This is a continuation of my previous question. I am trying to effectively compute a bound for the roundoff errors in some FEM computation (2d polygons, triangular meshes). Below I will write some of my computations which lead to a quantifiable bound (if I'm not mistaken), but the error is quite large for small mesh size $h$. Therefore, my questions are:
is there a better way to quantify this type of errors? Do you know references who do this type of computation?
is the error bound obtained below comparable to what is observed in practice? What is the general admitted threshold $h_0$ such that for $h<h_0$ the roundoff errors would be larger than the theoretical errors (when using $P_1$ piecewise affine elements with standard double precision arithmetic)?
Just to fix the framework, I have some particular polygon, meshed with equal triangles and I use $P_1$ Lagrange (piecewise affine) elements. Therefore, the stiffness and mass matrices $K$ and $M$ can be computed explicitly. Zero Dirichlet boundary conditions are applied and the nodes are eliminated from the system by considering submatrices of $K$ and $M$. When implementing the problem, roundoff errors appear, leading to perturbations $\Delta K$ and $\Delta M$ of these matrices. In particular, $M$ is symmetric, positive definite and $K$ is symmetric positive definite (after removal of lines and columns of Dirichlet nodes).
I am interested in solving the generalized eigenvalue problem $$ Ku = \lambda Mu.$$ The associated eigenvectors form a basis of $\Bbb{R}^N$, orthonormal w.r.t. the scalar product $x^TMy$.
Consider now a solution $\overline u$ of $(K+\Delta K)\overline u = \overline \lambda (M+\Delta M)\overline u$. Write $\overline u= \sum_{i=1}^N c_i u_i$ in the basis $(u_i)$, and suppose that $\overline u^T M \overline u=1$ leading to $\sum_{i=1}^N c_i^2 = 1$. After a few computations we have $$\sum_{i=1}^n (\lambda_i-\overline \lambda) c_i M u_i = (-\Delta K+\overline \lambda \Delta M) \overline u$$ Multiply with $\text{sign}(\lambda_i-\overline \lambda)c_j u_j$ and sum for $j=1,...,n$ to get $$ \sum_{i=1}^n c_i^2|\lambda_i-\overline \lambda| = \overline u_m^T(-\Delta K+\overline \lambda \Delta M)\overline u $$ where $\overline u_m = \sum_{i=1}^N \pm c_i u_i$ leading to $\overline u_m^T M \overline u_m=1$.
Now suppose $\Delta K$ and $\Delta M$ are machine erorrs for $K$ and $M$, in particular, componentwise inequalities of the form $$ |\Delta K|\leq \varepsilon |K| , |\Delta M|\leq \varepsilon |M|$$ hold (in particular, zero elements in K,M lead to zero elements in $\Delta K, \Delta M$)
In the following paper the authors describe how to estimate the eigenvalues of $K$ and $M$ (maybe other references exist, I found this well explained and nice to read, with quantifiable constants).
In particular, given the mesh size $h$, eigenvalues of $M$ are all of size $O(h^2)$, implying that $\|\overline u\|_2 = O(h^{-1})$. Suppose that $\overline \lambda$ approximates the smallest eigenvalue $\lambda_1$. Then it is straightforward to see (using the fact that the lines of $K,M$ have a small number of non-zero elements) that $$ |\overline u_m^T \Delta M u_m| \leq \varepsilon O(1), |\overline u_m^T \Delta K u_m| \leq \varepsilon O(h^{-2}) $$
Therefore $$ \sum_{i=1}^n c_i^2|\lambda_i-\overline \lambda| \leq \varepsilon O(h^{-2})$$ This immediately shows that $\min |\lambda_i-\overline \lambda| \leq \varepsilon O(h^{-2})$ Moreover, for $j>1$ if $v_1, \overline v$ are the $P_1$ FEM functions given by coefficients $u_1$ and $\overline u$ then $$ \|v_1-\overline v\|_{L^2}^2 = (\overline u-u_1)^T M (\overline u-u_1) = (c_1-1)^2+\sum_{j>1} c_j^2.$$ $$ \|\nabla v_1- \nabla \overline v\|_{L^2}^2 = (\overline u-u_1)^T K (\overline u-u_1) = \lambda_1(c_1-1)^2+\sum_{j>1} \lambda_j c_j^2.$$ The previous estimate also shows that these two quantities are of order $\varepsilon O(h^{-2})$.
These error estimates seem too pessimistic. For $h<1e-4$ machine precision error would dominate when using standard double precision. Also I would expect the $L^2$ norm to be better approximated than the $H^1$ one. Is there any way of improving them?
