Exponential hp energy convergence for weighted-analytic polygon corners

Fix $0<a<{\min }_i(1-{\beta }_i)$ as in [F1]. Call a cell a cap when its closure contains a corner. Since the corner set is finite and distinct corners have positive separation, for all sufficiently large $k$ each cap contains at most one corner; its diameter is at most $C_1{\sigma }^k$. On a noncap cell $T$ with $d_T=\mathrm{dist}(T,\mathcal{C})<R$, the mesh condition gives $\sigma \le h_T\mathrm{/}{d}_T\le {\sigma }^{-1}$, so [F1] gives uniform scaled analytic derivative bounds after subtracting a suitable constant. If $d_T\ge R$, the compact-away-from-corners bounds in [F1], together with bounded $h_T$ and the fixed $r_T=R$, give the same conclusion after enlarging constants. Thus each noncap reference pullback has derivative bounds $C{r}_T^a{A}_1^{\mathrm{\mid }\alpha \mathrm{\mid }}\mathrm{\mid }\alpha \mathrm{\mid }!$, where $r_T=\min (R,d_T)$.

For every mesh edge incident to a polygon corner, prescribe the linear interpolant of the two endpoint values of the continuous representative of $u$. All cells sharing such an edge are caps. On every other edge $E$, prescribe the single common endpoint-preserving derivative-projection polynomial $g_E$ from [F2], in one globally fixed edge orientation. The edge lies a distance comparable to its length from any nearby corner: for an edge opposite a cap corner this follows from shape regularity and the altitude bound, and for a noncap neighbor it also follows from the mesh ratio condition. Thus [F1]–[F2] give, on each noncap edge, $\mathrm{\parallel }u-g_E{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(E)}+{\mathrm{\ell }}_E^{1\mathrm{/}2}\mathrm{\parallel }{\mathrm{\partial }}_{\tau }(u-g_E){\mathrm{\parallel }}_{L^2(E)}\le C{r}_T^a{e}^{-b{p}_k}$ for each adjoining noncap cell, with comparable scales on both sides. Here the sup bound follows from the endpoint identity and $\mathrm{\parallel }e{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{e}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^1}\le \mathrm{\parallel }{e}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2}$ for the endpoint-zero reference error; [F6] justifies the last inequality. On an edge incident to a corner, both adjacent cells are caps, its prescribed linear polynomial shares the exact continuous vertex values, and no tangential regularity of $u$ at the corner is used. Every edge polynomial takes the value of $u$ at both endpoints. If $E\subset {\mathrm{\Gamma }}_D$, these endpoint values and the entire prescribed edge polynomial are zero.

Let $T$ be a noncap triangle and choose $q_T\in {\mathbb{P}}_{p_k}(T)$ by transporting the analytic approximation [F2] for the scaled pullback, then adding back its subtracted constant. Thus $\mathrm{\parallel }u-q_T{\mathrm{\parallel }}_{\mathrm{\infty }}+h_T\mathrm{\parallel }\mathrm{\nabla }(u-q_T){\mathrm{\parallel }}_{\mathrm{\infty }}\le C{r}_T^a{e}^{-b{p}_k}$. Correct its three vertex values by an affine polynomial to obtain ${\tilde{q}}_T$ matching the prescribed $g_E$ at all vertices; the correction has $H^1$ seminorm at most $C{r}_T^a{e}^{-b{p}_k}$. On each edge, $g_E-{\tilde{q}}_T{\mathrm{\mid }}_E$ has zero endpoints, degree at most $p_k$, and strong edge derivative norm at most $C{r}_T^a{e}^{-b{p}_k}$. Lift these three residuals separately by [F3]. Their sum does not change the other edge traces, so the resulting $v_T\in {\mathbb{P}}_{p_k}(T)$ has all three prescribed traces and $\mathrm{\mid }u-v_T{\mathrm{\mid }}_{H^1(T)}\le C{r}_T^a{e}^{-b{p}_k}\le C{e}^{-b{p}_k}\mathrm{.}$

Let $T$ be a cap at corner $c_i$, with diameter $h_T$; for large $k$ it contains no other corner. Start with the affine interpolant $a_T$ of the three exact vertex values. By [F1], $\mathrm{\mid }u-a_T{\mathrm{\mid }}_{H^1(T)}\le C{h}_T^a$. Its two edges incident to $c_i$ already have the prescribed linear traces. On the opposite edge $E$, every point has distance comparable to $h_T$ from $c_i$, so the gradient bound in [F1] and Cauchy–Schwarz [F6] give ${\mathrm{\ell }}_E^{1\mathrm{/}2}\mathrm{\parallel }{\mathrm{\partial }}_{\tau }u{\mathrm{\parallel }}_{L^2(E)}\le C{h}_T^a$. The interval derivative projection in [F2] is contractive, and the vertex estimate in [F1] gives the same bound for $g_E-a_T{\mathrm{\mid }}_E$. This discrepancy vanishes at both endpoints; add its lift from [F3]. The resulting polynomial has all prescribed traces and $\mathrm{\mid }u-v_T{\mathrm{\mid }}_{H^1(T)}\le C{h}_T^a\le C{\sigma }^{ak}$. No $H^2$ regularity is used on a cap.

Every pair of adjacent local polynomials has common trace $g_E$; the boundary traces vanish on ${\mathrm{\Gamma }}_D$. Thus they assemble to $v_k\in V_k$ by [F4]. Summing squared cell errors from steps 3.1 and 3.2 over at most $C_{0}k$ cells gives $\mathrm{\parallel }\mathrm{\nabla }(u-v_k){\mathrm{\parallel }}_2\le C\sqrt{k}(e^{-b{p}_k}+{\sigma }^{ak})\le C^{\mathrm{\prime }}{e}^{-\eta k}$ for some $\eta >0$, because $p_k\ge c_{0}k$ and $0<\sigma <1$. The finitely many earlier meshes are handled by $v_k=0$ and an increase of $C^{\mathrm{\prime }}$. Since $N_k\le C_N{k}^3$ by [F4], $k\ge C_N^{-1\mathrm{/}3}{N}_k^{1\mathrm{/}3}$ and therefore ${\inf }_{v\in V_k}\mathrm{\parallel }\mathrm{\nabla }(u-v){\mathrm{\parallel }}_2\le C{e}^{-b{N}_k^{1\mathrm{/}3}}$.

Under the additional Galerkin hypotheses, $u$ satisfies $a(u,v)=F(v)$ for every $v\in V$, and the Galerkin solution $u_k\in V_k$ satisfies $a(u_k,v_k)=F(v_k)$ for every $v_k\in V_k$. Thus Céa [F5] applies to this same $u$ and $u_k$. Apply it to the best-approximation estimate of step 4.1. Norm equivalence between $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_V$ and the gradient seminorm changes only the stated constant, and the actual $M\mathrm{/}\alpha$ factor is retained. This proves the energy-error conclusion.