# Extrapolation after successive finite element refinement

I wish to compute a certain function $\lambda$ (in my case an eigenvalue of the Laplacian) to a certain accuracy, which I wish to guarantee, if possible. I started with a finite element method. I know that in 2D there are way better methods, but my goal is to work on general surfaces, so that's why I work with finite elements for now. Let $h$ be the size of the elements.

I start from some $h_0$ and keep refining using the midpoint refinement procedure (replace each triangle with four triangle constructed using midpoints). This procedure gives me an approximation of $\lambda$ depending on $h$: $\lambda_h,\lambda_{h/2},...,\lambda_{h/2^n}...$ Due to the size of the computations involved I can only do $9$ or $10$ refinements which seem to get me $4$ digits of precision.

After reading a few chapters from the extremely useful book SIAM 100 digit challenge I decided to try applying a basic extrapolation technique to the sequence of approximations $\lambda_h,\lambda_{2h},...,\lambda_{h/2^n}$. To my surprise the extrapolation seems to double the number of correct digits.

I searched a bit and did not found references justifying such a behavior. So my question is:

Do you know any papers dealing with this kind of approximations:

• compute some finite element approximations for successive mesh refinements

• apply an extrapolation procedure

• guarantee the extra precision of the extrapolated result ?

• This doesn't directly answer your question, but another way to do what you want is to use adaptive refinement based on a goal-oriented error estimator. A certain class of estimators, so-called functional a posteriori estimators, will give you sharp upper bounds for the error. – cfh Apr 10 '17 at 6:08
• In the multigrid context there are some techniques that allow to increase accuracy by extrapolation. You may search for "multilevel tau-extrapolation" or similar to find references. I know there are some publications by Brandt, also by Ulrich Rüde and maybe others, but I don't know exactly if and how they relate to what you tried. – Christian Waluga Apr 10 '17 at 19:02
• Thank you for your comments. I'll search for the works you suggest. – Beni Bogosel Apr 12 '17 at 11:26