)^2 } \int _{-1}^{1} (1-x^2)^{\beta } \psi _p(x)^2 \ dx&= 2^{2m-1-2\beta } \frac{\varGamma (\gamma +1) \varGamma (\gamma ) p! }{ (m-1)! (m-1)! \varGamma (p+2\gamma +1) }. \end{aligned}$$

By Stirling’s formula, $\frac{p!}{\varGamma (p+2\gamma +1)} \sim p^{-2(2m-1+\beta )}$. $\square$

C Traces of $C^2$ vertex functions

Thanks to Corollary A.1, we only need to specify the boundary values of $\varPhi _{n_i}^{\alpha }$ on each $K \in {\mathscr {T}}$. We summarize them below: First, we label a general element K as in Fig. 1. Let $F : [-1,1] \rightarrow {\mathbb {R}}$ and $G : [-1,1] \rightarrow {\mathbb {R}}$ be given by

$$\begin{aligned} F(t) = \frac{\psi _{p-5,3,0}(t)}{\psi ''_{p-5,3,0}(-1)} \qquad \text {and} \qquad G(t) = \frac{\psi _{p-4,2,0}(t)}{\psi _{p-4,2,0}'(-1)}, \end{aligned}$$

where $\psi _{p,m,\beta }$ is given by Lemma B.1. Now, we define the traces f and g as follows: $\alpha = (2,0)$:

$$\begin{aligned} \begin{aligned} f_{i}(t)&= 0 \\ g_{i}(t)&= 0 \\ f_{i+1}(t)&= \frac{1}{4} |\gamma _{i+1}|^2 \left( t_{i+1}^x\right) ^2 F\left( 1- \frac{2t }{|\gamma _{i+1}|} \right) \\ g_{i+1}(t)&= -\frac{1}{2} |\gamma _{i+1}| t_{i+1}^x n_{i+1}^x G\left( 1-\frac{ 2t }{|\gamma _{i+1}|} \right) \\ f_{i+2}(t)&= \frac{1}{4} |\gamma _{i+2}|^2 \left( t_{i+2}^x\right) ^2 F\left( \frac{ 2t }{|\gamma _{i+2}|} - 1 \right) \\ g_{i+2}(t)&= \frac{1}{2} |\gamma _{i+2}| t_{i+2}^x n_{i+2}^x G\left( \frac{ 2t }{|\gamma _{i+2}|} - 1 \right) \end{aligned} \end{aligned}$$

(C.1)

$\alpha = (1,1)$:

$$\begin{aligned}&\begin{aligned} f_i(t)&= 0 \\ g_i(t)&= 0 \\ f_{i+1}(t)&= \frac{1}{2} |\gamma _{i+1}|^2 t_{i+1}^x t_{i+1}^y F\left( 1- \frac{2t }{|\gamma _{i+1}|} \right) \\ g_{i+1}(t)&= -\frac{1}{2} |\gamma _{i+1}| \left( t_{i+1}^x n_{i+1}^y + t_{i+1}^y n_{i+1}^x\right) G\left( 1- \frac{2t }{|\gamma _{i+1}|} \right) . \\ f_{i+2}(t)&= \frac{1}{2} |\gamma _{i+2}|^2 t_{i+2}^x t_{i+2}^y F\left( \frac{ 2t }{|\gamma _{i+2}|} - 1 \right) \\ g_{i+2}(t)&= \frac{1}{2} |\gamma _{i+2}| \left( t_{i+2}^x n_{i+2}^y + t_{i+2}^y n_{i+2}^x\right) G\left( \frac{ 2t }{|\gamma _{i+2}|} - 1 \right) . \\ \end{aligned} \end{aligned}$$

(C.2)

$\alpha = (0,2)$:

$$\begin{aligned}&\begin{aligned} f_i(t)&= 0 \\ g_i(t)&= 0 \\ f_{i+1}(t)&= \frac{1}{4} |\gamma _{i+1}|^2 (t_{i+1}^y)^2 F\left( 1- \frac{2t }{|\gamma _{i+1}|} \right) \\ g_{i+1}(t)&= -\frac{1}{2} |\gamma _{i+1}| t_{i+1}^y n_{i+1}^y G\left( 1- \frac{2t }{|\gamma _{i+1}|} \right) . \\ f_{i+2}(t)&= \frac{1}{4} |\gamma _{i+2}|^2 \left( t_{i+2}^y\right) ^2 F\left( \frac{ 2t }{|\gamma _{i+2}|} - 1\right) \\ g_{i+2}(t)&= \frac{1}{2} |\gamma _{i+2}| t_{i+2}^y n_{i+2}^y G\left( \frac{ 2t }{|\gamma _{i+2}|} - 1 \right) . \end{aligned} \end{aligned}$$

(C.3)

Since F and G vanish to order 2 and 1, respectively, f and g satisfy the compatibility conditions in Corollary A.1. Let ${\bar{\varPhi }}_{n_i}^{\alpha }$ be the extension of f and g given by Corollary A.1. First note that we have the desired interpolation properties:

$$\begin{aligned} D^{\beta } {\bar{\varPhi }}_{n_i}^{\alpha }(n_j) = \delta _{\alpha \beta } \delta _{ij} \end{aligned}$$

for all $|\beta | \le 2$ and $n_j \in {\mathscr {N}}$. Moreover, for $m \in \{0,1,2\}$ there holds,

$$\begin{aligned} h_K^{m-1} |{\bar{\varPhi }}_{n_i}^{\alpha }|_{H^m(K)}&\le C \sum _{i=2}^{3} \bigg \{ h_{K}^{-1/2} \Vert f_i\Vert _{L^2(\gamma _i)} + h_{K}^{1/2} \left( |f_i|_{H^1(\gamma _i)} + \Vert g_i\Vert _{L^2(\gamma _i)} \right) \\&\quad + h_K \left( \left| f_i'\right| _{H^{1/2}_{00}(\gamma _i)} + |g_i|_{H^{1/2}_{00}(\gamma _i)} \right) \bigg \} \\&\le C h_K^2 \left( \Vert F\Vert _{H^{3/2}_{00}(-1,1)} + \Vert G\Vert _{H^{1/2}_{00}(-1,1)} \right) . \end{aligned}$$

Now, thanks to inverse estimates [8] and Lemma B.1, we have

$$\begin{aligned} \Vert F\Vert _{H^k(-1,1)} + \Vert G\Vert _{H^{k-1}(-1,1)} \le C p^{2k} \left( \Vert F\Vert _{L^2(-1,1)} + p^{-2}\Vert G\Vert _{L^{2}(-1,1)} \right) \le C p^{-5+2k}. \end{aligned}$$

By interpolation,

$$\begin{aligned} |{\bar{\varPhi }}_{n_i}^{\alpha }|_{H^m(K)}&\le C h_K^{3-m} \left( \Vert F\Vert _{H^{3/2}_{00}(-1,1)} + \Vert G\Vert _{H^{1/2}_{00}(-1,1)} \right) \le C \frac{h_K^{3-m}}{p^2}. \end{aligned}$$

(C.4)