Finite Element Methods: A Priori, A Posteriori, and Adaptivity¶

Job: develop the Galerkin / Céa, the a priori (Bramble–Hilbert / Scott–Zhang), the a posteriori ($\eta_K$ / Dörfler adaptivity), and the mixed / Stokes (inf-sup / Babuška–Brezzi) — with the Poisson FEM as the worked example.

Core¶

The Galerkin framework and Céa¶

Galerkin / Céa / $V_h$	Description
Galerkin	$a(u_h,v_h)=\ell(v_h)$ $\forall v_h\in V_h$ (the weak form, $V_h\subset H^1_0$)
Céa lemma	$\\|u-u_h\\|_{H^1}\le C\inf_{v\in V_h}\\|u-v\\|_{H^1}$ (the coercive)
$V_h$	the $P_k$ / $Q_k$ finite element space (the mesh, the degree)

The a priori error estimates¶

A priori / Bramble–Hilbert / $P_k$	Description
A priori	$\\|u-u_h\\|_{H^1}\le Ch^k\\|u\\|_{H^{k+1}}$ ($P_k$ elements, $u\in H^{k+1}$)
Bramble–Hilbert	the interpolation error $\|u-I_h u\|_{H^1}\le Ch^k
Scott–Zhang	the $H^1$-stable interpolation (the $I_h:H^1\to V_h$, the non-nodal)

The a posteriori and the adaptivity¶

A posteriori / $\eta_K$ / Dörfler	Description
A posteriori	$\eta_K$ the element indicator, $\\|u-u_h\\|\le C(\sum\eta_K^2)^{1/2}$ (the reliability)
Dörfler	the bulk chasing (the marking the $\eta_K$ large, the refinement, the convergence)
Adaptivity	the AFEM (the solve $\to$ estimate $\to$ mark $\to$ refine, the convergence)

The mixed / the Stokes / the inf-sup¶

Mixed / inf-sup / Stokes / Taylor–Hood	Description
Mixed FEM	$V_h\times Q_h$, the inf-sup / the Babuška–Brezzi (the stable)
Stokes	the velocity-pressure, the Taylor–Hood ($P_2$-$P_1$), the inf-sup

See it¶

Poisson FEM a priori $H^1$ / a posteriori $\eta_K$ and the adaptivity. The Poisson $-\Delta u=f$ on $\Omega$, the $V_h\subset H^1_0$ the $P_1$, the $\|u-u_h\|_{H^1}\le Ch\|u\|_{H^2}$ the a priori (the Bramble–Hilbert). The a posteriori: the $\eta_K=h_K\|f+\Delta u_h\|_{L^2(K)}+\frac12 h_K^{-1/2}\|[\partial_n u_h]\|_{L^2(\partial K)}$ the residual, the $\|u-u_h\|\le C(\sum\eta_K^2)^{1/2}$ the reliability. The adaptivity: the Dörfler the bulk, the marking the $\eta_K$ large, the refinement, the convergence. The key point: the Poisson the FEM a priori / a posteriori, the adaptivity the Dörfler.

Check¶

Show that the Céa lemma gives the a priori error, and explain the Bramble–Hilbert.

The Céa: the $a(u-u_h,v_h)=0$ (the Galerkin orthogonality), the coercive $a(v,v)\ge\alpha\|v\|^2$, the $\alpha\|u-u_h\|^2\le a(u-u_h,u-u_h)=a(u-u_h,u-v_h)\le M\|u-u_h\|\|u-v_h\|$, the $\|u-u_h\|\le(M/\alpha)\inf\|u-v_h\|$ (the Céa). The Bramble–Hilbert: the $\|u-I_h u\|_{H^1}\le Ch^k|u|_{H^{k+1}}$ (the interpolation, the $P_k$, the $H^{k+1}$). So the Céa + Bramble–Hilbert the a priori.

Explain the a posteriori $\eta_K$ and the Dörfler bulk marking, and the AFEM convergence.

The $\eta_K$: the $h_K\|f+\Delta u_h\|_{L^2(K)}$ the interior residual, the $\frac12 h_K^{-1/2}\|[\partial_n u_h]\|_{L^2(\partial K)}$ the edge jump, the $\|u-u_h\|\le C(\sum\eta_K^2)^{1/2}$ the reliability (the a posteriori). The Dörfler: the mark the $\eta_K$ the bulk ($\sum_{\mathrm{marked}}\eta_K^2\ge\theta\sum_{\mathrm{all}}\eta_K^2$, the $\theta$ the bulk parameter). The AFEM: the solve $\to$ estimate $\to$ mark $\to$ refine, the convergence (the Cascon–Kreuzer–Nochetto–Siebert the contraction). So the $\eta_K$ the reliability, the Dörfler the bulk, the AFEM the convergence.

Pitfalls¶

The Céa the coercive (the $a(v,v)\ge\alpha\|v\|^2$, the PDE / elliptic) — the Céa the coercive, the non-coercive the Banach–Nečas–Babuška.
The Bramble–Hilbert the $H^{k+1}$ (the $u\in H^{k+1}$, the $P_k$) — the $H^{k+1}$ the regularity, the $P_k$ the degree.
The a posteriori the reliability (the $\eta_K$, the $\|u-u_h\|\le C(\sum\eta_K^2)^{1/2}$) — the reliability the upper, the efficiency the lower.
The mixed / Stokes the inf-sup (the Babuška–Brezzi, the Taylor–Hood) — the inf-sup the stable, the $P_2$-$P_1$ the Taylor–Hood.

Refs¶

Brenner–Scott, The Mathematical Theory of Finite Element Methods (2008); the Céa / the Bramble–Hilbert
Babuška–Rheinboldt, "Error estimates for adaptive FEM" (1978); the a posteriori
Dörfler, "A convergent adaptive algorithm for Poisson" (1996); the bulk marking
Brezzi–Fortin, Mixed and Hybrid Finite Element Methods (1991); the inf-sup

Next¶

NA index · Foundations · Geometric Integration · Iterative & Multigrid · High-Dimensional & UQ · PDE / elliptic

Galerkin / Céa / \(V_h\)	Description
Galerkin	\(a(u_h,v_h)=\ell(v_h)\) \(\forall v_h\in V_h\) (the weak form, \(V_h\subset H^1_0\))
Céa lemma	\(\\|u-u_h\\|_{H^1}\le C\inf_{v\in V_h}\\|u-v\\|_{H^1}\) (the coercive)
\(V_h\)	the \(P_k\) / \(Q_k\) finite element space (the mesh, the degree)