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I have this open exercise which is difficult for me to answer completely

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I know that

  • h - density of the mesh
  • C - shape of the area, no load function

Factors affecting the left-hand side of the equation (error term)

  • the norm term $||u||_2$ affected by load function

Factors affecting the right-hand side of the equation

  • $H^2(\Omega)$ regularity i.e. $u \in H^2(\Omega)$

If the solution is regular enough, it holds

\begin{equation} ||u||_2 \leq C ||f||_0. \end{equation}

The inequality can be deduced from Poincare's inequality and Cea's lemma so all assumptions there affect the inequality

  • Cea's lemma and dependences
  • interpolation and its assumptions (regularity assumption $u \in H^2(\Omega)$ and nodal interpolation operator)
  • Scaling argument and interpolation result $\to$ Bramble-Hilbert lemma (getting rid of geometry, affine mapping from estimated functions to reference element) so constant C affected (independent of the mesh $V_h$ and $u_h$)
  • dependences and assumptions

Factors affecting Convergence Theory

  • regularity assumption (interpolation error estimate)
  • regular = bounded with load functional
  • $\Omega$ convex, Dirichlet problem is $H^2$ regular

The FEM is limited either by the regularity of the solution or by the approximation properties of the finite dimensional space. So if solution is regular, use higher order polynomials but then the degrees of freedom of the system grow. i.e. it takes more time to solve the linear problem. Therefore, if the solution is not regular, the linear finite elements are a better choice, but here I think we can assume that the solution of the Poisson is regular (Can we?).

Shape regularity

The converge estimate is obtained by combining two equations of triangulation (one with interpolation and the other with $B_K$ terms. $B_K$'s are geometric parameters related to the shape of the mesh. Let $\rho_K$ be diameter of the largest ball contained in the element K. Let $h_K$ be diameter of the smallest ball containing the element K. These estimates depend on the shape of the triangle K. If the triangle is very flat, the error term can be large, regardless of the size of the triangle.

So there is shape regularity assumption of the mesh in relation to the parameter $h$:

\begin{equation} h = max_{K \in T} h_K. \end{equation}

The shape of triangles affect the error behaviour or at least approximation properties. So we need to place restrictions on the mesh, to require the family of triangulation to be shape-regular which guarantees the error term to be bounded in the whole family. Using this, the convergence is related only to the mesh density parameter $h$.

Is shape regularity necessary? The FEM converges if the largest angle stays bounded (triangles and mesh) so no shape regularity then required. No problem with small angles. The FEM can converge when the interpolation error does not.

Regularity in Poisson Problem

Regularity of the solution depends on the shape of the $\Omega$ and the load function $f$. Optimal convergences can be obtained if $u \in H^2(\Omega)$.

Assume load function $f \in L^2(\Omega)$, the regularity of the solution depends on geometric properties of $\Omega$. If $\Omega$ is convex, the solution will always be in $H^2(\Omega)$. For non-convex polyhedral domains, the behaviour of the solution depends on the angle between two boundary segments.

The only part of the convergence theory depending on the regularity assumption is the interpolation error estimate.

I want to be more verbal in this answer.

How can you answer better to the questions?

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