4 question made more precise

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.

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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 $\int_\Omega \nabla u_h\nabla v dA=\int_\Omega f\nabla v fv dA$. 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 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.

2 misprint corrected

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 $\int_\Omega \nabla u_h\nabla v dA=\int_\Omega f\nabla v dA$. 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 but I would be also grateful for counterexamples for other norms.

[Brass] 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.

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