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Let $\Omega$ be a convex polygonally bounded Lipschitz domain in $\mathbb R^2$, let $f \in L^2(\Omega)$.

Then the solution of the Dirichlet problem $\Delta u = f$ in $\Omega$, $\operatorname{trace} u = 0$ on $\partial\Omega$ has a unique solution in $H^2$ and is well-posed, i.e. for some constant $C$ we have $\|u\|_{H^2} \leq C \|f\|_{L^2}$.

For some finite element approximation $u_h$, say, with nodal elements on a uniform grid, we have the error estimate

$\| u - u_h \|_{H^1} \leq C h \| u \|_{H^2}$

It seems (maybe I am wrong with that) that people usually do not use the obvious error estimate

$\| u - u_h \|_{H^1} \leq C h \| f \|_{L^2}$

which we can obtain by combination of the above two inequalities. Instead, a posteriori error estimaters are developed in various forms. The only objection I can imagine against the above equation is that the constant $C$ might in practice be too pessimistic or not reliably estimatable.

hardmath
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shuhalo
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2 Answers2

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The reason why people prefer to use the first estimate, in my opinion, is that the first one arises naturally from the Galerkin orthogonality of the FEM, interpolation approximation property, and most importantly the coercivity of the bilinear form(for Poisson equation's boundary value problem, it is equivalent with the Poincaré/Friedrichs inequality for $H^1_0$ functions): $$ \begin{aligned} \|u - u_h\|^2_{H^1(\Omega)} &\leq c_1 \| \nabla (u - u_h) \|^2_{L^2(\Omega)} \\ \| \nabla (u - u_h) \|^2_{L^2(\Omega)} &= \int_{\Omega} \nabla(u- u_h)\cdot \nabla(u- u_h) \\ &= \int_{\Omega} \nabla(u- u_h)\cdot \nabla(u- \mathcal{I}u) \\ &\leq \| \nabla (u - u_h) \|_{L^2(\Omega)} \| \nabla (u - \mathcal{I}u) \|_{L^2(\Omega)} \\ \Rightarrow \| \nabla (u - u_h) \|_{L^2(\Omega)} &\leq \| \nabla (u - \mathcal{I}u) \|_{L^2(\Omega)} \leq c_2 h\| u\|_{H^2(\Omega)} \end{aligned} $$ where $c_1$ depends on the constant in the Poincaré/Friedrichs inequality for $H^1_0$ functions, $\mathcal{I}u$ is the interpolation of $u$ in the finite element space, and $c_2$ depends on the minimum angles of the mesh.

While the elliptic regularity estimate $\|u \|_{H^2(\Omega)}\leq c\|f\|_{L^2(\Omega)}$ is solely on the PDE level, has nothing to do with the approximation, plus above argument holds even when $f\in H^{-1}$ is a distribution.

Now move on to the reason why a posteriori error estimates are widely used, is mainly because:

  • It is computable, there is no generic constant in the expression of the estimates.

  • The estimator has its local form, which could be the local error indicator using in the adaptive mesh refining procedure. Therefore, the problem with singularities or really "bad" geometries could be dealt with.

Both of the a priori type estimates you listed are valid, they provide us the information of the orders of convergence, however none of them could be a local error indicator just for one triangle/tetrahedron, because neither of them are computable due to the constant, nor are them defined locally.

EDIT: For more of a general view of the FEM for elliptic PDEs, I highly recommend reading Chapter 0 in Brenner and Scott's book:The Mathematical Theory of Finite Element Methods, which consists only 20 pages and covers briefly almost every aspect of finite element methods, from the Galerkin formulation from the PDE, to the motivation why we would like to use adaptive FEM to tackle some problem. Hope this would help you more.

Shuhao Cao
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Your estimate is too pessimistic on two fronts. You've identified the first one already ($C$ now not only includes the interpolation constant but also the stability constant). The second one is that the error estimate really reads $$ \|\nabla e\|_{L_2} \le C \;h\; |u|_{H^2}. $$ Note that the right hand side has the $H^2$ seminorm, not norm. Of course you can bound the rhs by the full norm, but you lose again this way.

Wolfgang Bangerth
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