There are many known results proving convergence of finite element method for elliptic problems under certain assumptions on underlying mesh [e.g., Braess,2007]. Which of these common assumptions are indeed necessary? Can anyone recommend any exact reference to an example of a sequence of triangulations on which the finite element solutions do NOT converge to the real solution?
Let us give a model problem which is of particular interest. Let $\Omega$ be a convex polygon in the plane and $f:R^2\to R$ be a $C^2$-function. Let $u:\Omega\to R$ be a $C^2$-function such that $\Delta u=f$ inside $\Omega$, $u=0$ in $\partial\Omega$. Let $T_h$ be a sequence of triangulations of $\Omega$ such that maximal edge length of $T_h$ approaches zero. Let $u_h:\Omega\to R$ be a continuous piecewise-linear function on $T_h$ such that $u_h=0$ in $\partial\Omega$ and for any continuous piecewise-linear function $v:\Omega\to R$ on $T_h$ we have $\int_\Omega \nabla u_h\nabla v dA=\int_\Omega fv dA$. Suppose that there is a constant $\mathrm{const}>0$ (not depending on $h$) such that:
(1) the ratio of any two edges of $T_h$ is greater than $\mathrm{const}$;
(2) the angles of any triangle of $T_h$ is greater than $\mathrm{const}$.
Then $\max_{\Omega}|u_h-u|\to 0$ as $h\to 0$. [Ciarlet, Theorem 3.3.7]
Which of the assumptions (1) and (2) cannot be dropped here? What are counterexamples? I am mostly interested in uniform ($L_\infty$) convergence of values at the vertices but I would be also grateful for counterexamples for other norms.
[Braess] D. Braess, Finite elements. Theory, fast solvers, and applications in elasticity theory, transl. by L.L. Schumaker, Cambridge Univ. Press, 2007.
[Ciarlet] P.G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, 1978, 530 p.
Best Answer
The maximum and minimum angle conditions for meshes are needed to prove various bounds on the error of interpolation. In other words, the solution of the PDE is a secondary concern; what goes wrong is that one cannot control the interpolation error.
Of the two, the minimum angle condition is less restrictive. What one may observe is that deteriorating conditioning of the linear system to solve for the unknown coefficients of the approximant as the minimum angle condition is violated.
There's a famous paper by Babuska and Aziz in a 1976 SIAM J. Numerical Analysis v. 13, no. 2, On the angle condition in the finite element method. This also has a nice counter example showing why the interpolation error cannot be bounded unless the maximum angle is bounded away from $\pi$.
Please see On angle conditions in the finite element method for a survey and discussion of these ideas.