Constructive ordinary Sobolev polynomial approximation on triangles

On $I=(-1,1)$ define $S_{n}f(x)=f(-1)+(x+1)f^{\mathrm{\prime }}(-1)+{\int }_{-1}^x(x-t){\mathrm{\Pi }}_{n-2}{f}^{\mathrm{\prime }\mathrm{\prime }}(t)dt,$ where ${\mathrm{\Pi }}_{n-2}$ is the ordinary Legendre $L^2$ projection. Here $p\ge 2r-4$ implies $n\ge r-2\ge 3$. For $f\in H^2(I)$, the endpoint values are those of its absolutely continuous representatives from [F4]. The polynomial $S_{n}f$ has degree at most $n$ and matches $f,f^{\mathrm{\prime }}$ at $-1$. Its second derivative is an $L^2$ contraction; integrating twice and using the one-dimensional endpoint estimate gives $\mathrm{\parallel }{S}_{n}f{\mathrm{\parallel }}_{H^2(I)}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{H^2(I)}$ with absolute $C$. For every integer $s\ge 3$ with $n\ge s-1$, the error $e=f-S_{n}f$ has $e(-1)=e^{\mathrm{\prime }}(-1)=0$ and $e^{\mathrm{\prime }\mathrm{\prime }}=f^{\mathrm{\prime }\mathrm{\prime }}-{\mathrm{\Pi }}_{n-2}{f}^{\mathrm{\prime }\mathrm{\prime }}$. Apply the Legendre tail estimate to $f^{\mathrm{\prime }\mathrm{\prime }}$ with derivative order $s-2$ and projection degree $n-2$; the factorial ratio is ${\rho }_{n,s}={\left(\frac{(n-s+1)!}{(n+s-3)!}\right)}^{1\mathrm{/}2}\mathrm{.}$ Two integrations of $e^{\mathrm{\prime }\mathrm{\prime }}$, with Cauchy–Schwarz [F5], give $\mathrm{\parallel }e{\mathrm{\parallel }}_{H^2(I)}\le C{\rho }_{n,s}\mathrm{\parallel }{f}^{(s)}{\mathrm{\parallel }}_2$. The same inequality holds for each derivative of order at most two.

Put ${\mathbb{Q}}_n=\mathrm{span}\{x^i{y}^j:0\le i,j\le n\}$, apply $S_n$ in each coordinate on $I^2$, and write $S_n^x,S_n^y$. For smooth $U$, $U-S_n^x{S}_n^{y}U=(1-S_n^x)U+S_n^x(1-S_n^y)U\mathrm{.}$ Let $a+b\le 2$. For the first term choose $s_b=r-\max (1,b)$, so $s_b\in \{r-1,r-2\}$, $s_b\ge 3$, $n\ge s_b-1$, and $s_b+b\le r$. Step 1.1 on the $x$ sections of $D_y^{b}U$ bounds $\mathrm{\parallel }{D}_x^a{D}_y^b(1-S_n^x)U{\mathrm{\parallel }}_2$ by $C{\rho }_{n,s_b}\mathrm{\parallel }{D}_x^{s_b}{D}_y^{b}U{\mathrm{\parallel }}_2$. For the second term, the uniform $H_x^2$ bound in step 1.1 bounds its $D_x^a{D}_y^b$ norm by $C{\sum }_{j=0}^2\mathrm{\parallel }{D}_x^j{D}_y^b(1-S_n^y)U{\mathrm{\parallel }}_2$. For each such $j$, choose $t_j=r-\max (1,j)\in \{r-1,r-2\}$; then $t_j\ge 3$, $n\ge t_j-1$, and $j+t_j\le r$. Apply the $H_y^2$ error estimate of step 1.1 to $D_x^{j}U$; it bounds every $y$ derivative of order $b\le 2$ in this sum by $C{\rho }_{n,t_j}\mathrm{\parallel }{D}_x^j{D}_y^{t_j}U{\mathrm{\parallel }}_2$. Because ${\rho }_{n,r-1}\le {\rho }_{n,r-2}={\eta }_{n,r}$, all terms are bounded by $C{\eta }_{n,r}\mathrm{\parallel }U{\mathrm{\parallel }}_{H^r(I^2)}$. Fubini [F5] and addition over $a+b\le 2$ give $\mathrm{\parallel }U-S_n^x{S}_n^{y}U{\mathrm{\parallel }}_{H^2(I^2)}\le C{\eta }_{n,r}\mathrm{\parallel }U{\mathrm{\parallel }}_{H^r(I^2)}\mathrm{.}$ For the inputs used below, namely restrictions of $H^r({\mathbb{R}}^2)$ extensions, choose smooth whole-space approximants by [F4]. The estimate on their differences makes $S_n^x{S}_n^{y}U$ a well-defined $H^2$ limit. Every smooth approximant has image in ${\mathbb{Q}}_n$, so closedness [F7] gives $S_n^x{S}_n^{y}U\in {\mathbb{Q}}_n$ and the same bound for the limit.

Let $b=B_{r-1}u$ and $w=u-b$. By [F3], $\mathrm{\parallel }w{\mathrm{\parallel }}_{H^r(T)}\le B_r\mathrm{\mid }u{\mathrm{\mid }}_{H^r(T)}$. Set $U=(E_{r}w){\mathrm{\mid }}_{I^2}$ and define $A_{p,r}u=b+(S_n^x{S}_n^{y}U){\mathrm{\mid }}_T\mathrm{.}$ This map is linear. The first summand has degree at most $r-1\le p$; the second lies in ${\mathbb{Q}}_n\subseteq {\mathbb{P}}_{2n}\subseteq {\mathbb{P}}_p$. Since $E_{r}w=w$ on $T$, step 2.1 and [F2]–[F3] yield $\mathrm{\parallel }u-A_{p,r}u{\mathrm{\parallel }}_{H^2(T)}\le C{M}_r{B}_r{\eta }_{n,r}\mathrm{\mid }u{\mathrm{\mid }}_{H^r(T)}\mathrm{.}$ This identifies the constant as the absolute tensor constant times the fixed extension and moment bounds; no growth in $r$ is asserted.

Transport step 3.1 to a shape-regular affine triangle $K$. For $e=u-A_{K,p,r}u$ and $\widehat{e}=e\circ F_K$, two-dimensional affine pullback gives $\mathrm{\mid }\widehat{e}{\mathrm{\mid }}_{H^j(T)}{\asymp }_{j,\gamma }{h}_K^{j-1}\mathrm{\mid }e{\mathrm{\mid }}_{H^j(K)}$ for $j=0,1,2$, and $\mathrm{\mid }\widehat{u}{\mathrm{\mid }}_{H^r(T)}\le C_{r,\gamma }{h}_K^{r-1}\mathrm{\mid }u{\mathrm{\mid }}_{H^r(K)}$ by [F6]. This proves the displayed full scaled estimate. Finally, ${\eta }_{n,r}^{-2}$ is a product of $2(r-4)$ consecutive integers beginning with $n-r+4$. If $p\ge 4(r-2)$ and $n=\lfloor p\mathrm{/}2\rfloor$, then $n-r+4\ge p\mathrm{/}4$; taking square roots yields ${\eta }_{n,r}\le (4\mathrm{/}p{)}^{r-4}$.