Computing error towards reference solution

Question

In a term paper about Finite Elements, I need to compute empirical convergence rates without knowing the exact solution of an elliptic PDE.

Given a family of grids $\mathcal{T}_h$ with $h \in \{h_1, \dots, h_n\}$ decreasing sequence of meshwidths, let us denote the resulting numerical approximations by $u_1,\dots,u_n$.

Here is where I am stuck: I need to compute a reference solution $u^{\ast}$ on a reference triangulation $\mathcal{T}^{\ast}$ which is a refinement of all $\mathcal{T}_i$.

Can anyone help me with an easy-to-implement approach on this?

My idea: Assume that the corresponding FE spaces $V^{\ast}; V_1,\dots,V_n$ (Lagrangean, conforming and non-conforming) are nested, and there is a (trivial) inclusion $V_i \subseteq V^{\ast}$ for all $i$.

I want to compute the matrix of a projection $V^{\ast} \twoheadrightarrow V_i$ (Bases are hierarchical hat-bases everywhere, call the matrix of the projection $\mathbf{P}_i$) and compute $\|u_i - \mathbf{P}_i u^{\ast}\|$. But then I don't know what projection I should use to get the right rate. Norm will be $L^2$; $H^1$ for conforming; dG for nonconforming ("dG=$H^1$-Norm with jump terms").

Answer 1

I typically use the following approach which tends to give better looking convergence curves. Suppose $u_h$ is the solution with mesh $T_h$ and mesh parameter $h$ and $u_{h/2}$ is the solution with mesh $T_{h/2}$ and mesh parameter $h/2$ (e.g. each triangle split into four subtriangles). Then by triangle inequality $$\|u_h-u_{h/2}\|=\|u-u_h-(u-u_{h/2})\|\leq\|u-u_h\|+\|u-u_{h/2}\|.$$ Thus you should get same asymptotic convergence rate for $\|u_h-u_{h/2}\|$ as you would for $\|u-u_h\|$ and $\|u-u_{h/2}\|$. The term $\|u_h-u_{h/2}\|$ is know as the relative error.

For implementation you can simply evaluate $$\int_K (u_h-u_{h/2})^2\,\mathrm{d}x$$ in each element $K \in T_h$. Just construct the quadrature rule first for the mesh $T_{h/2}$ and combine the points in each subtriangle of $K$ together.