Consider the problem $-\Delta u = f$ in $\Omega$, with $u=0$ on $\partial \Omega$. Suppose that $\Omega$ is a polygon and that we approximate the solutions to the previous problem using Lagrange finite elements on a triangular mesh. When solving such a finite element problem there are multiple sources of error:

  1. The discrete mesh may not be an exact mesh for the polygon due to rounding errors (typically of the order of the machine $\varepsilon$). This gives slight perturbations for the mass and rigidity matrices.

  2. The discrete solution comes from solving a linear system. This linear system is solved numerically, so there is an additional error here.

  3. The difference between the analytical solution and the finite element one is usually quantified with a priori estimates depending on the mesh size $h$.

Usually only point 3. above is mentioned when dealing with error estimates from finite elements. I guess this is due to the fact that errors coming from points 1 and 2 above (rounding errors, linear systems) are usually orders of magnitude smaller than the discretization error in 3. However, I am interested in knowing more about how one can quantify the errors coming from the first two points and eventual references treating these aspects.

Can you indicate references in the literature or the main ideas that can be used in order to better understand errors coming from points 1-2 above?

  • $\begingroup$ If you are counting rounding errors for polygonal meshes, you could count the geometry approximation of the domain as well. $\endgroup$
    – nicoguaro
    Oct 14 '21 at 10:47
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    $\begingroup$ There may also be errors due to inexact quadrature (e.g., for nonlinear problems). $\endgroup$
    – cos_theta
    Oct 14 '21 at 11:17
  • $\begingroup$ @nicoguaro: If the domain is already polygonal, then the "exact mesh" (before rounding errors) already covers it. $\endgroup$ Oct 14 '21 at 11:39
  • $\begingroup$ @cos_theta: I agree that there can also be inexact quadrature errors. However, if I'm not mistaken, using Lagrange finite elements of given order, we can choose quadrature rules that are exact. $\endgroup$ Oct 14 '21 at 11:40
  • 1
    $\begingroup$ @BeniBogosel: Yes, for some problems quadrature will be exact. But consider a simple diffusion problem $-\nabla \cdot (a \nabla u) = f$ where diffusivity $a\colon \Omega \to \mathbb{R}$ is not polynomial (e.g., exponential or Gaussian random field). Or consider the convective term in Navier Stokes: $u \cdot \nabla u$. You'll need tailored or higher order quadrature formulae for exact integration (if such tailored formula can be computed / derived at all). $\endgroup$
    – cos_theta
    Oct 14 '21 at 11:50

The triangle inequality is your friend. Let's ignore the issue of boundary approximation for a moment, then you are computing a solution with inexact linear solver. Let's call it $u_h$. We will call the exact solution of the PDE $u$, and the exact finite element solution $u_\text{FEM}$; neither of these can be computed exactly (in the case of $u_\text{FEM}$ because we can not solve linear systems exactly, but accrue round-off and iteration error).

Then the total error satisfies $$ \|u_h - u\| = \|u_h-u_\text{FEM} + u_\text{FEM} - u\| \le \|u_h-u_\text{FEM}\| + \|u_\text{FEM} - u\|. $$ The first of the terms on the right results from inexact solution of the linear system, and we know how to estimate it. The second of the terms on the right results from finite element approximation (without having to take into account any other approximation), and again we know how to estimate this error.

If you had other error terms -- say, quadrature error or boundary approximation errors -- then you would simply have to add and subtract additional terms in the norm as I've done above with $u_\text{FEM}$ and expand the norm using the triangle inequality into more separate norms that measure exactly one error effect at a time. You don't have to consider the combined effects of all errors at once.

  • $\begingroup$ Thank you very much for your answer. $\endgroup$ Oct 15 '21 at 7:11
  • $\begingroup$ Do you have any reference in mind concerning the estimation of errors coming from the inexact solution of the linear system when dealing with matrices coming from FEM? I have references concerning general estimates for linear systems in terms of the residual and the condition number of the matrix. However, often matrices of mass and rigidity have a simplified structure and maybe more can be said in this case. $\endgroup$ Oct 15 '21 at 7:40
  • $\begingroup$ @BeniBogosel I remember a paper by Rannacher and Becker from the 1990s that treats the combined issue of discretization and iteration error. You might be able to find that. $\endgroup$ Oct 15 '21 at 12:13
  • $\begingroup$ But beyond that: You want to solve $AU=F$ but only have an approximate solution $\tilde U$. Then you can relate $U-\tilde U$ to the residual $F-A\tilde U$ via the condition number of matrix, and then all you have to do is relate $U-\tilde U$ to $u_\text{FEM}-u_h$. $\endgroup$ Oct 15 '21 at 12:14
  • $\begingroup$ Thank you for these additional comments. $\endgroup$ Oct 15 '21 at 12:50

I think that errors 1 and 2 classify as variational crimes, as also classifies approximating the domain by a mesh (but that's not your case). That being said, I am not aware of how big these errors are in comparison with the approximation error.

Now, regarding rounding errors I remember Nick's Trefethen quote [1]

If rounding errors vanished, numerical analysis would remain.


  1. Trefethen, L. N. (1992). The definition of numerical analysis. Cornell University.

  2. Strang, G. (1972). Variational crimes in the finite element method. In The mathematical foundations of the finite element method with applications to partial differential equations (pp. 689-710). Academic Press.

  3. Brenner, S. C., Scott, L. R., & Scott, L. R. (2008). Chapter 10: The mathematical theory of finite element methods (Vol. 3). New York: Springer.

  4. Holst, M., & Stern, A. (2012). Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces. Foundations of Computational Mathematics, 12(3), 263-293.

  • $\begingroup$ Thank you for the interesting answer. However, the variational crimes that are presented in Brenner-Scott, Chapter 10 are not really related to the problem I stated: they deal with smooth sets approximated by polygons and discontinuous non-conforming elements. The last reference seems to go in the same direction. $\endgroup$ Oct 14 '21 at 20:16
  • $\begingroup$ @BeniBogosel, that's right. That's why I have seen people have studied the most. I have not seen anything for 1 and 2 in your question. But adding a keywords to your search might help (or not). $\endgroup$
    – nicoguaro
    Oct 14 '21 at 20:34

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